*Hi, welcome to Educator.com.*0000

*This lesson is on expressions and variables; let's begin.*0002

*First thing we have to go over--variable; variable.*0009

*A variable is a letter or symbol that can stand for or represent one or more numbers.*0014

*In place of numbers, I have usually letters.*0021

*If I use A or B... I can use X, Y... those are all considered variables.*0026

*I can also use symbols.*0033

*Sometimes in math, we use Greek symbols; or I can use star as a symbol.*0035

*Anything that represents numbers would be considered a variable.*0043

*Expressions; expressions are numbers or mathematical phrases that includes numbers and variables.*0049

*Keep in mind that expressions do not have equal signs.*0058

*If it does have an equal sign, then it would be called something else.*0063

*It would be called an equation.*0066

*We have different types of expressions.*0069

*The first type is a numerical expression; numerical means numbers.*0072

*If we have a numerical expression, it would be an expression, mathematical phrase, that has only numbers.*0077

*Here is an example, 30 plus 5; this is considered a numerical expression.*0086

*There is no variables; we only have numbers.*0092

*The next one--algebraic expression--is the most common type of expression.*0096

*That is when you have both numbers and variables.*0102

*6 plus A is an example of an algebraic expression.*0107

*2X minus 10 is also another example.*0112

*We have A; that is a variable; X is a variable here too.*0115

*2X minus 10 is an algebraic expression.*0120

*The other type is a word expression.*0125

*Word expression is when you use words to express your expression.*0128

*Or you write out your expression using words.*0134

*P divided by 9 would be considered a word expression*0136

*because instead of writing the symbol out, you would write just--divided by--the words.*0140

*Let's do a few examples of evaluating expressions.*0147

*Evaluating just means to solve or simplify the expression.*0150

*If I have a numerical expression, 12 times 20, this is a numerical expression.*0157

*I can actually simplify this out; I can multiply 12 and 20 together.*0164

*12 times 20; 2 times 0 is 0.*0171

*Because it is 0, I can just move on.*0180

*2 times 2 is 4; 2 times 1 is 2.*0182

*When I evaluate this expression, I get 240.*0188

*This next one, M minus 8, that is my expression.*0197

*They want me to evaluate this expression when they tell me that M is equal to 8.*0203

*Here is the expression; let me use black.*0210

*M minus 8 is my expression; they told me that m is 8.*0214

*Remember variables; this is a variable that represents a number.*0220

*The number that it represents is 8.*0224

*This is the same thing as 8 minus 8; my answer is 0.*0228

*When I evaluate the expression M minus 8 when M is equal to 8, then my answer would be 0.*0237

*The next example, I am evaluating the expression, S divided by T.*0245

*S divided by T can also be written like this, as a fraction, S divided by T.*0252

*They are telling me that S is 28 and T is 4; S is 28.*0259

*Instead of writing S, I am going to write out 28*0267

*because S is a variable that represents the number 28 and T represents the number 4.*0270

*I can write it like this; or I can write it just 28 divided by 4.*0281

*It is the same thing; 28 divided by 4 is 7.*0286

*When I evaluate the expression, I get 7 as my answer.*0294

*The final example, example four, evaluate the expression CD.*0300

*When I have two variables next to each other like that, CD, this means C times D; C times D.*0306

*C represents the number 3; it is 3; D represents the number 200.*0316

*If CD means C times D, then it means 3 times 200.*0325

*When I multiply 3 times 200, I get 600.*0332

*When you evaluate the expression CD, when you are given the numbers that they represent, then the answer will be 600.*0342

*That is it for this lesson on Educator.com.*0353

*We will see you next time; thank you.*0356

*Welcome back to Educator.com; this lesson is on the introduction of exponents.*0000

*When you have a number with an exponent, that number, this number right here, 10, is called the base.*0010

*This 2 is called the exponent.*0019

*This can be read as 10 squared, or 10 to the power of 2, or 10 to the 2nd power.*0025

*Whenever you have a base with a number that is a little bit higher to the side of it,*0035

*then that is called the exponent; you read it 10 to the power of 2.*0041

*Or if it is a 2, then you can just say 10 squared.*0046

*Find the value of 6 to the 3rd power.*0054

*Exponent, what that means is you are saying that 6 is going to be a factor 3 times.*0059

*When we have 6 to the 3rd power, we are going to write this out as 6 times 6 times 6.*0070

*It is just this number multiplied by itself that many times.*0078

*Be careful; this is not 6 times 3; this is not 18.*0084

*We have to write this out as 6 times 6 times 6; then you just solve this out.*0088

*We have 6 times 6 is 36; we have to multiply that by 6 again.*0094

*If you do 36 times 6, then you are going to get 6 times 6, 36.*0101

*This is 18; plus the 3 is 21; 6 to the 3rd power is 216.*0107

*Another example, this is written out in expanded form.*0118

*It is 4 times 4 times 4 times 4 times 4.*0125

*When we write that in exponent form, we are going to write the 4 as a base*0129

*because all these numbers are 4s so the base is going to be a 4.*0135

*How many times did it multiply by itself?*0140

*1, 2, 3, 4, 5; there is 5 of them.*0143

*I am going to write that as my exponent.*0146

*It is going to be 4 to the 5th power.*0148

*4 to the power of 5 or 4 to the 5th power.*0152

*Write 125 using an exponent and the base 5.*0159

*That means we want the base to be a 5.*0163

*We are going to have to see what the exponent is going to be*0167

*so that when we solve that out, it is going to become 125.*0170

*Base is 5.*0176

*Again I need to find a number that goes there as my exponent*0178

*so that when I solve this out, it is going to become 125.*0183

*First, in order for me to do this, I have to see*0188

*how many times I have to multiply 5 to itself to get 125.*0191

*125 is going to be... this is called the factor tree.*0198

*We haven't gone over that yet; it is later on in the lesson.*0203

*But we are just going to break this up; 125 is 25 times 5.*0208

*25 times 5 is 125; this 25 is 5 times 5.*0216

*That means 5 times 5 times this 5 gives you 125.*0223

*This can be written out as 5 times 5 times 5.*0234

*5 times 5 is 25; 25 times 5 is 125.*0240

*To write this using base 5, my exponent is going to be... how many times did I multiply 5 to itself?*0246

*3 times; it is going to be 3.*0253

*My answer is 5 to the 3rd power.*0258

*The next example, write the equal factors and the value of 3 to the 4th power.*0268

*The equal factors just means for you to write it out in expanded form.*0274

*It is 3 times 3 times 3 times 3.*0281

*Again, whenever you are solving exponents, make sure you write it out like this so you don't multiply 3 times 4.*0286

*This is not 12; be careful with that.*0294

*Exponents tell you how many times you are going to multiply this base number to itself.*0296

*We are going to multiply 3 to itself 4 times.*0303

*It is 3 times 3 times 3 times 3.*0307

*When I multiply this out, I can multiply these two first.*0311

*This is going to be 9; we can multiply these two; that is 9.*0313

*9 times 9 is 81; 3 to the 4th power is 81.*0318

*That is it for this lesson on exponents; thank you for watching Educator.com.*0329

*Welcome back to Educator.com; this lesson is on the order of operations.*0000

*For order of operations... operations we know are multiplying, dividing, adding, subtracting, things like that.*0009

*Those are all called operations; we are going to look at the order.*0017

*When we have several different operations we look at within a single problem,*0023

*there is an order of which ones we have to do first.*0028

*To help you remember the order of operations, there is this phrase right here.*0034

*Please excuse my dear aunt sally.*0040

*That is just an easy way for you to remember the order of operations.*0044

*Try to say it out loud a few times; please excuse my dear aunt sally.*0049

*That just means p for parentheses, e for exponent, m to multiply, d for divide, a for add, and s for subtract.*0057

*No matter what, you are always going to solve within the parentheses first.*0077

*If you have parentheses, then always solve that first.*0082

*Then e for exponent; you are going to do the exponents next.*0086

*For m and d, multiplying and dividing, they are actually the same.*0091

*When it comes to multiplying and dividing, you are just going to multiply or divide across whichever ones come first.*0099

*Multiplying and dividing are actually the same; adding and subtracting are also the same.*0107

*For multiplying and dividing, there is no order.*0114

*It is just whichever comes first when it comes to these two.*0117

*For adding and subtracting, also the same.*0121

*If there is something you have to subtract before you have to add, then you just go ahead and do that.*0125

*They are the same; there is no order in here.*0130

*But please excuse my dear aunt sally is just an easy way for you to remember the order of operations.*0135

*Make sure you say it a few times and try to remember those.*0139

*We just went over variables so I have some variables here to help us with the order of operations.*0147

*A plus in parentheses B minus C plus D squared plus E times F.*0154

*First thing we have to do is parentheses; this right here would be number one.*0164

*That is the first thing you would have to do.*0172

*When you have a number B minus a number C, that is the first thing you are going to solve out.*0175

*The second thing you are going to solve out is exponent.*0181

*This; this will be the second thing for you to solve out.*0185

*Then you are going to just rewrite this problem with that part solved.*0189

*The next would be to multiply; E times F.*0196

*That is going to be the third step; you are going to multiply those two first.*0200

*All you are going to have left are adding and subtracting.*0208

*You are going to just do that last; you will get your answer that way.*0211

*Let's do a few examples; let's look at this; 2 plus 3 times 5.*0216

*My order of operations is P-E-M-D-A-S.*0224

*That is PEMDAS--please excuse my dear aunt sally.*0231

*I don't have parentheses; I don't have any exponents.*0237

*But I do have a multiplication; you are going to do this first.*0240

*Even though this comes first, you would have to multiply before you add.*0248

*You are going to do 2 plus... 3 times 5 is 15.*0256

*This is going to become 17.*0265

*If you were to add first, if we don't follow the order of operations rule, *0271

*let's say you just did 2 plus 3 and then you times 5.*0276

*2 plus 3 is 5; times 5 is 25.*0280

*See how that is a different answer; this is a wrong answer.*0288

*You have to make sure you follow this rule so that you can get the correct answer, 17.*0291

*Another example would be A plus in parentheses 4 minus 2.*0298

*We know always, always solve within the parentheses first.*0304

*This is going to be solved first.*0310

*I am going to write this because I am not going to do anything to that.*0313

*4 minus 2 is 2; 8 plus 2 now is 10.*0318

*When you follow the order of operations, you are going to get the correct answer of 10.*0328

*Another example, 9 minus 2 squared.*0334

*We just discussed in the last couple lessons on exponents.*0338

*2 squared is the same thing as 2 times 2; or 2 times 2 like that.*0345

*Exponents come after parentheses; we are going to have to solve this before subtracting.*0355

*2 times 2 is 4; I am going to write 9 minus the 4.*0365

*9 minus 4 is 5; 9 minus 2 squared is 5.*0375

*This next example, kind of long, I have a few operations I can perform to this.*0387

*But remember we have to stick with the order.*0392

*I am just going to write PEMDAS so I can see the order.*0394

*Always, always parentheses first; I have a parentheses right here.*0404

*I am going to write everything else out; solve the parentheses out.*0413

*5 minus 2 is 3; that is 3 squared divided by 9.*0419

*Here what is my next operation?--exponents.*0430

*Since I have an exponent right here, I would have to solve this out before I do anything else.*0435

*This is going to be 6 times... this remember is 3 times 3.*0440

*3 squared is the same thing as 3 times 3.*0448

*3 times 3 we know is 9; then divided by 9.*0451

*I am just rewriting this out; then multiplication and division right here.*0459

*When we have only these two, they are actually going to be the same.*0468

*There is no order for multiplication and division.*0473

*You are just going to solve out whichever comes first when it comes to multiplying and dividing.*0476

*For this problem, it just happens to be multiplication.*0480

*We are just going to solve this out; 6 times 9 is 54.*0485

*54 divided by 9; I am just rewriting this; 54 divided by 9 is 6.*0491

*My answer to this, 6 times 5 minus 2 squared divided by 9, as long as you follow the order of operations, your answer will be 6.*0504

*That is it for this lesson on order of operations.*0515

*Thank you for watching Educator.com.*0518

*Welcome back to Educator.com.*0000

*This lesson, we are going to be looking at decimals.*0002

*We are going to compare and order them from least to greatest.*0005

*Before we go into that, let's look at some place values.*0014

*When we have a decimal... let's look at this decimal right here.*0020

*This is read 1 million 2 hundred and 34 thousand 5 hundred and 67 and 89 hundredths.*0024

*Right here, after the second comma, since our first comma is right here, our second comma is million.*0036

*A million 2 hundred and 34 thousand 5 hundred and 67 and 89 hundredths.*0046

*The 1 will be the million; 2 will be hundred thousands; 3 is ten thousands.*0055

*The 4 is thousands; 5 is hundreds; the 6 is tens; ones.*0061

*The decimal right here is read AND.*0069

*We are going to say 5 hundred 67 and 89 hundredths.*0073

*This right here, the first number after the decimal, is going to be tenths place with a -TH.*0079

*The next number is the hundredths place.*0088

*To figure out which decimal has a larger value, we need to look at the whole number first.*0095

*The whole number is going to be the number before the decimal place.*0102

*Here the whole number is 10; 10 is a number before the decimal place.*0107

*Whole number is 10 here; here it is 100.*0114

*We know that 100 is bigger than 10.*0118

*The larger value between these two is going to be 100.5.*0124

*We don't care what the number is after the decimal place*0128

*because the whole numbers are going to determine the larger value.*0131

*The next one, 1.01 or 1.10.*0138

*Here our whole number is 1; here the whole number is 1.*0144

*In this case, since we have the same whole number, we have to look at the values after the decimal place.*0149

*Here I have 1.01; and this would be 1.10.*0158

*We are going to go the next number which is 0; here is 1.*0164

*Whenever the whole numbers are the same, we are going to go to the next value which is the tens place.*0172

*In this case, it is 0; in this case, it is 1.*0180

*Even though this sounds like this is 1 and this is 10, this can be the same thing as 1.1.*0184

*If I have a 0 at the end of my decimal, at the very end,*0196

*and it is behind my decimal place, then I can just drop it.*0203

*This can be the same thing as 1.1.*0206

*Or it could be the same thing as 1.100.*0209

*Or I can even add ten 0s behind it.*0214

*It would still be the same value.*0218

*As long as it goes after my decimal point at the very end of the number,*0221

*then all those 0s mean the same thing, doesn't mean anything.*0228

*In this case, since I know that, I am just going to look at this 1, this first place right here, the tenths place.*0233

*1 is bigger than the 0 so I know that this number is actually going to be bigger than this number.*0247

*1.1 or 1.10 is greater than 1.01.*0254

*The next one, again look at the whole number; they are both the same, 44.*0260

*I am going to take a look at the next place value.*0268

*Again they are the same; they are both 4s.*0273

*Then I look at the next one; this is the 0.*0275

*Remember I can add 0s here if I want to because it is after the decimal place at the very end.*0279

*This can also be 0; they are also the same.*0286

*In this case, even though before I had the 0... 40 sounds like it would be bigger than 4.*0292

*Since I can add the 0s right here, they have the same value.*0301

*In this case, they are the same; they are equal.*0306

*44.40 is the same thing as, equal to 44.4.*0311

*The next one is 18.6; and this is 16.8.*0320

*Again first look at the whole number before the decimal place.*0328

*18 is larger than 16.*0333

*Automatically without even looking at the numbers after the decimal place...*0337

*even though 8 is bigger than 6, the whole number itself 18 is larger than 16.*0341

*This automatically becomes the bigger number.*0349

*These examples have a few numbers.*0358

*We are going to order them from the smallest to the biggest of values.*0363

*Again since I am comparing five different numbers, I want to first just look at their whole numbers.*0370

*This one has a 4, 10, 5, 5, and 6.*0378

*Just the numbers before the decimal point; those are the whole numbers.*0384

*Since I am going from least to greatest, which one has the smallest whole number?*0389

*I know that this one does, 4.*0395

*This decimal right here would be the least; it would be the smallest.*0398

*It is going to be 4.1; that is the smallest.*0402

*Then I have 5s; I have two decimal numbers with whole numbers of 5.*0407

*I have to compare these two now because I know that these two numbers are going to go next.*0415

*Since they have the same whole number, I am going to look at the next value.*0419

*This one is 1; this one is 0.*0425

*I know that 5.1 is bigger than 5.01 so this number is going to go next.*0429

*And the next smallest, 5.01; and then this one, 5.1.*0437

*I have two numbers left; I have 10.01 and I have 6.0.*0446

*6 is a smaller number than 10 if you look at the whole numbers.*0451

*6.0 is going to go next; 10.01, the biggest number, is going to go last.*0457

*I have 1, 2, 3, 4, 5 numbers; and then 1, 2, 3, 4, 5 numbers in order from least to greatest.*0467

*Another example, again I am going to look at the whole numbers.*0477

*This has a 0 as a whole number.*0481

*1, there is a 2, 0, and an 8; going from least to greatest again.*0484

*My least numbers, my smallest numbers are going to be the numbers with no whole number, 0 as a whole number.*0492

*It is going to be this one, 0.6 or 0.99.*0498

*From those two, I am going to look at the next number, in the tenths place.*0504

*6 is smaller than 9 so I know that this one is going to be the smallest.*0509

*0.6 is first; and then it is going to be that one, 0.99.*0517

*From the remaining numbers, my whole number here is 1, this is 2, and this is 8.*0526

*I know that this one is going to go next, 1.32.*0532

*Then it is going to be 2.02; and then 8.3.*0540

*There are my numbers, my decimal numbers in order from least to greatest.*0551

*The next example, again just look at the whole numbers--100, 101, 111, 110, and another 100.*0557

*Just looking at their whole numbers, 100 is my smallest.*0571

*I have two of them.*0578

*I have to look at the next place value of the tenths.*0580

*0 is smaller than 9, just looking at that number alone.*0591

*This is going to be my smallest value, 100.07.*0598

*Then it is going to be 100.9.*0607

*Then from the three numbers, 101 is the next smallest... 0.4.*0612

*Then from these two, 110 is smaller than 111 so it is going to be 110.8.*0623

*This one right here is going to be the largest value, 111.1.*0633

*Let's do one more example; again we are ordering from least to greatest.*0641

*You are going to look at all of your decimal numbers.*0649

*We have five numbers we are going to be comparing.*0652

*Before you look at all the numbers, let's just look at the whole numbers.*0656

*That one has no whole number, 0.*0662

*Whole numbers are all the numbers before the decimal place.*0665

*This is 0; this is 1, 0, and 1.*0668

*I know that the decimal numbers with no whole number,*0675

*with 0 as a whole number, are going to be the smaller numbers.*0679

*Between this one, this one, and that one.*0684

*Between those three numbers, I am going to have my least, my smallest value.*0689

*From those, you are going to look at the next place value which is the tenths place value.*0696

*That is 1; this one is 0.*0704

*Not for this one because that has a whole number.*0708

*I am just comparing the ones with the same whole number of 0.*0711

*Within this place value, this is 1, this is 0, and this is 0.*0717

*I know that this one right here, this one, and one of those two is going to be the smallest number.*0723

*I look at the next place value since they are the same.*0733

*Again they are the same; they are both 0s.*0737

*Look at the next one; 1 is smaller than 9.*0741

*This is smaller than that number.*0749

*That is going to make it the smallest number, 0.001.*0754

*Then this is my next smallest, 0.009.*0761

*Then this one right here because that had no whole number, 0.1.*0770

*I have two numbers left between 1.0001 and 1.1.*0777

*Since they have the same whole number, you are going to look at the tenths place.*0786

*This is 0; this is 1; this is bigger than that number.*0793

*So this is going to go next, 0001.*0800

*Then your largest number, the greatest value is going to be 1.1.*0805

*That is it for this lesson; thank you for watching Educator.com.*0814

*Welcome back to Educator.com; this lesson is on rounding decimals.*0000

*To begin, we need to go over the place values.*0007

*If I have a number right here, 12,345 and 6679 ten thousandths, each number has a place value.*0012

*The 1 right here is ten thousands.*0030

*The 2, the one right before the comma, thousands; this is hundreds, tens, ones.*0033

*The important part is looking at the numbers after the decimal point.*0041

*The decimal point is read as AND.*0048

*6, this first number is not one'ths; it is actually tenths.*0051

*It starts off as the tenths value with a -TH.*0056

*The next one would be hundredths, thousandths, ten thousandths; keep this in mind.*0061

*Make sure you look over this and remember these place values.*0070

*When we round decimals, the first thing you are going to do is circle...*0079

*You don't have to circle--but it just makes it easier--the number in the place value that you have to round to.*0085

*If you look at this example right here, it says this number to the nearest tenth.*0094

*I know that tenths is the first number after the decimal point.*0099

*I am going to take that; I am going to circle that number, 5.*0103

*Look at the number after the circled number; behind; the number behind is a 6.*0111

*That number behind the circled number, if it is greater than 5,*0123

*5 or greater, then you add one to the circled number,*0127

*meaning you are going to round that number up.*0132

*If it is smaller than 5--4, 3, 2, 1, or 0--then you don't make any changes to that circled number.*0135

*You keep it as a 5.*0143

*This circled number is either going to stay a 5.*0144

*Or it is going to become a 6 if you round up.*0147

*After you determine that, you are going to rewrite the number, the whole number, the whole thing,*0152

*but replace all the numbers behind the circled number with 0s.*0158

*That is the point of rounding; you are going to stop at this number.*0164

*That is going to be the last number you are going to write.*0168

*The rest are going to become zeros.*0169

*This number 6, the number behind the circled number, is greater than 5*0174

*which means I have to take this circled number and round it up.*0179

*I am going to add 1 to the circled number.*0184

*This number is going to become a 6; this number is now a 6.*0187

*I am going to rewrite all the numbers, the whole thing, but replace the last numbers with 0s.*0196

*My new number, after rounding, it is going to become 1 thousand 2 hundred and 34 and 6 tenths.*0203

*Then I can put 0s at the end of them.*0219

*This is my new number, my rounded number.*0223

*Let's do a few more example; round each.*0229

*Here is the number; this is read 4 hundred 26 and 93 hundredths.*0232

*This number right here, 93, you read it as a hundredths because that is the last number that you see there.*0241

*That is the last place value; this is tenths; this is hundredths.*0248

*You are going to read this as 93 hundredths.*0252

*It says to round this to the nearest tenths... sorry, tens.*0256

*The tens... number the tens value; this is ones; this is tens.*0263

*I am going to circle that; the number behind it is a 6.*0270

*It is greater than 5.*0275

*The 6 value is greater than 5 which means the circled number becomes a 3.*0278

*I have to round up by adding 1.*0285

*All the numbers before the 2 stay the same.*0291

*The numbers behind the 2 are replaced with 0s.*0294

*Don't forget the decimal though; you still have to have that decimal.*0298

*This number is before the 2; so 4.*0304

*The circled number... again remember we were going to round up.*0309

*That becomes a 3; the 6 becomes a 0; everything else becomes 0s.*0313

*If I round to the nearest tens, then this number is going to become 430 or 430.00.*0323

*The next one; round to the nearest tenths; the tenths.*0332

*Be careful, the tenths is the first number after the decimal point.*0339

*That is this 0; 0.*0343

*Then I look at the number right behind it which is a 9.*0348

*It is greater than 5; my zero rounds up.*0350

*I do 31 point ...0 becomes a 1; everything else becomes 0.*0357

*The next few examples; we have this number right here to the nearest hundredths.*0372

*Tenths, hundredths; circle this number; I look at the number behind it.*0377

*Is it 5 or greater?--yes it is.*0383

*That means this 2 rounds up to become a 3.*0386

*All the numbers before it stay the same, 22.8.*0391

*Instead of writing the 2, I have to write the 3.*0399

*The numbers behind it become 0s; that is my answer.*0402

*The next one, round this to the nearest hundredths again.*0408

*Hundredths... tenths, hundredths; circle it.*0413

*The number after it, the number behind it is a 3.*0420

*That is not 5 or greater; it is smaller than 5 which means my 6 stays the same.*0424

*If it is smaller than 5, the number behind it, this number right here,*0432

*if it is smaller than 5, then I don't make any changes to that 6.*0435

*I don't subtract 1; I just leave it the same.*0441

*It is either going to be the same; or you are going to add 1 to it.*0446

*It is 44 and 96 hundredths; and then 00.*0452

*This number to the nearest tenths.*0467

*The tenths is again this number right here, the first number after the decimal point.*0469

*Look at the number behind it; is the number behind it 5 or greater?*0474

*Yes, it is 5 or greater; it is 5.*0478

*I have to add 1 to that circled number; this becomes a 1 now.*0481

*693 and 1 tenths; I can put these as 0s.*0487

*The next, nearest tens.*0498

*The difference between this one and this one--tenths, tens; this one has the -TH.*0503

*That means it is this place value right here after the decimal point.*0510

*Tens would be this number right here because it is ones and tens.*0514

*I look at the number right behind it.*0521

*It is smaller than 5 which means my circled number stays the same.*0523

*This would be 60 or 69; my 3 changes to a 0, decimal point, 000.*0533

*Keep in mind, when you have 0s at the end of a number*0547

*after the decimal point, then I don't have to write them out.*0552

*Again only the 0s that are at the end of a number and behind the decimal point.*0557

*In that case, I don't have to write them; you could; you don't have to.*0563

*If there is a 0 before the decimal point, in front of it,*0571

*then you have to because 69 we know is different than 690.*0575

*You have to make sure to have that 0 there.*0581

*But these 0s, as long as they are at the end of a number,*0584

*and it is behind the decimal point, then you don't have to write them out.*0587

*But you could; you could just leave it like this; this is fine.*0592

*This number to the nearest thousands; no -TH.*0600

*That means that the thousands is the number right before the first comma.*0604

*Right in front of the first comma is the thousands.*0613

*I am going to circle that.*0616

*The number before it is a 5; it is 5 or greater.*0618

*That means I have to change this 0 to a 1.*0622

*I am going to write all the numbers before it--1, 8, 9.*0630

*Instead of writing the 0, I am going to write the 1; write my commas.*0636

*Remember all the numbers after the circled number are going to turn into 0s.*0643

*This was my circled number; that changed to a 1; 0 to 1.*0651

*It is going to become 000.00.*0654

*Again because these 0s are at the end of a number,*0659

*and they are behind the decimal point, I don't have to write them out.*0663

*But I could if I want; I could just leave it like that.*0666

*This is the number when you round this to the nearest thousands.*0671

*Now this one is to the nearest thousandths with a -TH.*0678

*We know it is behind the decimal point; here is tenths, hundredths, thousandths.*0683

*I am going to round to that number and circle it.*0691

*Look at the number behind it; it is a 7.*0695

*It is 5 or greater; 7 is bigger than 5.*0697

*That means I change the circled number to a 5; I round up.*0701

*Again the circled number, it is either going to stay the same if this number behind it is smaller than 5.*0707

*Or it is going to become 1 bigger if the number behind it is 5 or greater.*0715

*When I rewrite my number, I am going to write all the numbers in front of it up to my circled number.*0721

*All the numbers behind it becomes 0s; 5.055; this becomes 0 right there.*0727

*That is it for this lesson on rounding decimals; thank you for watching Educator.com.*0746

*Welcome back to Educator.com; this lesson is on adding and subtracting decimals.*0000

*When you add and subtract decimals, there are some rules to follow.*0008

*The first thing, the most important thing, is to align the decimal points.*0013

*Whenever you have decimal numbers and you have to add or subtract them, make sure the decimal point is lined up.*0018

*For example, if you are going to add 4.1 with 3.2,*0027

*you have to make sure that the decimal points, this and this, line up.*0036

*We are adding these numbers together; this becomes 3; this is 7.*0047

*After you add them, you have to make sure you place a decimal point in the same place.*0055

*You just align them straight down right here.*0059

*4.1 plus 3.2 is going to be 7.3; check by estimating.*0065

*4.1 is very close to 4; 4 is the whole number that it rounds to.*0073

*4 plus 3; this 3.2 rounds to 3; so it is about 7.*0077

*That is what it means to check by estimating.*0085

*Let's do a few problems; add 3.45 plus 7.835.*0088

*Again the first rule, the very important rule, is to line up the decimals.*0099

*Make sure you align them; 3.45.*0103

*The next number, we don't see a decimal point here.*0110

*7 doesn't have a decimal point; it actually does; we just can't see it.*0114

*If we have a whole number that doesn't show a decimal point, then the decimal point is right behind it.*0118

*It becomes 7.0; 7 is the same thing as 7.0.*0126

*If you have a whole number with no decimal point, that does not show a decimal point,*0132

*just place the decimal point at the end of it, right behind it.*0136

*I need to make sure that the decimal point goes right there; align them.*0141

*This one, 0.835; again 0.835, align the decimal.*0149

*That is all that matters; when you add them or subtract them, this is the main thing.*0159

*You don't line up the numbers; you line up the decimals.*0165

*We are adding these together.*0169

*Here, whenever I have decimals, I can place 0s at the end of them.*0170

*As long as it is behind the decimal and it is at the end, we can add as many 0s as we want.*0176

*Here it is behind the decimal point and it is at the end.*0182

*I can put 0s here to fill in those blanks.*0186

*If I add straight down, this becomes 5; 5 plus 3 is 8.*0191

*4 plus 8 is 12; 7 plus 3 plus 1 is 11.*0197

*My answer is 11.285 or 11 and 285.*0206

*The next problem, find the difference.*0215

*This is--write the first number--351 and 4 tenths, minus... be careful here.*0217

*If I write it like this, this would be wrong; this is wrong.*0229

*I have to make sure not to line up the numbers but line up the decimals.*0242

*65 and 25 hundredths; I have space right here; I have to put something there.*0251

*It is behind the decimal point and it is after the number so I can place a 0 there.*0266

*I can only place 0s if it is behind the decimal point and it is at the very end.*0272

*0 minus 5, remember I have to borrow; this becomes 3; this becomes a 10.*0277

*Get 5; 3 minus 2 is 1; decimal point comes down.*0284

*Again I have to borrow here; this becomes 4; this becomes 11.*0291

*11 minus 5 is 6; 4, again I have to borrow; this becomes a 2.*0298

*14; this becomes 8 and then 2; this is the answer.*0305

*Make sure your decimals are lined up.*0314

*Don't forget again if you have any empty spaces, you have to place 0s in them*0318

*because it is behind the decimal point and it is after the number so you can place 0s there.*0325

*Let's do a few more examples.*0331

*The next one, we want to add these two numbers, 123.1 and then 140.*0334

*We have a whole number that does not show a decimal point.*0343

*In that case, we are just going to place it at the very end.*0346

*It is going to be 140.0; adding these numbers.*0350

*It is 1; bring the decimal point down; 3; 6; and 2.*0358

*Next example, Sarah has 90 dollars and 75 cents.*0371

*Running shoes cost 55 dollars and 45 cents.*0377

*How much money will Sarah have left after buying the running shoes?*0381

*If she is going to buy something that costs 55 dollars and 45 cents, this is how much she has.*0386

*We have to see how much she will have left; we have to subtract the numbers.*0392

*She has 90 dollars and 75 cents; she spent 55 dollars and 45 cents.*0401

*I am going to subtract them; this will be 0; 3.*0412

*Again I have to borrow; this is 8; this is 10.*0421

*10 minus 5 is 5; this is 3.*0424

*After buying the running shoes, Sarah will have 35 dollars and 30 cents left.*0429

*The next example, we are going to subtract these two numbers.*0442

*Just write it out; 7 minus... be careful here.*0449

*The decimal place has to line up with this decimal place.*0455

*I am going to write 9 under the 2; 91 and 386 thousandths.*0458

*Again I have an empty space right here that I have to fill in with a 0.*0469

*Don't forget that; the reason why you have to place that 0 there.*0476

*If I don't have a 0... let's say I don't have a 0.*0480

*When you subtract this, this seems like it would be a 6 right here.*0486

*It seems like you would write a 6 here.*0491

*But if you place a 0 there, then you know that you have to borrow.*0494

*This is actually 10 minus 6.*0497

*If I borrow this, this is going to become 6; 10; 4.*0501

*Borrow again; 7; 16; there is 8.*0507

*7 minus 3 is 4; bring down the decimal point.*0514

*I am going to make this a 10; this becomes 11; this becomes 5.*0520

*10 minus 1 is 9; 11 minus 9 is 2.*0527

*5 minus 0 is 5; there is my answer.*0533

*If you want to check your answer by estimating, this is about 600.*0540

*If you just round it to the nearest hundreds, it is 600.*0546

*Here, this is about 90 or maybe let's say 100.*0550

*600 minus 100 is about 500; we have 500 something; it sounds right.*0554

*If we had let's say 50 something as our answer, we know that is wrong*0562

*because we know that if we estimate, it should be around 500.*0566

*The last example, adding these decimals together.*0573

*We have 8 and 215 dollars and 49 cents and 75 cents.*0578

*Remember the rule when we add or subtract decimals is to line up the decimal point.*0586

*That is very, very important.*0591

*When we have a whole number that is not showing a decimal point,*0593

*then we can place a decimal point at the end of it.*0597

*Just because it is not showing a decimal point does not mean it does not have one.*0601

*This is 8 point... and then I can add 0s to it.*0607

*8 dollars is the same thing as 8.00.*0610

*Then I am going to add 215 and 49 cents; line up the decimal point.*0614

*It is going to go 215.49; 75 cents here is going to be 0.75.*0621

*Going to add these all up together; 9 plus 5 is 14.*0637

*1 plus 4 is 5; 5 plus 7 is 12.*0646

*Bring down the decimal point; come straight down.*0655

*1 plus 8 is 9; plus 5 is 14; 1 plus 1, 2; and 2.*0660

*We are adding money together; don't forget your dollar sign.*0669

*This is your answer--8 dollars plus 215 dollars and 49 cents plus 75 cents becomes 224 dollars and 24 cents.*0674

*That is it for this lesson on adding and subtracting decimals.*0686

*Thank you for watching Educator.com.*0689

*Welcome back to Educator.com; this lesson is on multiplying decimals.*0003

*When you multiply decimals together, it is very different than when you are adding and subtracting decimals.*0007

*The rules are very different; try not to get confused between the two.*0015

*Remember when you add and subtract the decimals, you have to line up the decimal point.*0020

*Then you add and subtract.*0025

*Then you bring the decimal point straight down into the answer.*0026

*When you multiply decimals, you don't worry about the decimal point at all.*0030

*If I am going to multiply two numbers, let's say 1.1 and 6.*0038

*I don't have to line up the decimal point; make sure you don't do that.*0048

*All you have to do is multiply the numbers without having any consideration for the decimal point.*0051

*I am just going to ignore it; I am going to multiply this.*0058

*It is going to be 6; and then 6; 66.*0061

*What you do is you count the total number of decimal places from the numbers you multiplied.*0066

*From these two numbers, this one and this one,*0071

*you are going to count to see how many numbers are behind the decimal point.*0074

*Here I have one number.*0080

*Here I have none because the decimal point is behind the 6.*0082

*From these two numbers, from 1.1 and 6, I only have one number behind the decimal point.*0087

*I go to my answer; I place one number behind the decimal point.*0098

*The last number is 6; I am going to put the decimal point right there, 6.6.*0103

*Let's do a few examples, 0.2 times 0.6.*0111

*Again I am going to multiply the numbers without considering the decimal points.*0117

*6 times 2 is 12.*0126

*I don't have to multiply those 0s together; it is just 12.*0129

*From this number, from these two numbers, the two numbers that I multiplied,*0134

*I am going to see how many numbers I have behind the decimal point.*0138

*From this number, I have one; from this number, I have another one.*0143

*I have two total; I go to my answer.*0147

*I am going to place two numbers behind the decimal point.*0151

*It is going to become 0.12 or 0.12 or 0.12.*0155

*I can put a 0 up here too.*0161

*This is the whole number; we don't have any whole numbers; it is just 0.*0163

*Another example, if Susan works 25.5 hours per week and she earns*0170

*9 dollars and 40 cents an hour, how much does she earn in a week?*0177

*This is how many hours she works in a week.*0184

*This is how much she earns per hour.*0186

*To figure out how much she earns in a whole week,*0189

*I have to multiply how many hours she worked with how much she makes per hour.*0191

*It is going to be 25.5 times 9 dollars and 40 cents.*0200

*Again when I multiply these numbers, I am just going to line up the numbers.*0210

*I don't care about the decimal point.*0215

*25.5 times 9.40; you are just lining up the numbers.*0218

*Let's multiply the 0; 0 times 0 is going to be all 0s.*0232

*I am just going to move on to the next number.*0237

*4 times 5 is 20; this is 22; 8, 9, 10.*0238

*9 times 5 is 45; 45... that is 49; 18... that is 22.*0248

*These are just 0s here; it is 0, 0, 7, 9, 3, and 2.*0266

*Here is my answer when I multiply these two numbers together.*0276

*Now I have to look at my decimal point.*0280

*The first number, I look at these two numbers, the two numbers that I multiplied.*0284

*I have one number here behind the decimal point and I have two numbers here.*0288

*How many numbers do I have total?--I have three.*0295

*I go to my answer; I count three numbers.*0300

*Make sure that I have three numbers behind my decimal point.*0305

*It is going to be 237.7.*0311

*I am dealing with money here because I am trying to figure out how much she earns in a week.*0317

*This is going to be in money; I am going to have a dollar sign.*0323

*This becomes 239 dollars and 70 cents.*0330

*This is how much she is going to earn in a week.*0338

*The next example, I have 0.21 times 2.1.*0346

*0.21 or I can read this as 21 hundredths because I have two numbers.*0356

*This is tenths; this is hundredths.*0363

*This would be 21 hundredths times 2.1.*0365

*Again I am not going to line up the decimal point; 2.1 or 2 and 1 tenths.*0372

*When I multiply this out, ignore the decimal point.*0383

*2 times 1; write that here; that is 2; 2 times 2 is 4.*0389

*You don't have to look at this number.*0395

*If you want, you can just write that down.*0397

*1 times 0 is 0; that goes there; then I add these down.*0399

*1 plus nothing is 1; 2 plus 2 is 4; this is 4.*0406

*Now I look at my numbers; how many numbers do I have behind decimal points?*0413

*Here I have two; here I have another one; I have three total.*0421

*I am going to go to my answer; I am going to count one, two, three.*0428

*My answer, I have to make sure that there is going to be three numbers,*0436

*the same number of numbers behind this decimal point.*0439

*It is going to be 0.441 or 0.441.*0442

*You can read this as 441 thousandths because I have three numbers and this is the thousandths place.*0451

*This is my answer when I multiply 0.21 times 2.1.*0460

*The fourth example, I am going to multiply these two numbers, 4.08 and 1.35.*0468

*4 and 8 hundredths... again when I multiply these decimals together, I am just going to ignore my decimal points.*0479

*I am just going to line up the numbers just like I do when I multiply whole numbers.*0488

*This just happens to line up because there is the same number of numbers.*0495

*Multiply this out; 8 times 5 is 40; 0, 4; this is 20.*0502

*This is 24, 0, 2; this is 12 here; this is 8, 0, and 4.*0512

*I add them down; 0; this is 8; this is 10.*0528

*2, 4, 5; this is also 5.*0539

*From here, after I multiply my two numbers, I am going to look at the actual numbers that I multiplied.*0545

*I am going to count how many numbers I have behind decimal points.*0554

*For this one, I have two numbers behind the decimal point.*0558

*Here I also have two; total behind the decimal points, I have four numbers.*0562

*You look at just these two numbers that you multiplied together.*0570

*I have four numbers total behind decimal points; I go to my answer.*0574

*I make sure that there is four numbers behind the decimal point.*0579

*That is going to help me place the decimal point.*0583

*One, two, three, four; there is four numbers; place the decimal point right there.*0587

*Since there is four numbers behind decimal points here,*0595

*there has to be four numbers behind the decimal point in the answer.*0598

*It is going to be 5.5080.*0601

*Or this number, it is a 0 at the end of a number behind the decimal point.*0605

*I can just drop it if I want.*0609

*Or you can leave it; it doesn't matter.*0611

*It could be 5 and 508 thousandths.*0613

*Either way, this can be the answer or this can be the answer.*0620

*That is it for this lesson on multiplying decimals; thank you for watching Educator.com.*0626

*Welcome back to Educator.com; this lesson is on dividing decimals.*0000

*Make sure, when you are dividing decimals, that you apply the correct rules for it.*0010

*Don't get confused between when you add and subtract decimals and when you multiply decimals.*0018

*Dividing decimals is actually very different.*0023

*Before we begin, let's go over some words--divisor and dividend.*0027

*When divide two numbers together, if I have let's say 10 divided by 2,*0034

*this top number is the one that is going to go inside the box.*0043

*That is called the dividend.*0050

*This top number, the number that goes inside the box, is called the dividend.*0052

*The bottom number, the one that goes outside the box, is called the divisor.*0057

*Here is a divisor; here is a dividend.*0063

*When we divide decimals together, we have to make sure that the divisor becomes a whole number.*0068

*If this number right here is a decimal, then we have to change it into a whole number.*0075

*The way you do that is by multiplying both the divisor and the dividend by the same multiple of 10.*0080

*We don't care if this number is a decimal; it is only the divisor.*0090

*Let's say that the divisor is 0.2; it is 10 divided by 0.2.*0096

*That is a little high; 0.2; this is not a whole number.*0105

*We have a decimal; we have a number behind the decimal point.*0114

*I count how many numbers are behind that decimal point.*0118

*There is only one; there is one number.*0122

*That means I need to multiply this number by 10.*0124

*0.2, multiply it by 10 so that this will become a whole number*0129

*because 0.2 times 10 will just become 2.*0138

*If I have 0.22, I have two numbers behind the decimal point so I have to multiply it by 100.*0143

*There is two numbers behind the decimal point.*0153

*I have to have two 0s here as a multiple of 10.*0155

*That way this will become 22 or 22.0, same thing.*0160

*We will do a few examples of those.*0170

*For this one, I have to change the divisor to a whole number.*0172

*0.2, I have to multiply this by 10.*0176

*But that means I have to multiply the dividend by 10 also.*0179

*I can't just multiply one of these numbers by 10.*0182

*If I multiply 0.2 by 10, it becomes 2; 10 times 10 is 100.*0185

*This becomes 2; this becomes 100; this will be the actual problem.*0193

*When you find the answer for this, it is still going to be the same answer as if you were dividing this.*0199

*That is the rule; you have to make sure that this becomes a whole number.*0208

*Place the decimal point in the same place right above the dividend; the decimal point here.*0213

*We don't see a decimal point because it is a whole number.*0219

*If it is a whole number that doesn't show a decimal point, it is at the end.*0222

*It is right there; let me make this longer.*0226

*As long as it is behind the decimal point and at the end of a number,*0231

*I can add as many 0s as I want.*0235

*I can add one 0; I can add ten 0s.*0237

*I can add a million 0s; it doesn't matter.*0239

*100.0 is the same thing as 100; here is a decimal point.*0243

*I am going to place a decimal point right above, right there.*0247

*I know that my answer is going to go on top here.*0251

*Let me give myself some room.*0257

*When I solve this, I know 2 goes into 10 five times.*0267

*0, bring down the 0; 2 goes into 0 zero times.*0275

*My answer just becomes 50; 2 times 50 is 100.*0280

*Or if I bring down this 0, again 2 goes into 0 zero times.*0285

*It is just going to be 50; the answer will be 50.*0291

*Again the first rule when you divide decimals together is to make the divisor a whole number by multiplying it,*0295

*multiplying this number and this number to the multiple of 10.*0305

*That decimal point will go behind the number; it becomes a whole number.*0309

*Once you multiply that same number to the dividend, you place a decimal point right above here.*0315

*Then you just divide it the same way.*0321

*Let's do a few examples; 26.2 divided 0.4.*0323

*The first number, this is the dividend; this is the divisor.*0331

*The dividend goes inside the box, 26.2; and then 0.4.*0335

*I have a decimal in my divisor; I have to make it a whole number.*0346

*In order to make this 0.4 a whole number, I have one number behind the decimal point.*0353

*I have to multiply it by 10; 0.4 times 10 is going to be 4.*0359

*Since I have one 0, I can move the decimal point one time to make it bigger.*0368

*It becomes 4.*0374

*If I am going to multiply this by 10, then I have to multiply the dividend by 10.*0377

*If you want to think of it this way, you can do that.*0384

*Or a shortcut would be just to move the decimal point once here.*0388

*Then move the decimal point for the dividend once also.*0393

*Let me just rewrite this; this becomes 4.*0401

*My dividend is 262 because the decimal point moved behind the 2.*0404

*It is right there; then I can place a 0 at the end of it.*0412

*The next thing I do is to make sure I bring up the decimal point so it is lined up.*0418

*Then I can just divide it the same way.*0424

*I know that 4 does not go into 2*0427

*I can put a 0 there; or you don't have to.*0429

*4 goes into 26; 26 divided by 4.*0431

*How many times does 4 go into 26?*0435

*4 times 6 is 24; I am going to write that number right there.*0439

*24 subtracted; I get 2; I bring down this 2.*0451

*4 goes into 22 five times; that becomes 20.*0457

*Subtract again; 2; bring down this 0.*0463

*This 0 was not there; I placed it there.*0468

*Because it is behind the decimal point and it is at the end of a number, I can place as many 0s as I want.*0471

*I can bring down the 2; 4 goes into 20 five times.*0477

*That becomes 20; then I have 0.*0483

*I am done with the problem; my answer then becomes 65.5.*0488

*This is my answer; 26.2 divided by 0.4 is 65.5.*0493

*Sharon bought six CDs for 42 dollars and 8 cents.*0505

*How much does each CD cost?*0508

*If she has this much and she buys the six CDs, each CD costs the same amount.*0512

*42 dollars and 8 cents divided by 6.*0520

*Do 42 dollars and 8 cents divided by 6.*0526

*I don't have a decimal point here.*0537

*It is at the end; but I have a whole number.*0540

*I don't have to worry about changing this number, changing my divisor.*0542

*I can go ahead and just divide.*0547

*The next step would be to bring out the decimal point; don't forget that.*0550

*Then I can divide; 6 goes into 42 how many times?*0555

*Seven; 6 times 7 is 42; that becomes 0.*0562

*I have to bring down the other number 0.*0568

*6 goes into 0 zero times; that is 0; 0.*0572

*Bring down the 8; 6 goes into 8 one time; 6; 2.*0577

*I can add a 0 at the end of this*0588

*because it is behind the decimal point and it is at the end of a number.*0590

*I can bring down another 0.*0593

*I don't have to; I can just leave it like this.*0595

*But if I want to round this number, then I can just do the next step.*0597

*I can just do one more time; 6 goes into 20 how many times?*0604

*Three; that becomes 18; 2; I can just stop there.*0610

*Since I know I am dealing with money, I want to see how much each CD costs.*0617

*I know it is going to be in dollars.*0622

*Dollars only go to my hundredths place.*0624

*I only have two numbers after my decimal point.*0627

*How much is each CD going to cost?--7 dollars and 1 cent.*0632

*The only reason why I did one more number here was to see*0639

*if this was a 5 or greater, then this can round up to 2.*0643

*But it is smaller than 5; I can keep this as a 1.*0648

*It becomes 7 dollars and 1 cent; that is the cost of each CD.*0652

*My next example, 77.44 or 77 and 44 hundredths divided by 11.*0660

*77.44 divided by 11.*0668

*Rule number one, make sure this divisor is a whole number.*0676

*It is a whole number.*0679

*Step two, raise up my decimal point right there; now I can divide.*0682

*11 goes into 77 seven times; I subtract; this becomes 0; bring down the 4.*0689

*Fits into this number zero number of times; that becomes 0; subtract it.*0701

*I get 4; bring down this 4; 11 goes into 44 four times.*0707

*I get 0; 77.44 divided by 11 becomes 7.04.*0716

*My next example, I am going to do 45.218 divided by 0.23.*0730

*Or I can read this as 45 and 218 thousandths divided by 0.23 or 23 hundredths.*0737

*Remember the first rule in dividing decimals is to make sure that this number, my divisor, is a whole number.*0756

*It is not a whole number because there is numbers behind the decimal point.*0763

*I am going to count to see how many numbers I have behind the decimal point.*0767

*It is two; I have two numbers here.*0772

*What I am going to do is take that number, 0.23,*0775

*and multiply it by a multiple of 10 with this number many of 0s.*0783

*There is two numbers; multiply it by 100 with two 0s; this becomes 23.*0789

*If multiply my divisor by that number, I have to multiply my dividend by that number also.*0795

*I can just take this number, move it two places this way.*0804

*That is the same thing as multiplying it by 100.*0807

*Then I have to take this decimal point; I have to move it two places.*0810

*This was where my decimal point was originally.*0817

*It moves right there, two numbers.*0821

*Whatever you do to one number, you have to do to the other one.*0825

*If I write this over, it is going to be 4521.8 divided by 23.*0830

*When I divide this and get my answer, it is going to be*0847

*the same thing as if I divide that and get my answer.*0850

*My second rule is to bring up the decimal point; then I can go ahead and divide.*0854

*23 goes into 45 one time; this becomes 23.*0862

*I subtract it; I get 22; I bring down the 2.*0871

*How many times does 23 go into 222?*0876

*If you round this to 25, I know that 25 goes into 100 four times.*0881

*This is 200 and something; I can... let's see.*0888

*If I just try let's say 9; 23 times...*0891

*Or if I do 23 times 10, it is 230; that is too big.*0900

*I know it is going to be a little bit less than that which is 9.*0905

*23 times 9 is 27; 18, 19, 20... 207.*0910

*9 goes there; 207 goes there; subtract it; I get... I have to borrow here.*0919

*This is 12; this is 1; you subtract it; I get 5; this is 1 and 0.*0931

*What is my next number?--1; I am going to bring down the 1.*0941

*Again let's see how many times does 23 fit into 151?*0945

*Again 25 into 100 is four times; let's say 4; let's try 6.*0952

*On the side, I am going to do 23 times 6; 18.*0962

*Then let's try the next one; 23 times 7; that is 21... 14, 15, 16.*0970

*Which one do you think it is?*0978

*Is it going to be 6 or is it going to be 7?*0979

*We know it is going to be 6 because 7 is too big.*0982

*This number is too big; it can't be bigger than this number.*0986

*I am going to write the 6 here; it is 138.*0990

*If I subtract it... let me give myself some more room.*0997

*If I subtract it here, this will be 3... I am just borrowing.*1010

*This is 4; this becomes 11; 13; what happens next?*1017

*I have to bring this 8 down; 23 goes into 138 how many times?*1025

*Look at this; it is the same number.*1035

*I know that 23 times 6 is 138; 6 there.*1037

*Let me rewrite this right here; 138; 23 times 6 was what?*1044

*138; if I subtract it, then I get 0.*1049

*I have no more numbers to bring down.*1054

*I have no remainder; my answer is 196.6.*1056

*That is it for this lesson on dividing decimals.*1065

*Thank you for watching Educator.com.*1068

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over measures of central tendency.*0002

*The measures of central tendency are just three different types of ways you can describe data.*0008

*If you have a set of numbers, if you have some numbers,*0018

*then there are three ways you can represent the measures of those numbers.*0022

*The first one, the first measure of central tendency is the mean.*0031

*The mean is the sum of all the numbers divided by however many numbers you have.*0035

*Another word for it is average.*0043

*You are looking for the average of all the numbers in your data.*0044

*The next one is median.*0049

*Median is when you list out all the numbers in order from least to greatest,*0051

*you are going to find the middle number, the one that is right in the middle.*0058

*That is called the median; the key word here is middle.*0062

*The third one is the mode; the mode is the number that occurs the most.*0067

*It is the number that you see the most in your set of data.*0072

*The keyword here is going to be most.*0075

*Let's say if I have a set of numbers, let's say 1, 2, 3, 4, and 5.*0080

*The mean, keyword average, we are going to find the average of all those numbers.*0087

*We are going to add them all up; 1 plus 2 plus 3 plus 4 plus 5.*0095

*You are going to divide by however many numbers you have.*0102

*Here we have five different numbers; you are going to divide that sum by 5.*0107

*1 plus 2 is 3.*0115

*I am just going to write that number on top like that.*0118

*That is 3 plus 3 is 6; 6 plus 4 is 10; 10 plus 5 is 15.*0122

*This looks like a fraction... if I write it like... I am sorry; wrote the wrong number. *0130

*5... if write it like that, it looks like a fraction.*0138

*But fractions are division; you can just think of that as 15 divided by 5.*0141

*15 divided by 5; we know that 5 goes into 15 three times.*0148

*15 divided by 5 is 3; the mean is 3.*0154

*That is the average of those five numbers.*0160

*The median of that set of data is going to the middle number*0164

*but only when you list it out in order from least to greatest.*0170

*You must list it out.*0174

*Here it is already listed from least to greatest; 1, 2, 3, 4, 5.*0177

*The number in the middle will be this number right there; the median is 3.*0184

*When you have two numbers in the middle, let's say you have an even number of numbers.*0196

*Say it is just 1, 2, 3, and 4.*0202

*If that is your data, you have two numbers in the middle.*0207

*Then you are going to find the average between those two numbers.*0212

*We are going to add those two numbers and divide it by 2.*0216

*That will be 2 plus 3 divided by... there is only two numbers there so it is 2.*0220

*That is going to be 5/2.*0227

*You can usually leave it as a fraction.*0231

*If you want, you can change it to a mixed number.*0234

*How many times does 2 fit into 5?--2 fits into 5 two times.*0236

*You have 1 left over; keep your same denominator.*0243

*That is how you change... this is called an improper fraction*0248

*when the number on the top is bigger than the number on the bottom.*0251

*You can change it to a mixed number where you are going to have a whole number and then a proper fraction.*0255

*Again 2 fits into 5 two times; that becomes your whole number, 2.*0263

*Your leftovers is 1 over... your denominator is 2.*0267

*You can leave it like that.*0271

*Or if you want, you can just do 5 divided by 2, and change it to a decimal.*0273

*Remember 5, this top number, goes inside; that is on the outside.*0283

*Put a decimal point at the end of that number; bring it up.*0290

*2 fits into 5 twice; that is a 4; we subtract; get 1.*0293

*I can add 0s there at the end of that number behind the decimal point.*0301

*Bring that 0 down; 2 goes into 10 five times.*0307

*That is 10; my remainder is 0.*0314

*Your median here, when we find the average of that, will either be 2 and 1/2 or 2.5.*0318

*You could just think of it as halfway between 2 and 3.*0330

*That is the average; between 2 and 3 is going to be 2 and 1/2; 2.5.*0334

*The third one, the mode, remember the keyword here is most.*0342

*It is the one that you see the most.*0346

*Here with our set of data, 1, 2, 3, 4, 5, you only see each of the numbers one time.*0350

*In this case, we have no mode.*0359

*If you had 1, 2, 2, and 3, then you know the mode would be 2 because you see that number the most.*0363

*It occurs the most; that is the mode.*0373

*Again mean is average; median is middle; mode is most.*0377

*First example, using this set of data, we are going to find the mean, median, and mode.*0386

*Mean, we are just going to add up all the numbers.*0393

*For mean, it doesn't if the numbers are in order because when you add, the order doesn't matter.*0397

*If I add 1 plus 2, it is going to be the same thing as 2 plus 1.*0403

*Here just add up all the numbers; 3 plus 5 plus 3 plus 8 plus 6 plus 10 plus 4.*0407

*Then we are going to divide that number by 1, 2, 3, 4, 5, 6, 7, seven numbers.*0420

*3 plus 5 is 8; plus 3 is 11.*0431

*That is 19; that is 25; that is 35; that is 39.*0436

*It is going to be 39; that is the sum; divided by 7.*0444

*You can either leave it like this as long as it doesn't simplify.*0453

*As long as there is no factors that goes into 39 and 7, you can just leave it as an improper fraction.*0456

*To change it to a mixed number, we ask ourselves how many times does 7 fit into 39?*0467

*I know 7 times 5 is 35; 7 times 6 is 42; that is too big.*0474

*My whole number is going to be 5 because 7 fits into 39 five times.*0481

*I have 4 leftovers; 4 over... keep the same denominator.*0487

*That will be our mean.*0495

*Again if you want to change this to a decimal instead, just do 39 divided by 7.*0499

*39 inside; divided by 7; put the decimal point at the end; bring it up.*0506

*I can add a 0 there if I want; I can add two 0s.*0514

*I can add three; it doesn't matter.*0518

*7 fits into 39 five times; that is 35; subtract it; I get 4.*0520

*Bring down this 0; 7 goes into 40 again five times; that is 35.*0528

*Subtract it; I get 5; I can bring down another 0.*0538

*7 goes into 50 seven times; that is 49.*0544

*Usually as long as you have one or two numbers behind the decimal point,*0552

*you can probably just stop there and write that as your answer.*0556

*Maybe like 5.57 or 5 point and then what you can do is maybe you can round this number.*0559

*This number is 5 or greater.*0566

*What you can do is you can round this number up to be 5.6.*0571

*That is the mean; I am just going to write 5.6.*0578

*Either one will be your answer.*0585

*The next one, median; remember the median, the keyword is middle.*0588

*Be careful here, the most common mistake for this one*0595

*is just finding the middle number from your data set.*0599

*Make sure you have to write the number in order from least to greatest.*0605

*My smallest number here I see is 3; I have another 3.*0609

*I have this is 4, then 5, 6, 8, and 10.*0616

*Make sure I have one, two, three, four, five, six, seven numbers.*0628

*The number in the middle, I can cross out the outside numbers one more time.*0632

*My median will be 5.*0639

*The last one, the mode is most; the mode is most.*0645

*What number do you see the most?--what number occurs the most?*0652

*That would be the 3 because 3 you see it twice.*0657

*The other numbers, you only see them once; 3 is going to be the mode.*0662

*The next example, same thing.*0674

*Find the mean, median, mode for the following set of data.*0677

*We have four numbers here for the mean; this is average.*0683

*We are going to add up all the numbers divided by however many numbers we have.*0692

*It is 15 plus 12 plus 19 and plus 10.*0696

*Divide that by... I have four numbers.*0703

*15 plus 12 is 27; write that there.*0708

*27 plus 19... 7 plus 9 is 16; bring up the 1.*0715

*I am going to write that 6 right here; 2, 3, 4.*0722

*27 plus 19 is 46; add the 10; you are going to get 56.*0727

*Divide that by 4; 56 and 4; I want to just divide it.*0734

*56 is going to go on the inside for 56 divided by 4.*0748

*4 goes into 5 one time; that gives you 4; subtract it.*0754

*Get 1 left over; bring down this number, 6.*0762

*4 goes into 16 four times; my mean is 14.*0765

*My median, that your middle number.*0777

*Let's write our numbers in order from least to greatest.*0784

*That is 12, then... forgot the 10; 10, 12, 15, 19.*0787

*The middle number, we have two middle numbers.*0805

*We are looking for the middle right in between 12 and 15.*0811

*We are going to find the average; we can add those two numbers together.*0815

*It is 12 plus 15 divided by 2; this becomes 27 divided by 2.*0819

*We can again change it to a decimal or leave it as a fraction.*0830

*27... I don't know why I wrote that.*0835

*27 divided by 2; 2 goes into 2, this first number, one time.*0840

*That is 2; subtract it; get 0; bring down the 7.*0849

*2 goes into 7 three times which is a 6; subtract it; get a 1.*0854

*From here, since I have a remainder, I can just go ahead and add my decimal point.*0863

*Bring it up; add the 0; bring down the 0.*0867

*2 goes into 10 five times; that gives me 10; I get no remainders.*0872

*My median here is going to be 13.5.*0881

*The last one is mode; the mode is the number that occurs the most.*0890

*15, we only see it once; 12 only once; 19 once; 10 once.*0900

*For the mode, we have none; we can just write none.*0906

*The next example, Sarah's test scores for the last five chapters are 90, 92, 86, 97, and 90.*0913

*Find the mode, mean, and median of her scores; let's start with the mode.*0923

*The mode, keyword most; we look at what number occurs the most.*0931

*The 90, we see 90 twice; my mode is going to be 90.*0939

*The next one is mean; mean is the average.*0949

*We are going to add up all the numbers.*0959

*90 plus 92 plus 86 plus 97 plus 90; all over... 1, 2, 3, 4, 5... 5.*0961

*Let's do this one right here; 90 plus 92 is... 2 and then 18.*0981

*Then I am going to add the next number, 86; plus 86.*0991

*You can do it this way.*0995

*Or you can just maybe list them all out and then add them up like that.*0995

*86; this is 8; 8 plus 8 is 16; that is 2.*1001

*We got this, this, this; now we have to add 97.*1011

*That is 15; this is 1 plus 9 is 10; plus 6 is 16; this is 3.*1017

*The last one, 90; this is 5; this is 15; this is 4.*1028

*When I add up all the numbers, it becomes 455.*1040

*Divided by... I have five numbers.*1048

*I know that 5 is going to go into this number evenly because it ends in a 5.*1052

*The number ends in a 5 or 0, then it is going to be divisible by 5.*1057

*455, let's divide it; 5 doesn't go into 4; 5 goes into 45 nine times.*1063

*That is going to give you 45; subtract it; get a 0.*1076

*Bring down the 5; 5 goes into 5 one time; that is a 5.*1080

*My answer is 91; that is my mean, the average; mean.*1090

*That means her test scores, if she scored these scores, her average is 91.*1099

*She is averaging pretty well; that is an A.*1106

*The last one is the median which is the middle.*1112

*The middle number, let's list our numbers in order from least to greatest.*1120

*The smallest number is 86.*1125

*Then we have 90; then 90 again; 92; and then 97.*1130

*Our median, our middle number, is 90.*1144

*The fourth example, the daily temperature for the last few days were 72, 70, 83, 75, 81, and 75.*1153

*Find the three measures of central tendency.*1164

*We have the mean, the median, then we have the mode.*1166

*First, mean; we know the keyword for the mean is average.*1180

*We have to add up all the numbers and divide it by however many numbers we have.*1184

*That is 70... 72 is our first one.*1190

*72 plus 70 plus 83 plus 75 plus 81 plus 75.*1194

*I have one, two, three, four, five, six numbers.*1211

*I am going to divide this sum by 6 because I have six numbers.*1214

*Let's add up the numbers; 72 plus 70.*1222

*I am just going to add up just like how I did before.*1228

*2 plus 0 is 2; this is 14; I am going to take this number.*1232

*I got this; I got that one; add this number, 83.*1238

*This is 5; this is 12; and then 2.*1244

*Add the 75; this is 10; 7; 9; 10; this is 3.*1251

*Add this one, 81; 1; 8; 3.*1263

*The last one is 75; this is 6.*1271

*8 plus 7 is 15; 3 plus 1 is 4; 456; 456 divided by 6.*1278

*Let's divide this number by 6; 56 divided by 6.*1294

*I know that 6 cannot fit into 4.*1308

*6 is going to fit into 45, this number here.*1311

*6, let's see; 6 times 6 is 36; 6 times 7 is 42.*1316

*6 times 8 is 48; we know that it is 7; this is 42.*1321

*If I subtract it, I get 3; bring down this number here, 6.*1328

*6 goes into 36 six times; that is 36; 0.*1333

*My mean here is 76; that is the average.*1340

*Let me just write this a little bit lower.*1351

*The next one is median; median, we know the keyword is middle.*1356

*We are going to look for the middle number after we list the numbers out in order from least to greatest.*1366

*The smallest number is 70; then let's see, 72.*1373

*Then 75; then again 75; then 81; and 83.*1384

*I have my six numbers; the middle number now.*1401

*I am going to cross out the last numbers; cross those out.*1406

*Then I have two numbers here.*1411

*Normally when you have two numbers, you are going to have to find the average between those two numbers.*1414

*You are going to have to find the middle number between those two.*1419

*You would add them; divided by 2.*1423

*But I know since they are both 75, the number in the middle of 75 will just be 75.*1425

*Median will just be 75.*1436

*It is the same number so then our median has to be that same number.*1439

*The last one, the mode, the keyword here is most.*1446

*What number from all the six numbers on our data, what number do we see the most?*1451

*That number would be 75.*1458

*It is the number that occurs the most; that is 75.*1464

*That is it for this lesson; thank you for watching Educator.com.*1470

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over histograms.*0002

*A histogram is a bar graph that shows the frequency data that occurs within intervals.*0006

*It looks like almost identical to a bar graph.*0016

*It pretty much is a bar graph; but the difference is that the bar graph...*0018

*it just shows you the relationship between two different variables, two different things.*0025

*We know a bar graph looks like this.*0035

*Something like that where this represents something and this represents something.*0043

*A histogram is almost identical to that except a histogram shows the frequency.*0049

*It shows how many times something occurs, how frequent something occurs.*0058

*And it is also going to be within intervals.*0065

*On this side, this horizontal side right here, this is going to show the intervals.*0070

*This right here is going to show the frequency, how many times.*0078

*Let's say I have a set of numbers; I just have my data.*0084

*It is going to be 7, 11, 1, 12, 5, and 14.*0089

*I have six numbers.*0100

*The first thing I want to do to create a histogram is to find what I want my intervals to be in.*0104

*I am going to set intervals here.*0113

*Meaning it is going to be numbers between 1 and 5, then 6 and 10, then so on.*0116

*You are going to create the groups.*0125

*For this, let's say I am going to do exactly that.*0128

*It is going to be 1 through 5; my interval is going to be 1 through 5.*0132

*Then 6 through 10; then 11 through 15; it is going to be every five.*0136

*Let's say I am going to do 1.*0141

*From here to here is going to be 1 through 5.*0144

*Then from here to here, it is going to be 6 through 10.*0149

*Then from here to here, it is going to be 11 through 15.*0154

*Another thing about histograms is that the bars that you are going to draw, they are going to be stuck together.*0160

*There is not going to be any space in between them like the bar graph because it is in intervals.*0170

*All the numbers are going to fall in between one of these numbers.*0176

*All the bars are going to be stuck together.*0180

*Here it is going to be the frequency.*0185

*This is the frequency, how many times those numbers occur.*0190

*Step one was to create the intervals.*0198

*Then what I want to do just to make things a little bit easier for me to graph,*0201

*let's tally up how many times the numbers fall into their intervals.*0208

*I am going to just do 1 through 5... write my intervals... 6 through 10, 11 through 15.*0215

*Let's say I am going to tally up.*0223

*Every time a number falls under that category, that group, I am going to tally it up.*0228

*7, the first one, is right there; that is a tally mark for this group.*0233

*11, it is going to fall under there; 1 is going to fall here.*0239

*12 here; 5 here; and 14 is going to be there.*0243

*The frequency from this group 1 through 5 is going to be just two.*0252

*6 through 10 is going to be just once; 11 through 15 is three.*0257

*The frequency, here I am going to just do one, two, three, four.*0262

*One, two, three, four because the most I see a group is going to be is three.*0271

*That is the most; now all I have to do is create my bars.*0279

*The first one, 1 through 5, what is the frequency for that?--two.*0284

*I am going to graph that; I can shade it in.*0290

*The next group, 6 through 10, is just once.*0303

*It is going to be like that.*0310

*My third group frequency, three times; I go up to three; shade that in.*0320

*That is it; that is your histogram.*0339

*Again the first thing you do when you have a set of data like that,*0343

*the first thing you want to do is look at your data, look at your numbers and decide on your intervals.*0347

*How do you want to group up those numbers?*0354

*Once you do that, tally up how many times those numbers fall into that group.*0357

*Once you do that, you are going to create your intervals here,*0364

*create the frequency here, and then you are just going to draw your bars.*0368

*That is a histogram.*0372

*The first example, we are going to draw a histogram for the frequency table.*0376

*Here it gives us the intervals and the frequency.*0380

*All we have to do is draw the histogram.*0383

*Remember that this right here, the horizontal part of it, we are going to create the intervals.*0390

*I have four different groups, four intervals; one, two, three, and four.*0400

*This will be 1 through 4; this will be 5 through 8; 9 through 12; 13 through 16.*0410

*The frequency is going to be here; look at the biggest number is eight.*0421

*Then I know I have to show up to 8 on this part right here.*0426

*It will be 1, 2, 3, 4, 5, 6, 7, 8.*0432

*This is eight; 1, 2, 3, 4, 5, 6, 7, 8.*0441

*1 through 4, 1 through 4, the frequency is five.*0449

*I am going to go all the way up to five.*0454

*The next one is 5 through 8; 5 through 8 is three; go up to three.*0463

*Next one is eight; 9 through 12, the interval is eight; up to eight.*0477

*I don't have any more colors so I am just going to use black again.*0498

*13 through 16 is seven; go up to seven.*0502

*It would be nice if you have a ruler, you can draw these straight lines.*0515

*That is it; that is the histogram; again interval is the frequency.*0522

*The age of each child that attended the summer camp is given.*0529

*Create a histogram of the data.*0532

*We have all the ages of the children that attended the summer camp.*0535

*Again the first step is to create your intervals.*0542

*You want to create how you want to group up the eight, all the numbers.*0545

*It is really up to you.*0551

*If you want to create more intervals, then you can make the groups smaller.*0551

*You can make numbers for each group smaller.*0556

*Or if you want to create less than intervals, then you have to make the groups a little bit larger.*0562

*To create the interval, you want to look at the biggest number.*0569

*The biggest number here or this case the oldest child is 17.*0575

*The smallest number here is 2; youngest is 2; the oldest is 17.*0583

*You have to create your intervals based on those numbers.*0588

*Let's see, I want to create my intervals in every five.*0594

*Let's say 1 through 5, then 6 through 10, 11 through 15, 16 through 20.*0600

*Remember whenever you create your intervals, make sure that each interval has to be the same.*0612

*It has to be the same for each number of numbers for each group.*0618

*1 through 5 and then see here.*0624

*It is 1 through 5 and then through 10 and then through 15 and then through 20.*0626

*I know that is the same for each group.*0631

*Next step once you create your intervals is to see how many fall under each group.*0635

*Let's tally up the numbers; 5 is going to go into this group.*0643

*11 is going to fall under 11 through 15.*0648

*3 is right there; 9 in there; 6 is going to go right there.*0652

*7 in there; 8; 15; 10.*0660

*Once you have four, just make that the fifth one.*0669

*4 in there; 14, 10, 5, 8, 17, 14, 2, 2, 12, and 12 right there.*0676

*I have 5, 10, 15, 16, 17, 18 numbers total.*0706

*1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18.*0711

*Now that I have my intervals and I have the frequency of each interval, I can create my histogram.*0718

*Just draw that; and then you are going to draw that.*0726

*These numbers here are going to be the age.*0732

*This is going to be the frequency.*0740

*I am going to need four intervals.*0751

*It is 1 through 5, then 6 through 10, three, and four.*0753

*6 through 10, 11 through 15, 16 through 20.*0763

*Here the most that occurs, this one, seven.*0773

*I know I have to list out up to seven.*0777

*One, two, three, four, five, six, seven, eight.*0782

*Here is seven; one, two, three, four, five, six, seven, eight.*0789

*The first one, 1 through 5, that interval I have five; draw up to five.*0797

*The next one is seven; up to seven.*0812

*11 through 15, another five; I am going to use black again.*0827

*16 through 20, the last one is just only one person that falls under that age group.*0840

*That is it; that is your histogram.*0852

*The next one, create a histogram of the following test scores.*0858

*We have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, twelve scores.*0863

*For this one, I probably want to create my intervals maybe in tens.*0871

*50, I know that my smallest number is 50 here.*0882

*My biggest number that I see is 96.*0886

*Because these are test scores, it will help me see how many fall under...*0891

*how many scored within the 50s and the 60, 70s, 80s, 90s.*0895

*Let's just create our intervals; 50 through 59.*0900

*Then 60 through 69; 70 through 79; 80 to 89; then 90 to 99.*0907

*The first one, 50, right there; 56; 82.*0928

*Again the first thing I did was create my intervals to see*0937

*how many groups and how many numbers will fall under each group.*0941

*Now I am taking the numbers in the data.*0946

*I am going to tally it into the groups to see the frequency of each interval.*0948

*The next one is 79, 71, 65, 90, 85, 83, 64, 96, 92.*0956

*This way I can see that means if you scored within 50 to 59, that is an F.*0987

*Two Fs; this is a D; two Ds; two Cs; three Bs; and three As.*0994

*That is good; we have majority of the scores being Bs and As.*0999

*This helps you see; it helps you see how many of each there are.*1006

*Once you have this done, you can go ahead and create your histogram.*1012

*Going to make this a little longer.*1022

*I have five intervals, five groups.*1028

*It is going to be first right there; two, three, four, five.*1031

*This will be 50 through 59; 60 through 69; 70 through 79... 90 through 99.*1040

*These are all the scores; label that scores.*1058

*This right here is always going to be the frequency.*1062

*Here the group that occurs the most is three.*1073

*One of these, just three; one, two, three.*1077

*Or I just always like to do one more than I need; one, two, three, four.*1083

*Let's just create our bars now; the first group, 50 to 59 is two.*1091

*You are going to draw the bar up to two.*1096

*The next one, 60 to 69 is two; also two; two.*1104

*The next one is also two; two Cs; the Bs, we have three of them.*1120

*I only have three colors so I am going to go back to black; go up to three.*1133

*Then I am going to use blue again; 90 to 99, the As, three of them.*1145

*That is our histogram.*1160

*That way we can see our data a little bit clearer rather than when they are listed out like that.*1161

*If they are like this, then I can see since these are all the different scores,*1167

*F, D, C, B, A, I can see how frequent each of them are.*1172

*That is the whole point of the histogram.*1177

*That is it for this lesson; thank you for watching Educator.com.*1180

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over the box and whisker plot.*0002

*A box and whisker plot looks like this.*0009

*It displays the distribution of data items along a number line.*0012

*We are going to use the box and the whiskers to represent certain items in the data.*0018

*Before I explain this, I want to give you the data set.*0027

*The numbers that I am basing this plot on are 2, 3, 4, 5, 6, 7, and 9.*0032

*Let's say that that is my data set.*0051

*From here, I want to find the median.*0056

*Remember median; the key word for median is middle.*0061

*If the numbers are in order from least to greatest, then I want to find the middle number.*0065

*2, 3, 4, 5, 6, 7, and 9.*0069

*Let me just rewrite it here so I can give myself some room to work with... 6, 7, and 9.*0073

*To find the median, because it is in order from least to greatest, I can just find the middle number.*0083

*3 on this side; 3 on this side; that is my median.*0091

*This is called the median.*0095

*Not including the median, on the lower group of numbers,*0106

*the smaller group of numbers, I want to look for the median again.*0110

*Just from this group right here, just from those three numbers,*0115

*the median, the middle number, is going to be 3.*0119

*That number is called the lower... because it is part of the lower set... lower quartile.*0124

*Same thing for the other side.*0138

*Again between those three numbers, the median is going to be 7.*0141

*That is called your upper quartile.*0148

*All three numbers represent the median.*0157

*This 5 right here, that is the median within the whole set of numbers.*0161

*The lower quartile is the median among the lower half.*0167

*The upper quartile among the upper half.*0171

*The smallest number in your data which is the 2, that number there is called your lower extreme.*0179

*That means this number, the biggest number in your data, is going to be your upper extreme.*0194

*Again we have our median.*0208

*That is the first thing you want to look for, median.*0210

*From your lower half, you are going to find the median again.*0212

*That is your lower quartile; in the upper half, upper quartile.*0215

*Smallest number is your lower extreme.*0221

*Your largest number, the biggest number, is your upper extreme.*0224

*Once you find those numbers, we have five numbers circled there.*0227

*Those are the numbers that are going to be represented in this box and whisker plot.*0232

*The box right here, these three lines in the box,*0241

*this one, this one, and this one, represent your three medians.*0246

*The number right here, this right here is your median of the data.*0254

*This is your median; this right here which is 5; this your median.*0260

*These two right here, this one and this one, the 3 and the 7, are your quartiles.*0272

*This is the lower quartile; this one right here is your upper quartile.*0281

*Those three, your quartiles right here, that is what forms the box.*0293

*All you have to do is once you have the lower quartile,*0299

*you have the upper quartile, then you just draw lines connecting them like that.*0302

*That is how you create your box from the box and whisker plot.*0308

*These right here are called your whiskers.*0312

*Here is you whisker here; the whisker here.*0318

*This number right here is your lower extreme.*0321

*That is the 2; see how it is under the 2.*0330

*This one is your upper extreme.*0334

*That is how we form the box and whisker plot.*0341

*Median, your lower quartiles, that forms the box.*0344

*The whiskers are going to go to your extremes, the lower extreme and then the upper extreme.*0348

*Once you have a set of data, just look for those numbers.*0355

*Once you have them, you can just go ahead and just draw your box and your whiskers.*0358

*The first example, name the median, lower quartile, upper quartile, lower and upper extremes.*0365

*Remember within our box, we look for these numbers here.*0373

*This right here, this is your median.*0379

*The median I am going to say is 8.*0385

*This number here is my lower quartile; lower quartile is 6.*0396

*My upper quartile is 12.*0410

*Then your extremes, that number right there, the whiskers, is going to the 4.*0421

*My lower extreme is 4; my upper extreme is 13.*0428

*We got median, lower quartile, upper quartile, lower extreme, and upper extreme.*0446

*The next one, the information is given to us.*0457

*We just have to draw the box and whisker plot.*0462

*Remember the lower extreme and the upper extreme, that is my smallest number and my biggest number.*0466

*Those are the numbers that are going to be the numbers that my whiskers are going to be drawn to.*0472

*I know my lower extreme is going to go like that.*0482

*I am going to draw a point right there.*0486

*My upper extreme is going to be 12.*0488

*The easiest way to draw the box and whisker plot is to draw the box first.*0493

*My lower quartile is 5.*0499

*I am going to draw a little line like this on 5.*0504

*My upper quartile is 10; that is the median of the upper half.*0509

*Upper quartile is 10; draw line like that.*0516

*My median is 8; 8 right there.*0523

*Remember you want to draw your box; that is going to be the box.*0532

*You don't have to use it in different colors; it could just be like that.*0539

*Just make sure you have these three under the correct numbers.*0543

*Now you can just draw your whiskers.*0550

*From here, you can draw the whisker going out to the 4.*0553

*Here the whisker to the upper extreme; that is 12.*0558

*That is it; that is your box and whisker plot.*0565

*The next example, we are going to find the median,*0572

*the lower and the upper quartiles, the lower and the upper extremes.*0575

*Then we are going to graph it using the box and whisker plot.*0578

*The first thing to do since we have to find the median and the quartiles and the extremes,*0584

*I need to list my numbers out in order from least to greatest, the smallest number to the biggest number.*0592

*Smallest here is 5, then 6, then 7, 8, 8, 9, and 10.*0599

*Let's see, one, two, three, four, five, six, seven.*0613

*One, two, three, four, five, six, seven; just to make sure I didn't miss a number.*0615

*Then I need to find my median; let's see.*0620

*My median will be this right here; this is a median.*0627

*Then I want to find my quartiles.*0635

*Among these numbers, that will be the lower quartile.*0638

*This from here, there is my upper quartile.*0652

*Then my lower extreme; my upper extreme.*0660

*Now I need to graph the box and whiskers plot.*0680

*I need a number line first.*0685

*My largest number is 10 so I have to make sure I cover up to 10 on the number line.*0694

*1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.*0709

*My median is 8; draw a little line like that under 8.*0721

*My lower quartile is 6; my upper quartile is 9.*0729

*Once you have your three little lines right there, draw the box that connects those quartiles.*0739

*My lower extreme, from the box, I am going to draw a whisker out to that lower extreme, 5.*0749

*From here, whisker out to the upper extreme which is 10.*0757

*That is it for this one.*0765

*The fourth example, we are going to draw a box and whiskers plot for this set of data.*0769

*Again we are going to find the median.*0776

*The way we do that is to list numbers out from least to greatest.*0778

*The key word for median is middle.*0782

*Smallest number, 2; we got 2; then the 3; we have 4; another 4.*0787

*Then we have 6; then two 7s; 7, 7, 8, 9, 10, and 11.*0801

*I have one, two, three, four, five, six, seven, eight, nine, ten, eleven numbers.*0820

*One, two, three, four, five, six, seven, eight, nine, ten, eleven.*0825

*Now I need to look for my middle number, the median.*0830

*Let's see, one, two, three, four, five; we count five this way.*0835

*One, two, three, four, five; that leaves me with this number in the middle.*0838

*Remember if you have two numbers in the middle,*0845

*you have to find the average of those two numbers.*0850

*Let's say I had an extra number on this side.*0852

*6 and 7, if those were my middle numbers,*0857

*I would have to find the middle number between 6 and 7.*0859

*Again this is when you have an even number of numbers in your data.*0863

*Then your median, you have to find the middle number between the two numbers in the middle.*0868

*You can do that by adding those two numbers together and dividing it by 2.*0873

*You are finding the average, the mean, within those two numbers.*0877

*In this case, we have only one number so that is our median.*0881

*Then from the lower set of numbers, I look for my median again.*0890

*The middle number is 4; remember that becomes our lower quartile.*0900

*From the upper set, I have the 9; that is going to be my upper quartile.*0912

*My smallest number which is 2 is my lower is my extreme.*0928

*My largest number is my upper extreme.*0938

*Once I have those five numbers, I can draw my box and whiskers plot.*0947

*I am going to draw a number line.*0952

*My largest number is 11.*0962

*I have to make sure I cover up to 11 on this number line.*0964

*1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, one more, 12.*0968

*1, 2, 3, 4, 5, 6, 7...*0978

*I want to first draw my median.*0989

*On 7, I am going to draw that little line right there under 7.*0993

*Then your quartiles; the 4 like that; another one under 9.*0999

*Remember the quartiles are going to form your box.*1009

*Let's just draw a box like that.*1012

*Once you have your box, we need to draw our whiskers.*1021

*From this end, I am going to draw a whisker out to the lower extreme which is 2.*1025

*Draw a whisker out there.*1033

*On this side, my upper extreme is 11.*1038

*Draw the whisker out to 11 right there; that is it.*1041

*Remember these three are going to represent your medians.*1046

*This is the median of your data.*1051

*These two are the medians of the lower and upper halves.*1053

*That is going to form your box.*1057

*Your whiskers, draw it out to the lower extreme, the smallest number.*1059

*This whisker, upper extreme which is the largest number.*1064

*That is it for this lesson; thank you for watching Educator.com.*1069

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over stem and leaf plots.*0002

*A stem and leaf plot is a way for you to organize your data so that it is easy to see.*0007

*It lists them out in order from least to greatest.*0015

*The stem of a stem and leaf plot is usually the tens digit.*0020

*It is the number on the left.*0027

*The leaf, red for the leaf, is the digit on the right which is the ones digit.*0031

*Let's say I have a number 52.*0039

*I am going to separate the number 52.*0044

*I am going to separate the digits into a stem and then a leaf.*0046

*The tens digit, the 5, I am going to make my stem.*0050

*The ones digit, my 2, is going to be my leaf; stem and leaf.*0057

*I am going to separate them by drawing a little line like that.*0063

*All the stems are going to go on the left side.*0069

*My leaves are going to go on the right side.*0071

*That is a number.*0074

*If I have these numbers here, I have 50, 52, and 55.*0075

*The stem, the tens digit, is all the same.*0088

*Since it is the same number, I am going to separate the stem and the leaves.*0093

*But I only have to write the stem one time because it is the same stem.*0098

*Then when I write my leaves, I am going to write the 0, the 2, and the 5.*0105

*It becomes 0, just like I separated this number 52.*0115

*Again because the stem is the same number,*0122

*I am going to just put the leaf, that 2, right there next to the 0.*0124

*Then again the next leaf, 5.*0130

*This represents the tens digit and then all the ones digits.*0134

*This is 50, 52, and then 55.*0139

*That is how you do a stem and leaf plot.*0145

*Using these numbers here, we are going to create a stem and leaf plot.*0151

*The first stem, the smallest stem that I see is the 1.*0157

*The smallest tens digit is the 1.*0164

*I have a 2; that is another stem.*0167

*I have a 3; I think that is it for my stems.*0170

*I am going to list out my stems; stems and my leaves; stem and leaf.*0175

*My stem, the smallest stem, 1; then 2; and then 3.*0186

*I don't think I have any more stems; that is it for my stems.*0195

*My leaves, the other side, I am going to write them out.*0200

*Before you do this, it will be a little bit easier... you don't have to do this.*0210

*But it is a little bit easier if you arrange your numbers in order from least to greatest.*0214

*Let's just do that; my smallest number here is 11.*0218

*Then it is 12; then 15; 18; let's see; then my twenties so it is 22; 25; 29.*0227

*Next is 30; 32; I have another 32; 33; and 39.*0260

*I have one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve.*0276

*One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve.*0281

*I have all my numbers.*0285

*Now my stem, the 1, that is a tens digit; here is a tens digit.*0287

*Then my 1, my leaf here is going to be a 1.*0297

*This number here represents tens; ones; together, it becomes 11.*0302

*My next one, because it is the same stem, I am going to write it under here, the same stem.*0307

*The leaf here is 2 like that.*0312

*You don't have to write commas; just leave a little space; 1, 2.*0316

*Next is 5; the next leaf under the same stem is 8.*0320

*That is it for my ones, my tens digit right here.*0330

*Next is the 2, the 20s; that leaf is 2.*0336

*The next leaf is 5; next leaf is 9.*0344

*Moving on to the thirties, 3-0; 3-2; 3- another 2; 3-3; and 3-9.*0352

*This is the stem and leaf plot that represents all of these numbers here.*0369

*It represents my data; then I can see... what is my smallest number?*0372

*My smallest number with the smallest stem and the smallest leaf together is going to be 11.*0378

*My biggest number is going to be the biggest stem with the biggest leaf together is 39.*0385

*To find the median here, it doesn't ask me to find the median.*0392

*But I can find the median by eliminating because each leaf represents a number in the data.*0395

*I can just go ahead and eliminate starting from this.*0402

*This is the start; this is the last; the first and the last.*0405

*I can just start eliminating until I get to a middle number, till I get to a middle leaf.*0409

*Then I am going to use that stem and that leaf to create the median.*0415

*We are going to do that in the next example, example three.*0420

*Here we have a stem and leaf plot.*0427

*We have to list all the numbers in order from least to greatest.*0430

*It is like what we just did, except we are just going to do the opposite now.*0434

*Instead of creating the stem and leaf plot with our data,*0437

*we are going to use this to come up with our data, our list of numbers.*0441

*The smallest number here, this stem with this leaf.*0447

*This is the smallest leaf with the smallest stem, together makes 40.*0451

*That is my smallest number, 40.*0458

*My next is going to be 44; then 52; 55; 57; 58.*0460

*Here this is listed out because all the stems from 4 to 7 have to be listed out.*0475

*I don't have any leaves here with this stem.*0482

*There is no number here that is in my data.*0487

*I don't have sixties number.*0491

*Then I just move on to 71; 73; and then 73.*0495

*Notice, back to this 40, if I have a number 40, then I have to list out the 0 as my leaf.*0505

*Here be careful not to represent this as 60 because there is no 0 here.*0516

*Just because there is nothing there doesn't mean that you have a 60.*0524

*There has to be a 0 in order for you to have a 60 in your data.*0527

*This is it; these are all my numbers in this stem and leaf plot.*0532

*For this one, some chapter test scores are given.*0542

*Create a stem and leaf plot of the data.*0544

*Then find the median and the mode.*0546

*Again the first thing I want to do is list out my numbers in order from least to greatest.*0552

*These are all chapter tests; the lowest score is 40.*0556

*Then let's see, 48; 52; 67; 72; 83; and then... oh, I forgot 71.*0565

*Let me go back to here, 72, 83; 71, 72, and then 83.*0598

*After 83, 90... I forgot 79; back to 79.*0609

*Where was I?--83; after 83 is 90; 91; 98; and 100.*0624

*I have one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve.*0643

*One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve.*0648

*I have all my numbers in order from least to greatest.*0652

*Now let's create our stem and leaf plot.*0657

*My stems and my leaf, going to separate like that; going to use black for this.*0660

*My smallest stem is a 4.*0671

*My biggest stem is going to be this number here.*0677

*Remember it is the numbers on the left.*0685

*My ones digit is going to be my leaf.*0687

*The remaining numbers to the left are going to be my stem.*0693

*It is going to be my stem.*0696

*Here I am going to separate this like that.*0698

*10 is going to be my stem; the 0 is going to be my leaf.*0700

*It is OK if you have more than one digit.*0704

*If you have two or three digits as your stem, that is fine.*0706

*4, 5, 6, 7, 8, 9, and 10; the leaf here is 0 and then 8.*0710

*For the next one, the stem is 5; the leaf is 2.*0727

*That is it for that.*0733

*Stem is 6; leaf is 7; here, stem 7, leaf 1; and 2 and 3. *0735

*This one's leaf is 3; for the nineties, 9-0; 9-1; 9-8.*0751

*For the last one, 10-0.*0762

*Those are all of the numbers represented in the stem and leaf plot.*0766

*I need to find my median and my mode; here let's see.*0773

*My median, because I know that the smallest stem with the smallest leaf is my smallest number in the data,*0784

*and then my biggest stem with the biggest... this right here, the last leaf is going to be my biggest number,*0792

*I know that I can start eliminating numbers from this right here and from the first and the last.*0801

*If I keep doing that, eliminate this and eliminate that.*0813

*The next one over is 48; the next one down from this is 98.*0816

*Then this; then this; then this; then this; then 71; then 83.*0822

*Then I have two numbers left in my data.*0832

*The whole point of doing this right now is to find the median.*0838

*I am trying to find the middle number.*0841

*Median is always middle; that is the keyword.*0842

*It has to be from order of least to greatest.*0846

*You can look for the median from here because we already listed them out.*0849

*But also be able to find the median from your stem and leaf plot.*0853

*This is the start; small stem with the first leaf all the way down to the biggest stem with the last leaf.*0859

*Start eliminating numbers until you get the middle; middle one or two middles.*0867

*In this case, because we have an even number of numbers,*0873

*we are going to have two numbers in the middle.*0876

*To find the median, we have to find the average of those two numbers.*0880

*For the median, it is going to be 2 plus the 9.*0885

*We are going to find the middle number between 2 and 9.*0893

*We are going to find the average or the mean between 2 and 9.*0896

*I am going to add them together and divided by 2.*0900

*2 plus 9 is 11; over 2.*0906

*To change this to a decimal, you can change it to a fraction or a decimal.*0912

*Remember that; 2, to change it to a mixed number...*0916

*This is called an improper fraction where the top number is bigger than the bottom number.*0920

*Let's change this to a mixed number.*0926

*2 fits into 11 how many times?*0928

*2 fits into 11 five times because that is going to make it a 10.*0931

*It only fits into it five times; that is our whole number.*0937

*How many leftovers do you have?--I have 1; over... keep that same denominator.*0940

*It is going to be 5 and 1/2.*0946

*To change this to a decimal, you are going to take the top number.*0949

*Place it inside like that; 2 on the outside.*0954

*You are going to put the decimal point at the end of that number.*0959

*Bring it up; 2 fits into 11 five times.*0963

*That gives us 10; subtract it; I get 1.*0968

*I am going to add a 0 at the end of this; bring that down.*0972

*2 goes into 10 five times; that gives us 10; subtract it; I get 0.*0976

*Your median is going to be either 5 and 1/2 as a fraction or 5.5 as a decimal.*0984

*It is just 5 and 1/2; that is the middle number between these two.*0992

*That is our median.*0996

*The mode, now mode, let's see.*0998

*Do we see any leaves written out twice for the same stem?*1002

*No, they are all different; I don't have a mode for this one.*1009

*Let's say for example, if I have another 8 right here,*1015

*then I know within the same stem, I have the same leaf written twice.*1019

*That would mean that I would have 98 and 98 again.*1024

*That would be my mode because remember the keyword for mode is most.*1028

*We are looking for any repeats, any numbers that occur more than one time.*1032

*Since I don't have any numbers that are repeating,*1042

*any numbers that are occurring more than once, I have no mode.*1045

*For mode, I am going to write none.*1050

*That is it for this lesson; thank you for watching Educator.com.*1059

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over the coordinate plane.*0002

*This right here is called the coordinate plane; it consists of two lines like this.*0008

*This right here, the horizontal line, is called the x-axis.*0016

*The vertical line is the y-axis; together they make up the coordinate plane.*0024

*Think of the coordinate plane as like a map.*0034

*It is a map; we are going to place points along this map.*0037

*We have to label each point based on the x-axis and the y-axis.*0042

*That is the coordinate plane.*0048

*These axes, the x-axis and the y-axis, break the coordinate plane into four sections.*0052

*One, two, three, four, those four sections are called quadrants.*0064

*The first quadrant is this space right here; this is called quadrant one.*0074

*Going over to this side, this is quadrant two; then down, this is quadrant three.*0086

*The last one is quadrant four; those are the four sections of the coordinate plane.*0097

*Each axis is numbered; it is labelled.*0108

*Here, this point right here is going to start off at 0.*0112

*It is going to go 1, 2, 3, 4, 5, and so on; 1, 2, 3, 4, 5.*0116

*Again this is 0; this way, then it is going to go negative.*0126

*See how it goes positive this way.*0133

*If you go the other way, it is going to go negative; -1, -2, -3, -4, -5, and so on.*0134

*Same thing for the y-axis.*0143

*If you go upwards, it is going to go positive.*0146

*It is going to go 1, 2, 3, 4, 5.*0151

*The arrows represent continuation; it is going to keep going on forever.*0158

*It doesn't just stop at 5; it is going to keep going.*0165

*Down this way, it is going to be negative; -1, -2, -3, -4, -5.*0168

*Next right here, a number on the x-axis paired with the number on the y-axis is going to give us an ordered pair.*0183

*You are going to look for the number on the x and the number on the y.*0198

*Together it is going to map out your location on the coordinate plane.*0202

*That point is called an ordered pair; it is going to be written like this.*0209

*In parentheses, you are going to have two numbers.*0216

*The first number is going to be your x number; that is called the x-coordinate.*0219

*The second number is going to be your y coordinate.*0231

*It is always going to be x and then y.*0239

*X always has to come first; (x, y) for short.*0242

*(x, y), that is going to be your ordered pair.*0247

*That is going to map out your location on the coordinate plane.*0252

*This point where the x-axis and y-axis meet, that is called the origin.*0259

*That is right there.*0267

*If you look at the x-number, the x-number, it is 0.*0269

*The x-number is 0; the y-number is also 0.*0275

*The origin is (0, 0); that is the ordered pair for the origin.*0280

*If I put a point right here on the coordinate plane, that ordered pair, what is the x-number?*0287

*If you look, this point is paired with an x-number and a y-number.*0301

*This point is going to have a x-number of 3 and a y-number of 1.*0309

*The ordered pair for this point is (3, 1).*0321

*Be careful that you don't mix those up, switch them up.*0324

*It is not (1, 3); that is not the same thing.*0328

*(1, 3), if I say (1, 3), because I said 1 first, that means I am talking about 1 on the x-axis.*0331

*(1, 3) would be 1 on the x; 3 on the y.*0341

*Where do they meet?--right there, that would be (1, 3).*0348

*(1, 3) and (3, 1) are not the same thing; be careful with that.*0352

*Always make sure the first number is your x-coordinate on the x-axis, the horizontal line.*0356

*The second number is your y-axis, the number on your vertical line.*0363

*That is the coordinate plane.*0369

*Again we have the x-axis, the y-axis; each of the four sections are called quadrants.*0370

*It starts off with this right here, quadrant one, quadrant two, quadrant, three, and quadrant four.*0379

*Remember when you are writing out the numbers, this number is always 0.*0385

*It is going to go positive to the right and negative to the left.*0392

*On the y-axis, positive when you go up, negative when you go down.*0396

*Everything together, the coordinate plane which is like the map, the ordered pairs, the quadrants,*0402

*all of this that have to do with these points, the plane, all that is called the coordinate system.*0408

*It is like a system of all these things together.*0418

*Again a coordinate plane, each of these are just marked for you.*0425

*We have three points, A, B, and C.*0431

*Here is A; here is B; here is C; we want to write the coordinates.*0435

*In other words, we want to know the ordered pairs.*0439

*Find the ordered pairs of A, B, and C.*0442

*We know that this is the x and that is the y.*0446

*Let's start with A, this point right here.*0451

*What is the x-number that makes up this point?--1.*0455

*What is the y-number?--1.*0459

*For A, my x-number, the number on my x-axis, is 1.*0461

*The number on the y-axis is 1.*0468

*The ordered pair or the coordinates for A is (1, 1).*0473

*I am just going to write... because this is the name of the point.*0480

*A is what that point is labelled.*0483

*You can just write it in front of that ordered pair.*0486

*Next is B; again I am looking for the x-number first.*0491

*The x-number, the number on this x-axis that makes up this B is -1.*0497

*The y-number, -2.*0505

*They are negative numbers; but it is fine.*0513

*You are just listing them out like that, same way as when you have positive numbers, (-1, -2).*0515

*The next one is C; the x-number for C is 2.*0525

*The y-number for C is -1.*0533

*Those are the coordinate for A, B, and C.*0540

*Let's do our examples now; graph each point on the coordinate plane.*0543

*The first one, A is (4, 2).*0549

*Remember this is the x-number; this is the y-number.*0554

*I am going to label this as x; this as y.*0559

*4, you are going to look for 4 on the x-axis.*0564

*4; this is -4; here is 4.*0567

*Then look for 2 on the y-axis; here is 2.*0572

*Where do they meet?--right there.*0575

*This is A; I can label that point as A.*0581

*Next, B is (-3, 0); the x is -3; the y is 0.*0586

*-3 on the x is right here; 0 on the y.*0596

*Where is 0?--0 is right there.*0602

*(-3, 0), that means I am not moving any up or any down because there is no y.*0606

*If the y was 1, I would have to move up 1.*0611

*If y was -1, I would have to move down 1.*0614

*But y is 0 so I stay here; that is my B.*0617

*Next, x, y; 2 is x; y is 1.*0629

*Where do they meet?--right there; that is C.*0636

*Next is (-4, -2).*0643

*-4 on the x is right there; -2 on the y is right there.*0647

*Right there, they intersect at that point right there; that point is called D.*0654

*That is it for this example.*0663

*The next one, write the coordinates and quadrant for each point.*0667

*It is like the same thing that we just did a couple examples ago.*0672

*First let's start with A; remember this is my x, this is my y.*0676

*Look for the x that makes up this point right here.*0682

*A is -1; that is my x; y is a -3.*0688

*We also have to state what quadrant it is in.*0698

*Remember that this right here, this section, any points that fall in this section right here is quadrant one.*0702

*This section is quadrant two; quadrant three; and quadrant four.*0709

*If a point is on let's say right here, that doesn't fall under any quadrants.*0720

*That wouldn't be considered quadrant one or quadrant four.*0727

*In that case, you are just going to say that it is on the x-axis.*0729

*Or in this case, if a point lands right here.*0734

*Then it is not in quadrant three or four; it is actually on the y-axis.*0739

*This quadrant, point A is in quadrant three; let's just say quadrant three.*0745

*B, point B is... x is 1, y is 2; (1, 2); that is in quadrant one.*0753

*C is right there; 3 as my x; -1 as my y.*0768

*That is in quadrant four.*0776

*D, x is -3; y is 3; that is quadrant two.*0780

*Example three, name two points from each of the four quadrants.*0795

*It is a little bit hard to do when you don't have the coordinate plane.*0801

*Let's draw out a coordinate plane.*0805

*We don't have to draw it too big because we are not going to be plotting any points.*0812

*Here is the x; here is the y.*0817

*I know this is quadrant one; quadrant two; quadrant three; and quadrant four.*0819

*The trick here is to figure out whether my x-coordinates are going to be positive or negative*0830

*and my y-coordinates are going to be positive or negative.*0840

*Quadrant one, if you look here, all the points that have this as my x and this as my y,*0843

*any points that pair up with this part of my x, that part of my y, is going to be in quadrant one.*0852

*I know that here as I get to these numbers, it is going to be positive.*0861

*This is +1, +2, +3, so on.*0866

*My x-coordinate is going to be positive; it is a positive number; positive.*0870

*My y-coordinate is also going to be positive because these are positive numbers here; positive, positive.*0878

*Any ordered pair that has a positive number paired with a positive y-number is going to be in quadrant one.*0888

*For quadrant one, I have to name two points.*0898

*Any ordered pair with a positive x, positive y, is in quadrant one.*0904

*(1, 2), that is in quadrant one; what else?*0911

*(3, 4), that is also in quadrant one.*0916

*Look at quadrant two; quadrant two, all points with this part of my x.*0920

*Those are my negative numbers.*0931

*A negative x with this part of my y is going to be in quadrant two.*0934

*Then when I list my points in quadrant two,*0944

*it is going to be a negative x number with a positive y number.*0948

*Does that make sense?--it is negative.*0955

*All these numbers are negative here.*0957

*This is -1, -2, -3; it has to be a negative x-coordinate.*0961

*Then paired with a positive y coordinate is going to be in quadrant two.*0968

*(-1, +2) is right there; it is in quadrant two.*0976

*Again negative x, positive y; (-3, 4).*0984

*Quadrant three, again any numbers or any ordered pairs paired with a ?x with this part of the y-axis.*0995

*That is negative also; negative and a negative is going to be in quadrant three.*1007

*This is going to be a negative x, negative y.*1014

*(-1, -2) here is -1; here is -2.*1020

*Another one, (-3, -4).*1026

*For quadrant four, again positive x; how about the y?*1033

*It is going to be any ordered pairs paired with this side and this side.*1041

*Positive x, negative y; positive x, 1; negative y, -2.*1045

*Positive x, 3; negative y, -4.*1056

*Those two points are going to fall under quadrant four.*1061

*The fourth example, we are just going to graph more points on this coordinate plane.*1069

*I didn't number them; go ahead and number them right now.*1075

*1, 2, 3, 4, let's go up to 5; -1, -2, -3, -4, -5; 1, 2, 3, 4, 5.*1078

*The first point, A; I know this is going to be my x-number; this my y-number.*1098

*0 on my x-number is right here; on this right here, 0.*1106

*3 on my y-number is right here.*1113

*That means my y is going to be 3 and my x is going to be 0.*1116

*Meaning I am not going to have -1.*1121

*I am not going to have 1 as my x; I am going to have 0.*1124

*It is going to be that point right there; this is A.*1126

*The next one is (-2, -1); -2 on my x; -1 on my y.*1132

*They meet right there; here is B.*1140

*C is going to be -5 and then 0 as my y.*1147

*Remember if my y is 1, I go up 1.*1154

*If my y is -1, I go down 1; right there.*1156

*But it is 0; my y is 0; that means I don't move up or down.*1161

*I stay put; that is my C.*1164

*My last point D is going to be 4 and -6.*1168

*-6 on my y-axis is right here; they meet right there.*1177

*That point is labelled D.*1187

*That is it for this lesson; thank you for watching Educator.com.*1195

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over organizing possible outcomes using compound events.*0002

*A compound event is when you have two or more different events that will affect the possible outcome.*0010

*When you have different options, that is going to affect the actual outcome.*0018

*An example is say we want to pair up the type of shirt*0027

*that we are going to wear with the type of pants we are going to wear.*0033

*My two events is going to be my top, my shirt, with my bottom.*0037

*Let's say my options for my shirt.*0050

*Say I am debating on whether I should wear a t-shirt or a collar shirt.*0053

*For the bottom, let's say I am either going to pair it up with a skirt or say pants or shorts.*0065

*My two events are the shirt and the bottom.*0082

*You are going to pair them up.*0089

*The different pairs are going to create the different possible outcomes.*0091

*When you list out all the possible outcomes, that is here.*0096

*That is what you are doing; you are creating the organized list.*0101

*If I say t-shirt with skirt, t-shirt with pants, t-shirt with shorts.*0105

*If I list them all out, that is a way for you to just list out all the possible outcomes using a list.*0110

*The next one, drawing the tree diagram.*0120

*When you draw the tree diagram, you are going to first list out the first event.*0123

*That would be the shirt.*0128

*You are going to do t-shirt with the collar.*0132

*Those are your two options for the first event.*0142

*Then you are going to branch out from the t-shirt.*0148

*How many options do I have to pair up with this?*0152

*I have three; you are going to do one, two, three.*0154

*You are going to write the skirt here; pants; and shorts.*0160

*Same thing with this.*0169

*It is going to be the skirt with the pants with the shorts.*0172

*Skirt, pants, and then shorts.*0180

*The t-shirt with the skirt is one option.*0185

*T-shirt with the pants; and then t-shirt with the shorts.*0192

*Then collar shirt with the skirt; collar shirt with the pants.*0198

*And then collar shirt with the shorts; would be your other options.*0203

*Total, we have six possible outcomes; that is the tree diagram.*0207

*The fundamental counting principle is when you just take all the possible outcomes for this first event M*0217

*and then all the options for your second event N and you multiply them together, M times N.*0229

*You are going to get the total possible number of outcomes.*0241

*M, let's say M in our example is the type of shirt that we have, the options.*0245

*We have two; M is 2; that is the first event; M is 2.*0252

*Our second event is type of skirts, pants, bottoms, whatever we are going to wear.*0259

*That is N; how many options do we have there?*0265

*We have three; N is 3.*0267

*We multiply them together; we are going to get 6.*0271

*I know that I have 6 possible outcomes total.*0277

*That is the fundamental counting principle.*0282

*Let's do some more examples.*0287

*The Jackson family plans to travel in July or August to San Francisco, San Jose, or San Diego.*0290

*Create a list of all the possible outcomes.*0297

*We have two events. The first event is going to be when the family is going to travel, July or August.*0300

*That is the first event.*0307

*The second event that is going to affect the outcome will be the place; where.*0309

*Let's see... I am just going to create a list of all the possible outcomes.*0318

*My possible outcomes can be in July to where?--San Francisco.*0324

*July to San Jose; and then July to San Diego.*0336

*It is easiest when you have to list them out, to list out the first event*0350

*and then the different possible places in your second event.*0356

*See how this is the first option for event one.*0361

*The second one will be August to San Francisco.*0367

*August to San Jose; and then August to San Diego.*0375

*We have six possible outcomes.*0389

*For this example, Samantha cannot decide if she is going to order*0396

*chicken sandwich, turkey sandwich, or a club sandwich on either white or wheat bread.*0403

*Create a tree diagram that lists out all the possible outcomes.*0408

*The first event, the first option, is the type of sandwich.*0412

*The second event is going to be the type of bread.*0426

*The first part of doing this is to figure out your events, the different things,*0435

*the different events that are going to affect your outcome, the sandwich and then the bread.*0443

*For the first event, I am going to list it out.*0449

*Again I am drawing the tree diagram.*0451

*The first event is going to be between chicken...*0453

*Give yourself some space between each... turkey, and the club.*0460

*Then I am going to branch out from chicken.*0472

*What are the possible types of bread on the chicken?*0479

*It is going to be white or wheat; for the turkey, white, wheat.*0483

*On the club, we can get the club with white or wheat.*0498

*The chicken with the white, that is one option.*0506

*I can just say this is the white; the chicken with the wheat.*0515

*The turkey with the white; turkey with the wheat.*0531

*Club with the white; and club with the wheat.*0540

*See how you went from chicken to white, chicken to wheat,*0550

*turkey to white, turkey to wheat, club to white, and club to wheat.*0554

*Those are all of your possible outcomes here.*0558

*You can also make your bread your first event and then your sandwich the second event.*0565

*You would just do white and wheat; you list those two out.*0572

*Then you would branch out to the three options for the sandwich.*0578

*It works either way.*0583

*It doesn't matter which one you label as your first event and your second event.*0584

*As long as you make sure that you are going to pair up*0587

*each of the first events with each option for the second event.*0590

*You have six different options.*0596

*The next one, draw a tree diagram showing the possible outcomes*0600

*for the choice of vanilla, strawberry, chocolate yogurt in a small, medium, or large cup.*0604

*First event is going to be the yogurt.*0612

*The second event is going to be the size, type of cup.*0620

*Yogurt is going to be either vanilla, strawberry, or chocolate.*0628

*You can get the vanilla yogurt in small, medium, or large.*0646

*You can get the strawberry in small medium or large.*0657

*You can order the chocolate yogurt in small, medium, or large.*0664

*The different possible outcomes is going to be vanilla to the small; small vanilla.*0671

*Or you can just do vanilla small; vanilla medium; vanilla yogurt in a large cup.*0678

*Or strawberry small; strawberry medium; strawberry large.*0687

*Then chocolate small; chocolate medium; and chocolate large.*0697

*These will be your possible outcomes.*0702

*We have one, two, three, four, five, six, seven, eight, nine.*0705

*Remember the fundamental counting principle.*0713

*We have M as our first event and N as your second event.*0716

*If this is M, this is N, how many options do we have for our M?*0725

*We have three different options for the yogurt.*0729

*If we were to do M times N, we have three options for our yogurt, that is 3.*0734

*Times how many options do we have for the size?*0740

*Small, medium, large; we have 3; N is 3.*0743

*To find the total possible number of outcomes, it is going to be 9.*0748

*3 times 3; 9; we have all 9 here.*0753

*Three, four, five, six, seven, eight, and nine.*0756

*We are going to use that fundamental counting principle again for this one*0763

*to find the total possible outcomes for rolling a number cube three times.*0767

*We have three different events.*0775

*Just the three different times you are going to be rolling the number cube.*0778

*Let's say a number cube... we are going to say first.*0786

*Because it is going to be rolled three times.*0791

*The first time, the second time, and the third time.*0792

*The first time we roll it, how many different options are there?*0800

*How many different possible outcomes for just that first time you roll the number cube is going to be 6*0805

*because the number cube has 6 sides and each side has a different number.*0812

*We have 6 different possible numbers that can show up within our first roll.*0817

*Within the second roll, how many options do we have there?--we also have 6.*0825

*Then for the third, we also have another 6 because there is 6 different numbers.*0834

*To find the possible number of outcomes, we know that we have to do M times N.*0842

*That is if you have two events.*0852

*In this case, we have three events; we just multiply all three together.*0854

*We can just label this as M, this as N, and the third one whatever you want, P.*0859

*We are going to do times P.*0867

*That is going to be 6 times the 6; 6 times 6 is 36.*0871

*Then we are going to multiply this by 6.*0880

*This is 36; 6 times 3 is 18; 21.*0888

*There are 216 different possible outcomes when you roll the number cube three times.*0895

*Just to list out a couple, the first time you roll it, you can roll a 2.*0907

*The second time you roll it, you can roll a 1.*0913

*The third time you roll it, you can roll let's say a 1.*0916

*That is just one of the 216 different possible outcomes; my answer is 216.*0921

*That is it for this lesson; thank you for watching Educator.com.*0933

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over probability of compound events*0002

*and those events being independent and dependent.*0007

*Before we go over these events, let's first review over probability.*0012

*Probability is talking about the chances of something happening.*0017

*What is the probability of picking a card from a deck?*0024

*Or what is the probability of rolling a 2 if you roll a die?*0030

*Probability talks about your chances of something occurring.*0035

*To find probability, we are looking at a ratio; a ratio is like a fraction.*0040

*It is comparing the top number with the bottom number.*0055

*Probability is talking about your desired outcome or the outcome that you are looking for*0058

*over the total possible number of outcomes or just total for short.*0072

*It is desired outcome over the total--is the probability.*0081

*Probability, once you have this, you can leave it as a fraction.*0085

*You can change it to a decimal; it is just fraction to decimal.*0089

*You just divide the top number with the bottom number.*0094

*Or you can change it to a percent.*0098

*Usually probability is left as a fraction, desired outcome over the total.*0101

*If I have a die, a number cube... we are going to talk about number cubes in our examples later.*0112

*A number cube has different numbers of dots on each side.*0122

*There is 6 sides; each side has a different number, 1 through 6.*0131

*If I said what is the probability, what are the chances of rolling a 1?*0135

*My desired outcome then, I am going to say my probability of rolling a 1.*0142

*That is how we write it out, probability of rolling a 1.*0148

*That is my desired outcome; how many sides has a 1?--only 1 side.*0154

*My desired outcome, there is only 1; over... how many total sides are there?*0162

*How many total possible outcomes are there?*0170

*There is 6 sides; my total is 6.*0173

*The probability of rolling a 1 is 1/6.*0176

*Same thing if I said probability of rolling a 3.*0181

*Be careful, this number is not going to be the number on top.*0186

*How many sides have a 3?--only 1 side.*0190

*My desired outcome would be just 1 because there is only 1 possible side that is going to be a 3.*0196

*It is 1 over... how many total sides are there?--6.*0205

*The chances of rolling a 1 is the same as the chance of rolling a 3.*0210

*That is probability.*0214

*When we talk about two events, each of this, this is one event.*0218

*You are rolling a 1; this is another event, rolling a 3.*0224

*Each of those are events.*0228

*We are talking about when we have two events or two or more events.*0231

*When we have two events, when there is two things going on,*0236

*those two events can be either independent events or dependent events.*0240

*That is what we are talking about now.*0246

*Independent events is when the outcome of the second event does not depend on the outcome of the first event.*0248

*Think about what the word independent means.*0256

*It doesn't depend on it; it is not affected by anyone else, anything else.*0259

*The first event happens; the second event happens; they don't affect each other.*0266

*When you have two events that are independent, then we write each of those events as A and B.*0273

*It is probability of A, that is the first event.*0280

*The probability of B, that is the second event.*0285

*When we have two events that are independent, all we have to do is multiply*0289

*the probability of that first event with the probability of the second event.*0294

*Let's say I have a bag of marbles.*0301

*In this bag, I have 1, 2, 3 red marbles.*0309

*Let's say I have 2, 3, 4 blue marbles.*0316

*And let's say I have 2 green; I don't have green.*0323

*So I am going to just use black for green; I will put G for green.*0328

*3 red marbles, 4 blue, and 2 green marbles; I have a bag of marbles.*0333

*Just talking about one event, let's just say what is the probability of picking a red marble?*0339

*That is one event because I am going to pick up one marble.*0346

*The probability of picking a red, that red is my desired outcome.*0350

*That is what I am asking for.*0358

*How many reds do I have?--I have 3 reds.*0361

*That number, the desired outcome, is going to go on top.*0366

*3 over... how many total number of marbles do I have?*0369

*I have 1, 2, 3, 4, 5, 6, 7, 8, 9.*0375

*If you have a fraction, you always have to simplify.*0381

*I can simplify this by 3; divide each by 3.*0384

*It is going to become 1/3; the probability of picking a red marble is 1/3.*0390

*That is only when you have one event.*0399

*Talking about compound events, two events, if I ask for*0402

*the probability of picking a red and afterwards picking a blue...*0408

*Probability of picking a red, we already found that; that is 1/3.*0420

*It is going to be probability of picking red times the probability of picking the blue.*0425

*The only way both of these events, picking the red and then picking another marble the blue,*0440

*the only way these two events are going to be independent is if after we pick the red marble,*0447

*after this first event, after you pick the first marble, you have to place it back into the bag.*0455

*You are going to pick one out; put it back in.*0461

*Then pick the second one.*0464

*It will be independent because then picking this red or picking this one, it won't affect this one.*0466

*Probability of picking a red marble, we know that is 3/9 or 1/3.*0474

*Times probability of picking a blue.*0484

*Blue is my desired outcome; I have 4.*0489

*Over a total number of 4, 5, 6, 7, 8, 9.*0494

*Again after you pick the first marble, we put it back in the bag.*0503

*Now it is just original number of marbles,*0510

*the same number of marbles when we picked the blue one, when we picked the second one.*0515

*This is 1 times 4 is 4; over... 3 times 9 is 27.*0519

*This can't be simplified; this is our answer.*0527

*The probability of picking a red and then after replacing it, picking a blue, would be 4/27.*0533

*This is independent events.*0545

*When we have two events and the second outcome is affected*0553

*by the first outcome, then we have dependent events.*0560

*The second event depends on the first event.*0567

*Finding the probability of two dependent events is a little bit different.*0573

*Same thing here; when we have probability of the first event A*0581

*and then the probability of the second event B, we are still going to multiply them.*0586

*It will be the probability of A times the probability of B after A because remember this second event is affected.*0594

*It depends on the first event A.*0605

*Back to the bag of marbles; again 3 red, 4 blue, and 2 green marbles.*0612

*Same bag of marbles; but now the way it becomes dependent events.*0631

*I want to find the probability of picking a red and then my second event will be picking a blue.*0637

*But the difference is here after we pick the first marble, after we find the probability of picking a red,*0648

*we are not going to put the marble back in the bag.*0658

*We are going to take it out; we are going to leave it out.*0661

*Then the second event, the probability of picking a blue, is going to be*0666

*slightly different because the total number of marbles is different.*0670

*There is less marbles; that is why these would be dependent events.*0675

*Because the probability of picking a blue is not going to be*0681

*the same as if we were to place the marble back in.*0685

*This will be probability of red times probability of blue.*0692

*Again we are not going to replace it back in.*0699

*The probability of picking a red, how many reds do I have?*0703

*My desired outcome is 3; desired outcome goes on top.*0709

*3 over... total number of marbles is 9; you can simplify this to become 1/3.*0715

*Let's say that... let me just change this to 1/3.*0729

*Because this red is no longer there, we took it out.*0738

*That is the first event.*0743

*For the second event, since this marble was not replaced back in, it is left out.*0745

*This is going to be different.*0755

*Probability of picking a blue, my desired outcome is number of blue.*0756

*How many blues do I have?--4.*0760

*My total number of marbles is going to be different.*0765

*It is going to be 1, 2, 3, 4, 5, 6, 7, 8.*0768

*It is going to be 1 less than the total here.*0772

*This was originally 9 before we simplified.*0775

*Now it is going to be 8; there is 1 less marble in the bag.*0779

*Now we multiply these numbers.*0784

*It is going to be... 1 times 4 is 4; over... 3 times 8 is 24.*0787

*This can be simplified; 4 goes into both numbers.*0795

*Divide each number by 4; this is 1; this is 6.*0799

*The probability of picking a red and then picking a second marble blue without replacing marbles is going to be 1/6.*0806

*Let's go over some examples.*0824

*Determine if the two events are independent or dependent events.*0825

*The first one, rolling a number cube twice.*0831

*Remember for independent or dependent events, we have to have two events; at least two.*0835

*Here rolling a number cube twice.*0843

*The first event would be the first time you roll the number cube.*0846

*The second event is going to be the second time you roll the number cube.*0850

*Does the second event depend on the outcome of the first event?*0857

*Meaning if we roll a number cube, if we roll a die,*0865

*we get either 1 through 6, a number from 1 to 6.*0870

*If you roll it again the second time, does it change or is it affected?*0876

*If I roll a 2 the first time, does that mean I can't roll a 2 the second time?*0883

*The first time we roll a number cube, all the numbers...*0892

*let's say I want to find the probability of picking a 5.*0896

*How many 5s are there?--how many sides on the number cube is a 5?*0899

*There is only 1 side; it will be 1/6.*0904

*That would be the probability of my first roll.*0909

*Then for my second roll, what is the probability of picking a 5 or picking any number?*0914

*It is also 1; do the number of sides change?--no, still the same.*0922

*This roll and this roll, my second roll, they don't affect each other.*0930

*They have nothing to do with each other.*0936

*In this case, this would be independent.*0938

*The second one, drawing a card from a deck of cards and without replacing it, drawing another card.*0946

*There are 52 cards in a deck.*0954

*If I pull a number out or take a card, my total number of cards is going to be 52.*0958

*If I don't put it back in, then for my second draw,*0969

*when I draw my second card, my total is going to be different.*0976

*My probability will be different because it is always desired outcome over the total.*0981

*For my second draw, there is less cards in the deck.*0986

*In this case, this would be dependent because the second draw depends on the first draw.*0991

*The outcome of the second is affected by the first; dependent.*1003

*Picking two students in your class to be class representatives.*1015

*Imagine your class; there is let's say 30 students in the class.*1022

*You pick the first student.*1031

*Let's say you are picking the president and vice-president as class representatives.*1036

*If you pick the first student to be your president,*1041

*how many students do you have left to pick from when you pick the vice-president?*1045

*The total number of students, does it change?*1052

*It does change because you already picked one student and that same person can't be both.*1055

*You pick one student to be the president of your class.*1060

*Then for the vice-president, you have one less student to pick from.*1066

*You have 29 students; so this is dependent; this is dependent.*1074

*These two events would be considered dependent events.*1085

*Samantha rolls a number cube twice; find the probability of each pair of events.*1094

*Here rolling twice, two events, this is the first event; this is the second event.*1100

*We want to know the probability of rolling a 2 and then probability of rolling a 5 afterwards.*1108

*Probability of picking a 2; a number cube... let's say 1 here; 2 here; let's say 5 here.*1120

*There is 6 total sides; how many sides have 2?*1148

*Only 1 side does; my desired outcome is the 2.*1154

*But how many 2s are there?--only 1.*1159

*It is 1 out of a total 6.*1161

*What is the probability for my second roll, for my second event, probability of rolling a 5?*1169

*Again there is only 1 side with a 5.*1174

*1 over... still number of sides is the same, 6.*1177

*The probability of rolling both of those, I just have to multiply*1185

*probability of 2 times the probability of 5 occurring.*1189

*It is going to be 1/6 times 1/6.*1195

*1 times 1; 6 times 6 is 36.*1202

*The probability of rolling a 2 and then rolling a 5 afterwards is 1/36.*1207

*This one here, the probability of rolling a number that is not a 3.*1216

*Probability of not 3; that is my desired outcome; not 3.*1225

*How many numbers are not 3?*1236

*We have 6 of them; only 1 is a 3; the rest aren't.*1240

*There is 5 sides that are not 3; that is going to be 5 on the top.*1245

*Over... how many do I have total?--6.*1252

*That is my first roll; that is my first event.*1260

*My second event, my second roll, is probability of rolling a 6.*1262

*Again there is only 1 side with a 6; that is 1/6.*1267

*Probability of the first one times the probability of the second one.*1274

*Probability of that is 5/6 times probability of the second one 1/6.*1281

*5 times 1 is 5; over 36; that can't be simplified; that is my answer.*1288

*Here we have a spinner that we are going to use to find the probability of each.*1304

*The first one, I only have one event, only 1 spin.*1310

*I am looking for the probability of rolling a black; there are no blacks.*1317

*It is red, orange, yellow, green, blue, purple, light purple, and then another orange.*1324

*Probability of rolling a black, my desired outcome is black.*1332

*Do I have any black?--no; 0 on top.*1337

*Over... how many total do I have?--1, 2, 3, 4, 5, 6, 7, 8; over 8.*1343

*This 0/8 is always 0.*1354

*If have a 0 on top, that is going to make the whole thing 0.*1358

*Here there is no probability of picking a black; that is what the 0 means.*1364

*For the second one, we want to know the probability of spinning a red.*1372

*And then if we do a second spin, because there is two events...*1378

*First spin lands on red; second spin lands on green.*1384

*Probability of red; how many sections of red do I have?*1388

*I only have 1; 1 over... total number of sections, 8.*1397

*What about probability of green?*1406

*This would be considered independent events because if I spin the first time, I land on red.*1410

*That is not going to affect what my second spin is going to land on.*1420

*These would be independent; the probability of green is 1 out of 8.*1425

*I multiply them together; 1/8 times 1/8.*1433

*1 times 1 is 1; 8 times 8 is 64.*1440

*The probability of landing on red and then spinning again landing on green is 1/64.*1448

*The next one, the probability of landing on any color that is not yellow*1456

*and then for the second spin, landing on blue.*1464

*Probability of not yellow; how many are not yellow?*1468

*There is 8; there is only 1 yellow; 7 are not yellow; 7/8.*1475

*The probability of blue; there is 1 blue; 1/8.*1484

*We are going to multiply them together; 7/8 times 1/8 is 7/64.*1492

*If you notice these two numbers, the chances of this happening is greater than the chances of this happening*1505

*because here the chances of the spinner landing on a color that is not yellow is actually pretty high.*1519

*7/8, that is pretty high because there is so many spaces that are not yellow.*1529

*If this fraction is greater than this fraction, that means*1537

*the probability of this happening is greater than the chances of that happening.*1541

*For our fourth example, we have a bag of marbles; draw my bag of marbles.*1550

*I have 5 red marbles; 1, 2, 3, 4, 5.*1560

*I have 4 blue; 1, 2, 3, 4.*1566

*I have 6 green; I don't have green color.*1572

*I am going to use black G for green; 1, 2, 3, 4, 5, 6.*1575

*We are going to find the probability for each when the first marble is not replaced back in the bag.*1586

*Here we have two events; two things are happening.*1593

*We are going to pick two marbles.*1596

*After you pick the first marble, we are not going to put it back in the bag.*1601

*It is not going to be replaced back in.*1606

*We pick one; that one stays out of the bag.*1609

*Then we are going to pick our second marble; that is my two events.*1612

*Remember the second event, because after we pick the first marble, we are not going to replace it back in.*1618

*That is going to affect the probability of that second marble.*1626

*Both of these would be considered dependent events because the second one is affected by that first event.*1633

*Let's first talk about this event, the probability of picking a green.*1645

*That is our first pick, green.*1652

*Probability, we look at the desired outcome over the total possible number of outcomes.*1656

*How many green marbles do I have?--I have 1, 2, 3, 4, 5, 6.*1661

*6 is going to be my top number.*1670

*Over... how many marbles do I have total, in all?*1672

*1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15.*1676

*That is going to go right there; the probability of picking a green is 6/15.*1683

*It is still a fraction; I still have to simplify it.*1691

*What number goes into both 6 and 15?--they share a factor of 3.*1694

*This is going to be 2/5; the probability of picking a green is 2/5.*1702

*Now I have to pick my second marble; that is going to be red.*1710

*Because that first marble was not replaced back in the bag, this marble, one of the green, is now gone.*1719

*It is no longer in there.*1727

*Probability of picking the red, how many reds to I have?*1732

*I have 5 red marbles; 5 on top.*1735

*Over... how many marbles do I have now?*1739

*After that 1 is gone, I have 14 left.*1743

*It was 15 and then minus the 1 that we have already picked.*1748

*Now it is 5/14; this fraction I can't simplify.*1753

*That is the probability of picking the red.*1758

*Now to find the probability of both happening, I take this one and I take this one; I multiply it together.*1761

*It will be probability of the green times the probability of picking the red.*1769

*This is dependent; it is the red after green; 2/5 times 5/14.*1780

*2 times 5 is 10; 5 times 14; 14 times 5.*1792

*We do this; that is 20; 5 times 1 plus the 2 is 70.*1799

*Fraction, I have to simplify it; what number goes into both top and bottom?*1809

*10; I divide by 10 for each; I get 1/7 as my answer.*1814

*The probability of picking a green and then afterwards*1822

*without replacing it back in, picking a red marble, is 1/7.*1825

*I need to write this for this second problem.*1834

*Now we want to find the probability of picking a blue*1842

*and then afterwards without replacing it back in, pick another blue.*1845

*Again two dependent events.*1851

*Probability of picking the first blue, the first event, what is my desired outcome?*1855

*How many blue marbles do I have here in the bag?*1864

*I have 4 blue over a total of 15 marbles.*1866

*The probability of picking a blue marble is 4/15.*1874

*I can't simplify it; so that is the probability.*1878

*For my second pick, because again it is not being replaced in the bag.*1881

*This one is no longer in the bag; I have 1 less marble.*1891

*For my second pick, I want to look at how many blue marbles I have left.*1896

*I have 3 left; I had 4 but 1 is gone; now I have 3.*1904

*Over... I don't have 15 anymore; I have 14 now.*1913

*Probability of picking the first blue was 4 out of 15*1922

*because I had all my blue, just 4 of them, out of a total of 15 marbles.*1925

*For my second pick, I am also wanting to pick another blue one.*1933

*I only have 3 left because the first one wasn't replaced back in.*1937

*Out of a total of 14 marbles left.*1941

*Now I am going to take the first event and then*1945

*multiply it to the probability of the second event happening.*1951

*It is 4/15 times 3/14.*1956

*4 times 3 is 12; over... 15 times 14; you are going to multiply it.*1962

*This is 20; that is 4 times 1 is 4; plus 2 is 6.*1971

*I leave the space alone; 1 times 5 is 5; 1 times 1 is 1.*1978

*I am going to add them down; 0; this is 11; this is 2; 210.*1983

*I know I can simplify this fraction because this number is an even number and this number is an even number.*1998

*Let's divide each of these by 2; 12 divided by 2 is 6.*2007

*Over... this, if I take the 200 and I divide it by 2, I get 100.*2020

*This divided by 2 is 100; this divided by 2 is 5.*2032

*If I divide the whole thing, it will be 105.*2037

*It looks like 6/105; I also have another factor.*2046

*I can divide this again by... I know 3 goes into that one and 3 also goes into this one.*2051

*6 divided by 3 is... write it down here... 2.*2059

*Over... 105 divided by 3; let me show you that one.*2065

*3 goes into 10 three times; that gives you a 9.*2073

*Subtract it; I get 1 left over; bring down the 5.*2077

*3 goes into 15 five times; that gives you 15.*2082

*We subtract it; I have no remainders; my answer is 35.*2088

*Can I simplify this further?--no, I can't because this is not an even number.*2095

*This is my answer.*2100

*Again finding the probability of two events, you have to find the probability of each event occurring.*2102

*Then you are going to multiply them together, whether it is independent or dependent events.*2111

*That is it for this lesson; thank you for watching Educator.com.*2116

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over two events that are disjoint.*0002

*Two events are disjoint when those two events cannot occur at the same time.*0009

*It can't happen; it is not possible.*0017

*If we have two events A and B, then the probability or the chance that both are going to occur is 0.*0020

*That means there is no chance for them to occur together.*0030

*It is as if I look at my two events, event A occurring, let's say it is that right there.*0035

*Event B occurring would be like that; they don't overlap. *0047

*That means they cannot happen together; they cannot occur at the same time.*0052

*An example of this, if I were to say that right now, it is 5 o'clock pm.*0056

*It is 5pm; let's say this is A; event A.*0075

*Event B, if I say it is early in the morning.*0083

*See how event A, this statement here, it is 5pm, and in statement B, it is early in the morning.*0102

*They cannot occur at the same time.*0110

*It is not possible for it to be 5pm and for it to be early in the morning.*0112

*These two events would be disjoint.*0118

*These are considered disjoint events because they cannot occur at the same time.*0124

*They cannot occur together.*0130

*If I were to say the same statement here, it is 5 o'clock pm.*0133

*For my second event, my second statement, I am going to say it is dinnertime.*0147

*This is possible; you can have dinner at 5 o'clock pm.*0159

*In this case, probability of A and B would be not disjoint because this can occur at the same time.*0163

*It is not disjoint.*0177

*If two events cannot occur at the same time, they are disjoint.*0180

*If they can, then it is not disjoint.*0184

*Determine if each pair of events is disjoint or not disjoint.*0189

*The first one, statement A, Samantha is more than 10 years old.*0193

*That means she could be 11, 12 years old, 13, 20, 30.*0202

*She is older than 10.*0209

*Statement B, Samantha is less than 8 years old.*0211

*She can't be older than 10 and less than 8.*0217

*This is not possible; this would be disjoint.*0221

*The second one, the first statement, John is 6 feet tall.*0231

*For B, John is between 5'10 and 6'2.*0240

*This is possible; 6 feet is between 5'10 and 6'2; this is not disjoint.*0248

*The next example, a box weighs 10 pounds.*0265

*Name a pair of events that would make this statement disjoint and another pair that is not disjoint.*0268

*We are going to create our own disjoint events and then another pair of events that would be not disjoint.*0274

*That makes sense; that could occur.*0282

*The first statement, the one that is disjoint, let's make it that statement right there.*0287

*A, my first one, a box weighs 10 pounds.*0300

*For B, our second statement, to make it disjoint, the same box weighs less than 8 pounds.*0313

*It wouldn't make sense; this is disjoint.*0332

*For not disjoint, I can say my first statement, a box weighs more than 9 pounds.*0339

*For my next statement, a box weighs less than 11 pounds.*0366

*This is true; these two statements are true; it is not disjoint.*0381

*The third example here, determine if each set of events is independent, dependent, or disjoint.*0392

*Remember independent events, when we have two events that do not affect each other.*0398

*The outcome of the second event is not affected or does not depend on the first event.*0403

*Dependent events are the opposite.*0411

*The second event is affected by the first event.*0414

*The probability of the second event occurring is affected or is determined by the outcome of the first event.*0419

*Disjoint remember is when we have two events that cannot occur at the same time.0713.2.*0429

*It is not possible for them.*0433

*The first statement, a person picks a card from a deck of playing cards.*0436

*Without replacing it, another person picks another card from the same deck.*0441

*There are 52 cards in a deck; a person picks a card.*0449

*That is the first event; the first pick is the first event.*0455

*Without replacing it, another person picks another card from the same deck.*0461

*For the first event, when you pick a card, it is 1 card out of a total of 52.*0470

*This is the first event.*0483

*Then for the second event, for the second pick,*0487

*since the first card is not replaced back into the deck, there is o1ne card missing now.*0491

*There is no longer 52 cards.*0498

*The second person is going to pick 1 card out of a total of 51.*0503

*1 card out of 52 times 1 card out of 51.*0514

*Here all we want to know is if the two events together, is it independent, dependent, or disjoint?*0518

*See how the second event here is affected by this first event because the card was not replaced.*0530

*It is not put back in; so now there is less cards.*0536

*The card that the second person picks might be different; it is affected.*0539

*This would be a dependent event; these are dependent events.*0545

*The second one, Sarah received 100 percent on her chapter five math test.*0556

*Sarah failed her chapter five math test.*0562

*The first event is that she received 100 percent.*0565

*She got an A plus; nothing wrong.*0572

*Then the second event, Sarah failed her chapter five math test.*0576

*If you get 100 percent, is that failing?--no.*0584

*This event here with this event B here cannot occur at the same time.*0588

*It is not possible for both to be true.*0597

*This would be an example of disjoint events.*0600

*The third one, Susan rolled a number cube and got a 4.*0611

*She rolled again and got a 3.*0615

*A number cube is a die; we know that there are 6 sides.*0618

*Each side has a different number; 1, 2, 3.*0626

*She rolled a number cube at got a 4.*0633

*What is the probability of rolling a 4?*0637

*Desired outcome, how many sides on this number cube is a 4?*0641

*Only 1 side; that is 1 out of a total of 6 sides.*0647

*The probability of rolling a 4 is 1/6; that is the first event.*0654

*She rolled again and got a 3; what is the probability of rolling a 3?*0660

*How many sides has a 3?--only 1 side out of 6.*0669

*Here to find the probability of rolling a 4 and then a 3 for those two events is 1/6 times 1/6.*0677

*See how even though she rolled the first time and got a 4, she rolled again.*0691

*For the second event, rolling a 3, was that affected by what she got from the first roll?*0697

*No, just because she rolled a 4 the first time doesn't mean that*0704

*she can't roll a 4 again on the second time, on the second roll.*0707

*These two events are independent.*0711

*This second roll, the second event, is not affected,*0722

*does not depend on this roll here, the probability of getting the 4.*0725

*That is it for this lesson; thank you for watching Educator.com.*0730

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over how to find probability of an event not occurring.*0002

*We have gone over how to find the probability of an event occurring.*0009

*That would be the desired outcome over the total possible outcomes; it is a ratio.*0014

*We are comparing what we are looking for, the desired outcome,*0021

*over how many total possible outcomes there are, what is the total number.*0025

*This ratio is in the form of a fraction, top number over bottom number.*0032

*Probability can also be in the form of a decimal and a percent.*0038

*If I have probability of an event occurring, let's say it is 1/4.*0043

*That is a probability.*0050

*I can change this into a decimal and into a percent.*0051

*Let's just review over that.*0056

*To change this fraction to a decimal, I am going to take the top number and divide it to the bottom number.*0058

*This top number is going to go inside; the 4 on the outside.*0065

*Here I am going to add a decimal point.*0073

*I can always add a decimal point at the end of a number.*0075

*Then I can add 0s.*0080

*I can add 0s as long as it is behind the decimal point and at the end of the number.*0081

*Bring this decimal point up; 4 does not go into 1.*0088

*I am going to use one 0 to make 10.*0093

*4 goes into 10 twice; that is going to give me 8.*0096

*I am going to subtract the 10 with the 8; I get 2.*0102

*I can add another 0; bring that 0 down.*0107

*4 goes into 20 five times; that is going to give you 20.*0111

*Subtract it; I get 0; 1/4 is the same as 0.25.*0117

*You can also think of it as 1 out of 4... let's say I have 4 quarters.*0129

*1 out of the 4 quarters gives me 25 cents.*0134

*To change this to a percent, I take the decimal point.*0139

*I always move it to the right two spaces.*0145

*Think of the decimal as being small and a percent as getting larger.*0149

*Percents are bigger than decimals.*0154

*We have to move it to the right to make the number bigger.*0157

*I am going to move it to the right two spaces.*0159

*The decimal point is now going to be behind the 5.*0163

*Add the percent sign; that is how you change it from decimal to percent.*0168

*Three ways we can write probability; that is the probability of an event occurring.*0174

*To find the probability of an event not occurring is actually going to be 1 minus that number.*0185

*1, why 1?--1 is actually the biggest number we can get for probability.*0196

*It is like seeing the whole thing.*0201

*Say I have a bag of marbles.*0204

*In this bag of marbles, I have let's say 4 red marbles.*0209

*If I want to find the probability of picking a red marble,*0217

*probability of my event will be... I am picking a red.*0223

*How many red marbles do I have?--I have 4.*0229

*My desired outcome is 4 out of... how many total marbles do I have?--4.*0233

*4/4 simplifies to get 1; 1 represents the whole thing; it represents all of it.*0240

*The probability of something happening is 1.*0251

*That means it is 100 percent chance that it is going to happen.*0254

*That is why when you look for the probability of an event not occurring,*0262

*it is as if you are going to take that 1, the whole thing, and you are going to find the leftovers.*0269

*Back to the bag of marbles, let's say I add 2 blues.*0277

*I want to find the probability of picking a red.*0288

*I still have 4 red out of... the total number of marbles changed.*0297

*I now have 6 marbles.*0304

*Here my probability is... I can simplify this; divide each of these by 2.*0306

*I get 2/3; that is the probability of the red.*0313

*I want to find the probability of it not being red, the event not occurring.*0322

*The probability of not red, I am going to take the whole thing which is 1.*0329

*Subtract it from the event occurring, 2/3; it is like finding the leftovers.*0342

*From the whole thing, if I take away this much*0351

*which is the actual probability of the event occurring,*0353

*then it is like I am finding what is left over.*0357

*Here in order to subtract this, I need to make this the same denominator with this.*0362

*I can turn 1 whole into, as long as my top number and bottom number are the same, it is still 1.*0371

*3/3 minus 2/3.*0378

*I made is 3/3 because I want this denominator to be 3,*0381

*the same, for me to be able to subtract these fractions.*0385

*This becomes 1/3.*0388

*Another way to explain this, the probability of picking a red is 2/3.*0397

*The probability of not picking a red is all the rest of it which is 1/3.*0407

*Together the probability, it is either red or not red.*0414

*It is either going to be red or it is not red.*0418

*It is one of those two.*0421

*This is the probability of picking a red.*0424

*This is the probability of picking one that is not red.*0426

*Together they make up the whole thing because it is going to be one or the other.*0429

*The whole thing is just 1; the probability of this not occurring is 1/3.*0433

*The first example, we are going to use a spinner.*0446

*We want to find the probability of spinning or landing on a color that is not green.*0448

*We can do this two ways.*0459

*When we have something like this spinner or maybe a bag of marbles, it is a little bit easier.*0462

*We can make all the colors that are not green our desired outcome.*0469

*I can look for all of the colors that are not green.*0482

*I have 1, 2, 3, 4; 4 that are not green.*0487

*My desired outcome again is not green.*0492

*There is 4 of them; over total of 5.*0496

*The probability of not green will be 4/5.*0504

*The way we did it before, previous slide, it is like*0509

*finding the probability of actually having green and subtracting that from 1.*0513

*We can find the probability of green, subtract it from 1.*0524

*We are still going to get the same answer.*0529

*Here probability of picking a green, that is 1 green out of 5.*0531

*This will be 1 minus 1/5.*0538

*I can change this whole number into 5/5 to make the denominators the same.*0544

*5/5 minus 1/5; 5 minus 1 is 4; over... keep the denominator the same.*0549

*I am going to get the same answer.*0562

*This is obviously the easier way to do it.*0566

*If you can do it this way, then that is fine.*0569

*But you still have to understand that 1 whole would be the whole thing.*0571

*There is 1, 2, 3, 4, 5 out of 5; that would be 1 whole.*0578

*To find the probability of something not occurring would be taking the whole thing*0586

*and subtracting it by the actual probability of the event occurring.*0589

*This next one here, the probability of not red or orange.*0598

*Again we can just make this all the colors that are not red or orange be the desired outcome.*0604

*How many are not red or orange?--here is red; here is orange.*0610

*How many are not either of these?--I have 3.*0615

*My desired outcome would be all the colors that are not red or orange.*0620

*That is going to be my top number; that is 3; over total of 5.*0625

*That would be my answer.*0634

*But again we still have to understand that I can find the probability*0636

*of the red or orange and then take the whole thing, subtract it.*0642

*How many are red or orange?--I have 1, 2; 2 out of 5.*0656

*1 minus 2/5; again change this to 5/5 minus 2/5.*0662

*That is going to give me 3/5.*0671

*Given the probability of an event occurring, find the probability that the event will not occur.*0683

*Here probability of event A occurring is 1/4.*0694

*The probability of the event not occurring is 1 minus this number.*0702

*It is going to be 1 minus 1/4.*0713

*Again change this whole number; it is 1.*0720

*Top number and the bottom number has to be the same because I want to change this to a fraction.*0724

*The denominator has to be a 4; it will be 4/4 minus 1/4.*0730

*Again I made it 4/4 because it has to stay a 1 and the denominator has to be the same.*0737

*This is 4 minus 1 is 3 over 4.*0744

*If the probability of an event occurring is 1/4, then the probability of the event not occurring is 3/4.*0752

*It is like if 1 is the whole thing... 1/4, let's talk money.*0759

*1/4, 1 quarter out of 4 quarters.*0765

*If you use 1 quarter, how many quarters do you have left?*0769

*The remaining of it is 3 quarters; you have 3 left.*0775

*1/4 left over is 3/4; the rest of it is 3/4.*0780

*That is the probability of it not occurring.*0787

*Here 0.67, again 1 minus the probability of B occurring would be 1 minus 0.67.*0791

*Again if this is a dollar minus 67 cents, what do you have left?*0811

*To subtract decimals, I am going to do 1.00.*0818

*I am just turning this 1 into 1.00 because when you subtract decimals, you have to line up the decimal point.*0823

*It is going to be 1.00.*0833

*Again I added 0s because it is at the end of a number behind the decimal point.*0835

*Minus 0.67; this 0, I am going to change to a 10.*0841

*I borrowed it from this; that became a 9; this becomes 0.*0851

*It is 10 minus 7 is 3; 9 minus 6 is 3; nothing there.*0860

*Bring down the decimal point.*0867

*The probability of this not occurring is 0.33.*0870

*For this one, the probability of event C happening is 42 percent.*0878

*The chance of this occurring is 42 percent.*0883

*What is the chance of it not occurring?*0888

*This is a little bit different because it is a percent.*0891

*The whole thing in a percent would be 100 percent.*0897

*If you have a percent, then you would have to do 100 percent minus the 42 percent.*0901

*Or you can think of it as still 1 minus probability of C occurring.*0914

*Then since it is a percent, now we can change it here.*0920

*100 percent, we are changing this 1 whole into a percent.*0924

*Again decimal point at the end; move it two spaces over.*0929

*That is 100 percent minus 42 percent; 100 minus 42 is 58 percent.*0933

*The third example here, the probability of Susie passing the math test is 85 percent.*0953

*Find the probability of her failing the test.*0960

*The probability of an event occurring, which is her passing the math test, is 85 percent.*0968

*We have to find the probability of her failing the test.*0977

*It is out of a possible 100 percent.*0981

*We know that from 100 percent, we have to subtract*0985

*the percent of the probability that she is going to pass the test*0994

*to see what the probability of her failing the test is going to be.*1001

*From here, this is 15 percent.*1008

*This and this together have to make up the 100 percent because that is the whole thing.*1013

*She is either going to pass it; or she is going to fail it.*1018

*This is the pass; this is the fail.*1020

*Together they have to make up the 100 percent, the whole thing.*1025

*Probability of her failing the test is going to be at 15 percent.*1032

*The second one, the probability of Sam not picking the correct colored marble from a bag is 5/8.*1041

*Find the probability of him picking the correct marble.*1049

*The probability of Sam picking the marble, let's just say marble, the correct marble.*1056

*This is what they are asking for.*1067

*They want to know what the probability of him picking the correct marble is going to be.*1068

*He is either going to pick the correct marble or he is going to pick the incorrect marble.*1074

*The probability, what is given to us, of not picking the correct color is going to be 5/8.*1086

*The probability of not picking the correct marble, not correct marble, is 5/8.*1095

*To find the probability of actually picking the correct marble*1110

*is going to be 1 whole because the whole thing is 1 whole.*1114

*There is 8 marbles total; 8/8 is 1; 1 whole; minus 5/8.*1121

*Here to do this, 1 minus 5/8, I need to change this 1 into a whole number.*1136

*Again remember it is 8/8 because I need the denominators to be the same.*1143

*I have to have the top number and the bottom number be the same number for it to just be 1.*1147

*The denominators have to be the same; it has to be 8/8 minus 5/8.*1153

*See how those are the same whenever we subtract fractions; this becomes 3/8.*1162

*If the probability of him not picking the correct marble is 5/8,*1171

*then the probability of him actually picking the correct marble is going to be 3/8*1178

*because together they have to make up 1 whole.*1184

*1 whole is going to be 100 percent; it is going to be all of it.*1188

*He is either going to pick the correct one or he is going to not pick the correct one.*1192

*These two numbers together have to add up to 1 whole.*1198

*That is it for this lesson; thank you for watching Educator.com.*1202

*Welcome back; this is a lesson on prime factorization.*0000

*Before we begin, let's go over some terms: prime and composite.*0008

*Prime number we know is a number that has no factors besides 1 and itself.*0014

*For example, the number 5.*0022

*5 only has the factors 1 and itself so that is considered a prime number.*0025

*A composite number is a number that has more factors than 1 and itself.*0031

*For example, the number 10.*0036

*10 is a composite number because we know that the number 2 is a factor of 10 and 5 is a factor of 10.*0039

*So 10 would be considered a composite number.*0051

*A factor then would be all the number parts that go into a given number.*0054

*If we have again 10, the factors would be 1, 2, 5, and 10.*0062

*These are all considered factors of the number 10.*0071

*Product; product is a number where we multiply numbers together.*0076

*The product of 2 and 5 is 10.*0084

*So product, you just multiply the numbers together.*0092

*The method in solving prime factorization is called the factor tree.*0100

*The factor tree... we all know what a tree looks like.*0106

*It is going to branch out into a bunch of factors.*0109

*More specifically, prime factors, which is why it is called prime factorization.*0115

*Let's start off with the number 10.*0121

*I want to break this up into factor pairs; say 5 and 2.*0125

*Once I have a prime number, I want you to circle it.*0133

*We know 5 is a prime number; we know 2 is a prime number.*0137

*Once all the numbers are circled, we know that we have only prime numbers.*0142

*Then we are done; all we have to do is write out the answer.*0148

*10, the prime factorization of 10 would be 5 times 2.*0155

*Another example, 24; 24 has a few different factor pairs.*0164

*You can choose whichever one; let's go with 6 and 4.*0172

*6 and 4, we know that both are not prime numbers.*0178

*They are composite numbers; we have to break them up even further.*0182

*I am going to branch it out again.*0186

*6 is going to break up into 3 and 2.*0189

*This is a prime number; I am going to circle it.*0194

*This is a prime number.*0196

*Then 4, I am going to break up into 2 and 2.*0199

*A prime number, I circle this and that.*0205

*All I have left are prime numbers.*0211

*The prime factorization of 24 would be 3 times 2 times 2 times 2.*0215

*Since I have the same numbers here, I can write this in scientific notation.*0225

*This can be also 3 times 2 to the 3rd power because I have three 2s.*0232

*Here is an example.*0248

*If you can, I want you to pause the video and I want you to try this problem on your own.*0252

*Let's go over this now; 50.*0259

*I am going to break this up into a factor pair; let's say 5 and 10.*0264

*5 is a prime number; circle that one.*0271

*10, I am going to break this up even further to 2 and 5.*0275

*I end up circling those.*0283

*50 becomes 5 times 5 times 2 or I can write it as 5*^{2} times 2.0286

*Here is another example; 15.*0307

*15, I can only break it up into 5 and 3.*0313

*They are both prime numbers; this was an easy one.*0320

*15 becomes 5 and 3; let's do a couple more examples.*0324

*The number 12 becomes 4 and 3; you can also do 6 and 2.*0337

*You are still going to get the same answer.*0350

*3 is a prime number; I am going to circle that one.*0353

*4 becomes 2 and 2; circle those; I have nothing else left.*0356

*The prime factorization of 12 is 2*^{2} times 3.0363

*One more example, 36.*0373

*36, again you have a couple different options; let's go with 6 and 6.*0378

*6, I can break up into 3 and 2; I am going to circle those.*0387

*This 6 also to 3 and 2.*0395

*36 becomes 3 times 3 times 2 times 2.*0401

*This is the same thing as 3*^{2} times 2^{2}.0412

*That is it for prime factorization.*0418

*Welcome back; this lesson is on the greatest common factor.*0000

*Let's go over what a factor is.*0008

*A factor is a part of a number that can be divided out without leaving a remainder.*0011

*If we have the number 20, it is all the parts of 20 that can be multiplied into 20 without leaving a remainder.*0018

*An example of 20, factors would be 1... we know that 1 can be multiplied to get 20.*0030

*2 is a factor of 20; 4, 5, 10, and the number itself 20.*0038

*These are all considered to be factors of 20.*0052

*When we think of greatest common factor, we know that we have to find a common factor.*0062

*That means we are comparing two different numbers.*0070

*We have to find the biggest factor between the two numbers.*0073

*We can also call greatest common factor, GCF; it is also known as GCF.*0078

*If you look at this example right here between 15 and 30,*0084

*we are going to use these two numbers, compare them,*0088

*and find the factor that is the biggest between them.*0090

*There is two methods to solve this; the first method is very simple.*0095

*All you need to do is list out the factors.*0099

*15, I know the factors of 15 would be 1, 3, 5, and 15.*0107

*For 30, the factors are 1, 2, 5, 6, 10... I forgot 3... and 30 itself.*0118

*If I look at these... I have one more, 15.*0138

*I look at these; I think of all the common factors.*0144

*I know 1 is a common factor; 5 is a common factor.*0149

*15 is a common factor; those are all considered common factors.*0155

*But we are looking for the greatest common factor.*0159

*For this problem, the answer would be 15; the GCF is 15.*0163

*Another method to solve this would be to list out the two numbers.*0178

*Let's say 15 and 30 like this.*0184

*I am going to draw a little L shape around the two numbers.*0187

*From here, I just want to find any common factor, except for 1 of course.*0197

*Any common factor between the two numbers; let's say 5.*0204

*I am going to write that right outside the box.*0211

*If I pull 5 out of 15 or 15 divided by 5, I get 3.*0216

*I am going to write that right below the 15.*0224

*Right here, if I take a 5 out of 30 or 30 divided by 5, then I get 6.*0228

*I look at these two numbers, the two numbers that I just wrote, 3 and 6.*0237

*Do they have anything common?*0241

*I know that 3 can go into both of these numbers.*0244

*Since I know that they have a common factor, I am going to write another box under.*0248

*The common factor was 3.*0254

*I am going to write that common factor on the outside.*0259

*Again 3 divided by 3, if I pull a 3 out, it is going to be left with 1.*0263

*From 6, if I take a 3 out or 6 divided by 3, then I get 2.*0270

*I look at these two numbers, 1 and 2.*0279

*The only common factor between 1 and 2 is the number 1.*0281

*I am not going to consider 1 because all numbers have a common factor of 1.*0287

*So there, I am done.*0292

*I am going to take all the numbers that I have on the side.*0295

*5 and 3, I am going to circle them.*0300

*I am going to multiply them out; it is going to be 5 times 3.*0302

*My answer is 15; therefore my GCF is 15.*0307

*Let's do another example; I am going to be using method two.*0318

*If you want, you can pause the video; I want you to try both methods.*0323

*The first method was listing out all the factors for each number and then finding the greatest common one.*0327

*The second method was listing them out.*0335

*You are going to pull out one common factor at a time.*0337

*Then you are going to multiply all of those common factors.*0341

*Go ahead and pause; work on this problem right now.*0344

*I am going to use method two for these examples.*0350

*I am going to write out the two numbers, 6 and 18.*0356

*I am going to draw out this little L shape box around it.*0362

*I am going to think of a common factor; let's see, 6 and 18.*0367

*I know that they are both even numbers.*0374

*I am going to take out a 2.*0377

*Whenever they are both even numbers, they have a common factor of 2.*0381

*If I pull out a 2 from 6, that means I am going to do 6 divided by 2.*0387

*I am going to get 3; I write that right below the 6.*0393

*From 18, if I pull out a 2, 18 divided by 2, I get 9.*0400

*3 and 9, they do have a common factor.*0408

*I am going to draw another one of these; their common factor is 3.*0414

*If I do 3 divided by this number 3, then I get a 1.*0421

*I do 9 divided by 3; I get a 3.*0426

*Again these bottom numbers, 1 and 3, they only have a common factor of 1 which means that I am done.*0431

*I take these two numbers on the side.*0440

*You are going to take all the numbers; you are going to multiply them out.*0443

*It is going to be... the GCF of 6 and 18 is 2 times 3 which is going to be 6.*0446

*The next example with 36 and 27; these numbers are a little bit bigger.*0466

*You just have to take one common factor out.*0482

*We have 36; we have 27; let's see.*0485

*I know that 3 goes into 36 and 3 goes into 27.*0490

*I am going to take out a 3; 36 divided by 3 is 12.*0497

*27 divided by 3 is 9; again they have a common factor besides 1.*0509

*Draw another box; the common factor between 12 and 9 is 3.*0520

*When I divide 12 by 3, I get 4; 9 divided by 3 is 3.*0528

*With these two numbers, their only common factor is 1.*0537

*That means I no longer have to pull out common factors.*0541

*These two numbers on the side, I am going to multiply out.*0547

*The GCF is 9.*0550

*This next example with 4 and 42; they are both even numbers.*0560

*I can just take out a 2 because I know even numbers have a common factor of 2.*0575

*14 divided by 2 is 7; 42 divided by 2 is 21.*0582

*Do they have a common factor?*0591

*I think they do; we are going to draw this again.*0593

*We are going to pull out another common factor which is a 7.*0598

*7 divided by 7 is 1; 21 divided by 7 is 3.*0603

*Again they only have a common factor of 1.*0610

*That is when I take the numbers on the side.*0616

*I am going to multiply them out; the GCF is 14.*0619

*Last example, 54 and 36; again these numbers are a little big.*0631

*You can just think of a small common factor; these are both even numbers.*0645

*I know that a 2 goes into both of them.*0651

*I can just take that 2 out.*0654

*54 divided by 2 is 27; 36 divided by 2 is 18.*0658

*For this one, if you pull out another example, you will still get the same answer.*0672

*For example, a 6 goes into both of these numbers.*0678

*If you take out a 6, then your numbers are going to be different than mine.*0681

*But we will still get the same answer.*0685

*From these two numbers, 27 and 18, I know that a 3 goes into both.*0690

*27 divided by 3 is 9; 18 divided by 3 is 6.*0699

*9 and 6, they do have a common factor.*0710

*I have to do this one more time; their common factor is 3.*0714

*9 divided by 3 is 3; 6 divided by 3 is 2.*0719

*From these two numbers, 3 and 2, their common factor is 1.*0725

*That means I can take all the numbers on the side, not the bottom numbers, only the side.*0730

*I am going to multiply them out.*0738

*The GCF is going to be 3 times 3 times 2.*0740

*3 times 3 is 9; 9 times 2 is 18.*0749

*The GCF, the greatest common factor, between 54 and 36 is 18.*0755

*That was the lesson for greatest common factor.*0763

*Thank you for watching Educator.com.*0765

*Welcome back to Educator.com.*0000

*This lesson is going to go over some concepts of fractions and how to simplify them.*0002

*When we have a fraction, we are representing it as a part.*0013

*If we have something that is whole, we are going to take parts of it, break it up into parts.*0020

*That is how we write fractions.*0026

*Let's pretend it is your birthday; this is your cake.*0030

*We are obviously going to have to cut the cake.*0037

*Since it is your birthday, you would probably want the biggest piece.*0040

*If we take this cake, let's say we are going to cut it up into 4 equal pieces.*0045

*If you get 1 piece of this cake, you are getting 1 piece out of how many total pieces?*0055

*1, 2, 3, 4; you will be getting 1/4 of your cake.*0067

*Let's say I cut the cake up into smaller pieces.*0076

*Here if you take 1 piece of your cake,*0091

*then you are going to take 1 out of 2, 3, 4, 5, 6, 7, 8.*0099

*Out of 8 total slices, you are getting 1.*0107

*Let's say you want more cake; you ate 3.*0114

*I can write this as 3 out of 8.*0122

*Look at this one right here; pretend that this is a chocolate bar.*0131

*Your chocolate bar, you are going to break it up into pieces.*0137

*We can break up the chocolate bar into 8 pieces.*0139

*If you eat 1 little piece, you are eating 1 out of 1, 2, 3, 4, 5, 6, 7, 8 pieces.*0145

*Let's say you eat another one; that is 2 out of 8 pieces.*0159

*I can also write this in a different way.*0169

*If the chocolate bar is broken up into 4 pieces, you break it up into 4 pieces,*0182

*then this right here that you ate is going to be the same thing as 1 out of 4 pieces.*0188

*If you look at this, 1, 2 out of 8 that you ate here*0200

*is going to be the same as 1 out of 4.*0207

*That is when we are going to learn how to simply these fractions.*0220

*Here we have 8 pieces.*0226

*4 out of 8 would be 1, 2, 3, and 4.*0230

*This is 4 out of 8.*0244

*Instead of having 8 pieces, I am only going to have 4.*0251

*In this case, having 4 out of 8 would be the same thing as 2 out of 4*0265

*because I have 4 pieces and I ate 2 out of those 4.*0274

*If I wanted to show 4 out of 10, I need to have something... I can have a circle.*0284

*Or I can have something rectangular like the chocolate bar.*0291

*I need to 10 equal pieces; I can do this.*0300

*Let's say these were 10 equal pieces.*0318

*To show 4 out of 10... that would be 4 out of 10.*0321

*How can you represent this in a different way?*0337

*Instead of breaking this up into 10 pieces, let's say you broke it up into 5 pieces.*0341

*In that case, if it was 5 pieces, this would be 1.*0346

*This would be 2, 3, 4, and 5.*0351

*4 out of 10 could be the same as 2 out of 5.*0357

*This one, we have 6 pieces; 6 pieces.*0374

*If I do this, this would be 1 out of 6.*0381

*If I take this piece, then that would be 2 out of 6.*0389

*Maybe if I ate this piece, then that would be 3 out of 6.*0397

*If you think about it, if I have 6 pieces and I ate 3 of them, I ate 1/2.*0408

*I ate half of this chocolate bar.*0418

*Let's say I don't have a picture; I don't have a chocolate bar.*0425

*I just have a fraction, 5 over 25.*0427

*How can I write this in simplest form?*0430

*In this case, I can take a common factor, a number that multiplies into 5 and 25.*0437

*I can divide both numbers by that number.*0447

*5 and 25; I know that 5 goes into 5 and 5 goes into 25.*0452

*That means I can divide this number by 5 and divide this number by 5.*0458

*They have to be divided by the same number.*0469

*Or else you are not going to get the right answer.*0471

*5 divided by 5 is 1; 25 divided by 5 is 5.*0475

*Let's look at this fraction again, 1 and 5.*0484

*We have to make sure that they have no common factors besides 1.*0488

*For 1 and 5, that is the only common factor they have.*0494

*That means we are done; this is the answer.*0497

*The simplest form of 5 over 25 would be 1 over 5.*0501

*To find the simplest form of 14 over 49, we are going to look for a common factor.*0514

*Meaning we are going to look for a number amongst 14 and 49*0525

*that can be multiplied to get 14 and it can be multiplied to get 49.*0532

*A common factor between these two numbers would be 7.*0539

*In order to get the simplest form, I can take this fraction, *0544

*divide the top number and the bottom number by that common factor.*0548

*Again they have to be the same number.*0557

*You have to divide both numbers by that same number.*0560

*14 divided by 7 is 2; 49 divided by 7 is 7.*0566

*I look at these two numbers now.*0576

*I want to see if there is a common factor between 2 and 7.*0579

*The only common factor is 1; that means I am done.*0584

*The simplest form of 14 over 49 would be 2 over 7.*0590

*That is it for this lesson; thank you for watching Educator.com.*0599

*This next lesson is on finding the least common multiple.*0000

*To review, a multiple is a number or the numbers that the original number can multiply into.*0008

*If I have a number 4, the multiples would be 4, 8, 12, 16, 20, and so on.*0016

*Multiples would be numbers that the original number can multiply into.*0033

*The least common multiple is known as the LCM; we are comparing two numbers.*0044

*From the multiples of the two numbers, we are going to find the smallest common multiple.*0055

*In order to do this, there is two methods that we can use.*0065

*The first method is to simply list out the multiples of each number.*0069

*You are going to find the smallest one.*0076

*For 6, the multiples would be 6, 12, 18, 24, 30, 36, 42, 48.*0082

*I am going to just stop there for now.*0111

*List out the multiples of 10; for 10, 20, 30, 40, 50, and 60.*0115

*If you look at this, from these numbers, a common multiple is 30.*0136

*If I were to keep going, for the 6, this would be 54 and 60.*0146

*Another common multiple would be 60.*0154

*But again I want to find the least common multiple.*0158

*It is going to be the smallest common multiple which is 30.*0162

*The LCM of 6 and 10 would be 30.*0168

*Another method to finding the LCM is going to involve prime factorization.*0176

*If you don't remember prime factorization, you can go back to that lesson and just review over that.*0185

*I am going to find all the prime factors of 6 and 10.*0196

*To do that, I have to use a factor tree.*0201

*6, I am going to break down into 3 and 2.*0204

*Circle them because they are prime numbers.*0210

*10, the factor pairs of 10 would be 5... 5 is a factor... and 2.*0214

*That is also prime; I circle it.*0229

*Look at the prime factors of 6, 3 and 2.*0233

*I look at the prime factors of 10 which is 5 and 2.*0238

*I look for any common factors; the common factor between 6 and 10 is 2.*0242

*If I list this out, 6 is 3 and 2.*0249

*The prime factors of 10, 5 and 2; they have a common factor of 2.*0257

*When they have a common factor, I am going to take one of them and cross it out.*0264

*Cross out only one of them.*0273

*Then I take all the remaining numbers--3, 5, and then this 2.*0276

*I am going to multiply them out.*0281

*The LCM is going to be 3 times 5 times 2 which is equal to 30.*0285

*Whichever method you would like to use, you will still get the same answer of 30.*0300

*Let's find the LCM of 12 and 15; you can pause it.*0311

*I want you to try to use both methods to find the LCM.*0317

*Then just come back and we will go over it.*0323

*12 and 15; I am going to use the second method for all these examples.*0328

*I am going to use the factor tree method to find all the prime factors of these numbers.*0337

*For 12, I can use a factor pair of 6 and 2.*0344

*Or I can use 4 and 3.*0349

*3 is a prime number; I am going to circle it.*0354

*4, I am going to break up into two prime numbers, 2 and 2.*0357

*I circle those numbers.*0363

*For 15, the factor pair would be 5 and 3.*0365

*These are both prime numbers; I am going to circle them.*0373

*For 12, all the prime numbers would be 2 times 2 times 3.*0377

*The prime factorization of 15 is 5 and 3.*0387

*I look for any common numbers between 12 and 15.*0393

*They have a common number of 3.*0399

*I am just going to take one of them and cross it out.*0401

*These are common; there is a 2 here and a 2 here.*0407

*But that is within the same number, 12.*0411

*I don't want to cancel that out.*0413

*It has to be one from here and one from the other number.*0415

*I am going to take the remaining numbers--2, 2, 5, and 3.*0418

*I am going to multiply them out.*0424

*My LCM is going to be 2 times 2 times 5 times 3.*0428

*This is going to be 4; this is going to be 15.*0438

*My answer is 60.*0446

*The next example is finding the LCM of 16 and 20.*0458

*Let's use the factor tree; the factor pair of 16 would be 4 and 4.*0466

*You can also use 8 and 2; circle them because they are prime.*0474

*2 and 2, circle those numbers.*0484

*For 20, I can use 10 and 2; or I can use 5 and 4.*0489

*5 is a prime number; I am going to circle that one.*0498

*4 becomes 2 and 2; circle those.*0501

*For 16, the prime factorization would be 2 times 2 times 2 times 2.*0509

*For 20, 5 times 2 times 2.*0522

*Look at this; we have a common number here.*0532

*I am going to cancel one of these out; it doesn't matter which one.*0535

*That took care of that pair.*0541

*We have a 2 here and another 2 here, another common number.*0543

*Again I am going to cancel out one of them.*0548

*I am going to leave the other one.*0551

*You are going to take all the remaining numbers--2, this 2, 5, 2, including this one, 2.*0554

*I am going to multiply them all out.*0564

*The LCM is going to be 2 times 2 times 5 times 2 times 2.*0567

*This is 2 times 2; this is 4; times 10 times 2.*0583

*You are going to get 80 as your answer.*0592

*Example three, let's use 15 and 25.*0603

*The prime factors of 15, 5 and 3; for 25, we have 5 and 5.*0615

*I can write them out; then for 25.*0631

*They have a common number of 5.*0644

*I am going to cancel out one of those factors.*0646

*Even though I have a 5 here and a 5 here,*0652

*I am not going to cancel that out because they are both in the same number, 25.*0654

*I take the remaining numbers.*0660

*My LCM is going to be the product of those numbers.*0664

*3 times 5 times 5; my answer is going to be 75.*0669

*For this example of 12 and 18, we are going to find the LCM, the least common multiple.*0691

*In order to do that, I am going to use the factor tree method*0699

*to find all the prime factors which is the second method that we went over.*0702

*12, I am going to use the factor pair 4 and 3.*0708

*I can also use 6 and 2.*0716

*This is a prime number; I am going to circle this.*0719

*For 4, break this up into 2 and 2.*0722

*They are both prime; I am going to circle them.*0729

*Then do the same thing for 18.*0733

*For 18, I can use 9 and 2; or I can use 6 and 3.*0736

*This is a prime number; circle that one.*0743

*For 6, I am going to use 2 and 3.*0747

*Circle them because they are prime.*0752

*For 12, all the prime factors are going to be 2, 2, and 3.*0755

*For 18, 2 times 3 times another 3.*0767

*I am going to look for any common factors between 12 and 18.*0775

*I know that there is a common factor of 3 for both 12 and 18.*0780

*I am going to take one of the numbers and just cross it out.*0787

*Just cancel one of the numbers out.*0790

*I also have a common factor of 2; 2 and 2.*0794

*I am going to take that and cancel one of those out.*0801

*Even though I have a 2 here and a 2 here, they are within the same numbers.*0806

*I am not going to cancel that out.*0811

*I take all the remaining numbers--this one, this one, and these two.*0814

*I am going to multiply them out to find the least common multiple.*0823

*It is going to be 2 times 2 times 3 times 3.*0828

*I am going to get 4 times 9 which is going to be 36.*0837

*The least common multiple of 12 and 18 is 36.*0848

*Thank you for watching Educator.com.*0854

*Welcome back; this lesson is going to be on comparing fractions.*0000

*Since we are going to be comparing fractions, we are going to be able to order them least to greatest.*0007

*This symbol right here means greater than.*0017

*If I were to use it, if I said 5 with that symbol and another number 2,*0023

*then I can read this is as 5 is greater than 2.*0034

*This means less than; I can say 2 is less than 4.*0040

*You have to make sure that this opening is going to face the bigger number; the same with this.*0051

*We are going to use those symbols to compare fractions.*0062

*We have a fraction 2/4 and another fraction 3/4.*0069

*Let's say that this is me; this is you.*0075

*Let's say we both have the same number of candy pieces.*0086

*You have 4 pieces; I have 4 pieces.*0091

*If I ate 2 out of the 4 pieces and you ate 3 out of the 4 pieces, who ate more?*0094

*You did; my fraction, what I ate, is less than what you ate which is 3 out of 4.*0103

*We know that 2 out of 4 is less than 3 out of 4.*0115

*In the same way, 5 out of 8 with 2/5.*0124

*These denominators are different; we have different number of parts.*0131

*If I look at this, 5/8, I know that half of 8 is 4.*0141

*If I look at this, this would be greater than half because 4 out of 8 is half.*0152

*5 out of 8 is bigger than half.*0163

*2/5, if you have 5 candy pieces and you ate 2 out of those 5, then you ate less than half.*0168

*5/8 is greater than half; 2/5 is less than half.*0180

*Which one do you think is a bigger fraction?*0188

*This is bigger than half; this is less than half.*0191

*I know that this one right here, 5/8, is going to be greater than 2/5.*0197

*Let's compare this one; 1/2 and 3/4.*0211

*Again these parts, the number of parts, this is 1 out of 2 parts; this is 3 out of 4.*0219

*The total number of parts are different between the two fractions.*0226

*What I can do is I can either look at this as 1/2 and this as 3/4, 3 parts out of the 4.*0230

*That is bigger than 1/2; I can compare it that way.*0241

*Or I can make these denominators the same.*0245

*Try to make them the same equal number of parts.*0249

*That way we can just compare the number on top.*0253

*1/2, I can change this 2 to a 4 because a 4 is a multiple of 2.*0260

*If I change the 2 to a 4, to get it from 2 to 4, I multiplied by 2.*0271

*I can do the same thing for the top numbers.*0283

*If I take the top number and I multiply it by 2, then I get 2.*0285

*If you want to demonstrate this fraction becoming this fraction...*0292

*Say I have a circle, I can either cut it in half that way.*0299

*This would become 1 out of 2 parts.*0306

*Or I can cut this into 4 pieces and say that I am going to eat 1, 2 out of 4 parts.*0312

*Either way I represent this, they are the same fraction.*0325

*I changed 1/2 to 2/4; I am going to compare it to 3/4.*0335

*Again 4 pieces, I ate 2; out of 4 pieces, let's say you ate 3.*0342

*Who ate more? 2/4 is less than 3/4.*0350

*I am going to write this symbol to show that this fraction is less than this fraction.*0359

*Another example, 3/9 with 4/12.*0369

*3/9, I can take this fraction; I can change it to look different*0380

*because this fraction and this fraction, they have different number of parts.*0393

*I am going to change them so that I can look at this in a more simpler way.*0399

*3/9; if I divide a common factor of 3 to both numbers, this is the same thing as 1/3.*0405

*Out of 3 parts, this is 1.*0421

*For 4/12, I also know that there is a common factor between the top number and the bottom number.*0426

*I am going to take the common factor which is 4, divide it to the top and to the bottom.*0436

*4 divided by 4 is 1; 12 divided by 4 is 3.*0447

*3/9 became 1/3; 4/12 also became 1/3.*0456

*These two fractions are actually the same; this is 1/3, 1 out of 3 parts.*0465

*This is also 1 out of 3 parts.*0472

*I am going to write equals; these two fractions are the same.*0475

*This next example, 8/11 and 21/100; these numbers are of big.*0483

*What I can do is I can look at these fractions and see how they compare with each other.*0495

*8/11, if I have 11 total number of parts and I use up 8,*0504

*or let's say you have 11 pieces of candy and you are going to eat 8 of them.*0513

*Did you eat more than half or less than half?*0522

*I know about 5 and 1/2 is half of 11.*0525

*8 is more than half; 8 is more than half of 11.*0529

*Let's look at this one, 21/100; 21 is smaller than half of 100.*0536

*Half of 100 is 50; 21 is less than that.*0547

*This one right here is bigger than half.*0552

*This one right here is smaller than half.*0555

*If I compare them, I know, even though these numbers are smaller than these numbers,*0558

*this fraction is actually more than this fraction.*0563

*This 8/11 is going to be greater than 21/100.*0568

*This example, we are going to compare these four fractions.*0580

*We are going to order them from least to greatest.*0586

*Since we are able to compare fractions, let's see which fraction is the smallest and which fraction is the greatest.*0593

*Let's look at these--3/4, 1/6, 1/2, and 4/4.*0604

*3/4, if I think of this fraction, 3/4 is 3 out of 4 parts.*0615

*Say you have a cake that is cut up into 4 pieces.*0624

*You ate 3 out of the 4 slices; that is more than half.*0630

*1/6, if you take that same cake and you cut it up into 6 pieces.*0635

*now the pieces are smaller and you ate 1 of those.*0642

*Then you are eating less than half of your cake.*0645

*You are eating a small piece; same thing for this fraction.*0651

*You are going to take the same cake.*0655

*You are going to cut it up into 2 pieces.*0657

*You are going to eat 1 of those.*0661

*Imagine, is that going to be a big slice or a small slice?*0663

*It is going to be half of your cake.*0667

*This last one right here, 4 out of the 4.*0671

*If you cut your cake into 4 pieces and you eat all 4, you are eating the whole cake.*0674

*Which fraction is the smallest?*0684

*3/4, 3 slices out of 4, 1 slice out of 6, 1 out of 2, or 4/4?*0689

*Which one represents the smallest piece of cake?*0699

*1/6 would actually be the smallest because if you cut it up into 6 pieces,*0707

*that means you are cutting them up into smaller pieces.*0713

*You are only going to be eating one of those.*0715

*The smallest fraction is going to be 1/ 6.*0719

*The next one... I am done with this one.*0726

*The other three, the next smallest would be 1/2*0731

*because I know this one is going to be more than half the cake.*0739

*4 out of 4, that is the whole cake.*0745

*Half the cake would be the next smallest.*0748

*Then 3/4 is going to be the next smallest*0752

*because if you eat 3 out of 4 slices, you still have some cake left over.*0758

*Whereas if you eat 4 out of 4 slices, then you ate the whole cake.*0763

*The next fraction is going to be 3/4.*0768

*Then the greatest fraction is going to be 4/4.*0774

*This was the lesson on comparing fractions; thank you for watching Educator.com.*0784

*Welcome back to Educator.com.*0000

*We are going over mixed numbers and improper fractions and how to switch between the two.*0002

*If we look at fractions, there is three different kinds.*0012

*There is the mixed number, proper fractions, and improper fractions.*0015

*Mixed number is a fraction with a whole number.*0021

*If I have 1 and 1/2, the number in the front, 1, is a whole number and then 1/2 is a fraction.*0026

*The whole number with a fraction is called a mixed number.*0035

*Another example would be 5 and 3/4; that is called a mixed number.*0040

*A proper fraction is a fraction where the top number is smaller than the bottom number.*0047

*3/4 with no whole number, just 3/4, that would be a proper fraction.*0058

*Keep in mind that proper fractions, because the top number,*0067

*the numerator, is smaller than the denominator, these fractions are smaller than 1*0069

*because if you ate 3 out of 4 pieces, then you ate less than the whole thing.*0077

*A proper fraction would be a fraction that is smaller than 1.*0083

*Improper fraction would be the opposite.*0090

*It is when the top number, the numerator, is bigger than the denominator.*0092

*Like 4/3, this is an improper fraction; again there is no whole number.*0100

*If there was a whole number, it would be called a mixed number.*0106

*An improper fraction, the top number is bigger than the bottom number.*0109

*If you ate 4 pieces out of 3, then you actually ate more than 1.*0115

*You ate 1; and you ate a little more.*0122

*Improper fractions are actually bigger than 1.*0126

*Mixed number we know is bigger than 1 because you have a whole number and you have a fraction.*0131

*And improper fractions are bigger than 1.*0137

*Since proper fractions are smaller than 1, we can't do anything to that one.*0142

*That one, we can't change; that one has to stay the way it is.*0147

*But since mixed number and improper fractions are both bigger than 1,*0151

*we can actually change them from mixed number to improper fraction and improper fraction to mixed number.*0157

*For you to be able to switch between, let's start with this one--mixed number to improper fraction.*0170

*If I have a mixed number, 2 and 1/2, again that is a mixed number*0178

*because you have a whole number in the front and you have...this is like a proper fraction.*0183

*It is a whole number with a proper fraction.*0188

*Together it is called a mixed number.*0190

*If I want to switch a mixed number to make it look like an improper fraction,*0194

*the first thing I am going to do is take this number on the bottom which is the denominator.*0202

*Multiply it to the whole number; it would be 2 times 2 which is 4.*0208

*Then you are going to add the top number--plus 1.*0218

*Again take the bottom number, 2, multiply it to the whole number, and add it to the top number.*0223

*It is 2 times 2 which is 4; plus 1 is 5.*0229

*That number is going to be the top number of your improper fraction.*0237

*It is going to be the numerator; 5 over... the denominator stays the same.*0242

*The denominator of the improper fraction is going to be the same as the denominator of your mixed number.*0252

*That does not change; 5/2.*0257

*Again the top number, the numerator, is bigger than the denominator.*0260

*We have no whole number.*0265

*From a mixed number, we just change that to an improper fraction.*0268

*Again denominator times the whole number; then add the top number.*0274

*If you are going to go the other way, you are going to go from an improper fraction*0283

*and change it to a mixed number, say I have 10/3.*0286

*We know this is an improper fraction because the top number, the numerator, is bigger than the denominator.*0294

*In this case, I want to see how many times the bottom number can fit into the top number.*0302

*How many times can the bottom number go into the top number?*0310

*I know 3 times 3 is 9.*0315

*That means the 3 fits into 10 three times because 10 is bigger than 9.*0319

*I am going to write that number as my whole number because mixed number again has a whole number.*0328

*You are figuring out the biggest multiple of 3 that fits into 10.*0335

*Again 3 goes into 10 three times which makes it a 9.*0343

*How many do I have left over then?*0349

*If 3 times 3 is 9, but this number is 10, I have 1 left over.*0351

*My leftover is going to be the top number of the fraction.*0358

*Again my denominator has to stay the same; the denominator here is 3.*0365

*The denominator here is going to stay 3.*0370

*Again to change from an improper fraction to a mixed number,*0374

*you are going to see how many times the bottom number will fit into 10.*0378

*It fits in there three times with 1 left over.*0385

*Then denominators both stay the same.*0389

*Here are some examples.*0398

*This is an improper fraction because again the top number is bigger than the bottom number.*0401

*We know this is bigger than 1.*0409

*Since this is an improper fraction, I want to change it to a mixed number.*0413

*I don't have to, but if I want to, I can switch it over.*0420

*In order to switch it, I see how many times the bottom number will fit into the top number.*0424

*How many times does 3 fit into 5?*0430

*3 times 1 is 3; 3 times 2 is 6; but 6 is too big.*0434

*Only one time; that becomes my whole number.*0442

*If 3 fits into 5 one time, how many do I have left over?*0448

*3 times 1 is 3; I have 5 here; my leftover is 2.*0453

*What goes down here as my denominator?--the same denominator.*0461

*From improper fraction, I can change it to this mixed number.*0468

*These mean the same thing.*0472

*This fraction is the same fraction as this one right here.*0474

*You are just writing it in different form.*0477

*This example here, this fraction is... can you guess?*0481

*Good, it is a mixed number because we have a whole number with a proper fraction.*0486

*Here, since this is a mixed number, I can change it to an improper fraction.*0494

*I take my denominator; I multiply it to my whole number.*0501

*Then I add it to my numerator.*0509

*It is 2 times 4 which is 8; plus 1.*0513

*That is 9 over... my denominator stays the same as a 2.*0520

*4 and 1/2 is the same thing as 9/2.*0529

*Here is another example; this right here is a mixed number.*0539

*We can change it to an improper fraction.*0543

*You take the 4, the denominator; you are going to multiply it to the 5.*0547

*Then you are going to add the top.*0554

*It is 20; plus 3 is 23; it is 23.*0556

*Then the denominator stays the same; the denominator is 4.*0564

*5 and 3/4 is the same thing as 23/4.*0571

*Another example, 6/7; the top number is smaller than the bottom number.*0578

*If I look at this and ask myself how many times does 7 go into 6?*0589

*It doesn't go into it at all.*0596

*6 is smaller than 7; 7 doesn't fit into 6.*0597

*This fraction is a proper fraction; this is a proper fraction.*0602

*I can't switch it over to the other types of fractions.*0607

*I can't change this to a mixed number.*0610

*I can't change it to an improper fraction.*0612

*This fraction is smaller than 1; it is called the proper fraction.*0614

*This fraction here, the top number is bigger than the bottom number.*0624

*This is an improper fraction; therefore we can change it to a mixed number.*0629

*Again I ask myself how many times does 8 fit into 19?*0636

*8 times 1 is 8; 8 times 2 is 16; 8 times 3 is 24.*0644

*This number right here is 19; 8 fits into 19 only two times.*0652

*That becomes a whole number.*0662

*If 8 times 2 is 16, how many are left over?*0665

*We do 19; subtract the 16; I have 3 left over.*0671

*My denominator stays the same as an 8.*0677

*19/8 is the same thing as 2 and 3/8.*0683

*This fourth example, this fraction right here is called a mixed number*0693

*because we have a whole number and we have a proper fraction.*0697

*Since I have a mixed number, I can switch this over.*0703

*I can change this to make it look like an improper fraction.*0705

*The first thing I do here is I take the denominator of 5.*0711

*I am going to multiply it to the whole number.*0715

*Then I take that number and add it to the top number, the numerator.*0721

*I do 5 times 8 which is 40.*0726

*I am going to add the top number; it is going to be 44.*0730

*That goes in the numerator of my improper fraction.*0737

*Then the denominator has to stay the same.*0741

*The denominator for this fraction is 5; it is going to stay a 5 here.*0745

*This is an improper fraction because the top number is bigger than the bottom number.*0752

*8 and 4/5 would be the same thing as 44/5.*0760

*Thank you for watching Educator.com.*0767

*Welcome back to Educator.com; this lesson is on connecting decimals and fractions.*0000

*When we connect fractions to decimals and decimals to fractions, let's think about money.*0007

*For fractions and decimals, I can represent each with a dollar.*0017

*If I have half a dollar, 1/2 is half.*0024

*If I have half a dollar, I have fifty cents.*0029

*Keep in mind that 0.5 is the same thing as 0.50 or 50 cents.*0035

*When the numbers are after the decimal place, I can put as many 0s as I want.*0045

*The number does not change.*0051

*I know that half a dollar is the same as 50 cents or 0.5.*0056

*1/2 would be the same as 0.5.*0065

*If I have 1/4 from a dollar, how can I divide the dollar up into 4 parts?*0070

*4 of what makes a dollar?--the answer is quarters.*0082

*We know that 4 quarters make 1 dollar.*0089

*If I have 1 out of 4 quarters, if I have 1 quarter, then I have 25 cents or 0.25.*0093

*How about this one?*0107

*What would I have if I divide the dollar up into 10 parts?*0110

*I would have 1 dime.*0115

*1/10 would be the same thing as having 1 dime which is 0.1 or 0.10.*0118

*Remember this is the same as this.*0131

*How about this one?*0136

*If I divide up the dollar into 100 parts, then I would have 100 pennies.*0139

*Having 1 of those parts, having 1 penny, is going to be 0.01 or 1 cent.*0149

*1/100 is having 1 cent, 0.01.*0160

*Let's look at this again; this fraction right here, 3/4.*0170

*I am going to again relate it to the fraction 1/4; remember 1/4.*0177

*If I have 1 part out of the 4 that I divided the dollar into, then I have 1 quarter which is 0.25.*0183

*This is the value of 1 quarter.*0200

*If you look at this fraction, this is 3/4.*0203

*The dollar is still divided into 4 parts which is still the quarter.*0207

*If I have 3 quarters, how much do I have?*0213

*I have 0.75 or 75 cents.*0218

*If I want to do this mathematically, I can take the 3 and divide it by 4.*0229

*3, this right here means divide; 3 divided by 4.*0237

*That means I can take 3 divided by 4.*0242

*This is a whole number; I have to put a decimal after it.*0249

*Then I can add as many 0s as I want; let's make this longer.*0254

*4 goes into 3 zero times; I am going to raise up this decimal.*0263

*4 goes into 30 how many times?--let's see.*0268

*4 times 7 is 28; subtract this; this is 2; bring down the 0.*0273

*4 goes into 20 five times.*0286

*If I want to convert 3/4, this fraction into a decimal, I can just do 3 divided by 4.*0297

*I am still going to get the same answer of 0.75.*0305

*But if I know that I can take this and think of the dollar,*0309

*and I know I can divide the dollar into 4 parts which is 4 quarters,*0315

*and I have 3 of those parts, then I have 75 cents.*0319

*Right here, 0.7; 0.7 is the same thing as 0.70.*0327

*0.7 is the same thing as having 70 cents; what gives me 70 cents?*0344

*If I have 7 dimes, then I will have 70 cents.*0351

*Then I can say that this is having 7 over... the value of a dime is 10 cents.*0359

*7/10 would be the same thing as 70 cents.*0370

*If I want to just take this without relating it to the dollar*0377

*and just convert it back to a fraction, I can take this number right here, 7.*0382

*I am going count how many numbers I have after the decimal place.*0388

*It has to be after the decimal place; 7, I only have one number.*0392

*Take that number; place it over a 10.*0398

*One 0 because I have only one number after the decimal place.*0405

*So I am only going to have one 0 here.*0409

*Let's go over more examples.*0414

*Convert from fraction to decimal or decimal to fraction.*0417

*Here we have a decimal; I am going to convert it to a fraction.*0421

*If I look at this, if I have 25 cents, I have 1 quarter.*0429

*1 quarter would be having 1 quarter out of... how many quarters equal a dollar?*0438

*4; so having 1 quarter, 25 cents or 0.25, is the same thing as 1/4.*0446

*Another way to do this, I take this number, 25.*0457

*I am going to make that into my numerator.*0465

*Then I am going to count how many numbers do I have after the decimal place?*0468

*I have two numbers; I have one, two.*0475

*I am going to put that many 0s after my 1 as my denominator.*0479

*This becomes 25/100.*0487

*I have to simplify this because this fraction... I know that 25 can go into 100.*0491

*I am going to take this 25; I am going to divide by 25.*0499

*Then I have to divide the bottom number by the same number.*0503

*25 divided by 25 is 1; 100 divided by 25 is 4.*0509

*0.25 is going to be the same thing as 1/4.*0518

*The next example is a fraction 3/10; I can convert this into a decimal.*0527

*Again let's relate it to money.*0536

*If I have 10 as my denominator, then what do I have?*0538

*10 parts to make a dollar would be 10 dimes.*0546

*I know that I am working with dimes here.*0553

*If I have 3 dimes, then how much do I have?*0556

*How can I write that into a decimal?--I would have 30 cents.*0561

*Having 3 dimes is the same thing as 30 cents.*0569

*Or 3/10 is the same thing as 0.30 or 0.3 because again this is the same thing.*0576

*I can put a 0 here because it is after the decimal place.*0585

*Another way you can do this, just divide; 3 divided by 10.*0592

*I have to add 0s at the end of it.*0601

*10 goes into 30 three times; but I have to write it over the 0.*0605

*Then I have to bring up my decimal; this becomes 30.*0612

*If I subtract, I get 0.*0617

*3/10 or 3 divided by 10 is going to be 0.3.*0621

*This next example, 0.77, I am going to convert this into a fraction.*0630

*If I have 0.77 or 77 cents, I have the same thing as 77 pennies.*0637

*Again I can write that number at the top.*0653

*How many pennies are equal to a dollar?*0659

*I know that 100 are equal to a dollar.*0662

*This is the same thing as saying I have 77 pennies.*0665

*0.77 is the same thing as 77/100.*0671

*Again without thinking of money, I can just take this number, 77, as my numerator.*0677

*Count how many numbers I have after my decimal place which is 2.*0685

*That means I am going to add two 0s which is 100 as my denominator.*0690

*It is the same thing.*0697

*This fourth example here, I have a fraction and I am going to convert this into a decimal.*0704

*This top number tells me how many I have.*0712

*This bottom number tells me how many out of 100 I have or how many out of a dollar I have.*0716

*If I have 100 parts to make a dollar, what do I have?*0722

*I have the penny because 100 pennies equals 1 dollar.*0727

*I know that I have 9 of them.*0732

*9 out of 100 would be the same thing as having 9 pennies.*0734

*In order to write this as a decimal, if I have 9 pennies, how much do I have?*0743

*I have 9 cents which is written like that.*0748

*9 out of 100 is the same thing as 0.09.*0753

*You can also think of this as 9 divided by 100.*0761

*You can take 9 divided by 100.*0768

*But since this is 100, I can do a shortcut here.*0771

*Whenever my denominator is 10 or 100 or 1000, any number that is a multiple of 10,*0779

*all I am going to do is take that top number... let's write that out; the top number is 9.*0787

*I am going to count how many 0s I have here.*0794

*I have one; I have two 0s.*0796

*Since I have two 0s, I am going to take that number 2.*0800

*I am going to place this decimal point right there.*0806

*The decimal point always goes after the number.*0812

*I am going to move the decimal point two places because again there are two 0s here.*0815

*If I had a 10, I only have one 0.*0822

*I would only move it one place value.*0826

*If I had 1000, I would have three 0s.*0828

*I would have to move it three place values.*0833

*Again I am going to take this decimal point.*0837

*I am going to move it two place values going to the left.*0839

*I am going to go one and two.*0844

*There is where my decimal point is going to go.*0848

*Then I have a space right here; I have to put a 0 right there.*0852

*For this shortcut, you can only do that if the denominator is a multiple of 10.*0856

*It has to be 10, 100, or 1000, and so on.*0861

*Otherwise you are going to have to do 9 divided by 100.*0866

*Again this sign right here means divide.*0870

*We can do 9 divided by 100 to change that into a decimal.*0872

*But I also knew that this bottom number right here, it takes 100 pennies to make a dollar.*0878

*Since it takes 100 pennies to make a dollar and I have 9 of them,*0888

*I would have 9 cents; I can just write that as 0.09.*0892

*That is it for this lesson; thank you for watching Educator.com.*0898

*Welcome back to Educator.com.*0000

*This lesson, we are going to add and subtract fractions with common denominators.*0002

*Right here, this is a fraction; the top number, 1, is called the numerator.*0013

*This number right here is called the numerator.*0022

*This bottom number, the 2, is called the denominator.*0029

*When we add fractions, I have 2/6 plus 5/6; I am adding two fractions together.*0042

*Let's look at their denominators.*0050

*The denominator for this fraction is a 6; the denominator for this fraction is a 6.*0052

*Once the denominators are the same, now they have a common denominator, then I can add the fractions.*0059

*This 2/6 plus 5/6, I am going to add the numerators together.*0068

*The numerator for this fraction is 2; the numerator for this fraction is a 5.*0075

*2 plus 5; that is going to become my numerator for my answer*0080

*For my denominator, it is going to stay the same as a 6.*0088

*2/6 plus 5/6 is 7/6.*0096

*Again the denominators for each fraction has to be the same.*0100

*Then I take my numerators; I add them together; 2 plus 5 is 7.*0105

*I do not add my denominators; my denominator has to stay the same as 6.*0111

*2/6 over 5/6 is going to equal 7/6.*0116

*Here we are subtracting these fractions, 7/8 minus 1/8.*0124

*The denominator is the same; here is an 8 here.*0130

*The denominator for this fraction is an 8.*0133

*Therefore I can go ahead and subtract them.*0136

*I am going to do 7 minus 1 which is 6.*0139

*I take my denominator; that is going to stay the same.*0147

*Do not subtract your denominators; it is 7 minus 1 which is 6.*0150

*My denominator must stay the same as 8; 7/8 minus 1/8 is 6/8.*0156

*My next example, 9/10 plus 3/10.*0166

*I am going to take my numerators, add them together.*0174

*It is 9 plus 3 which is 12 over... do not add your denominators.*0177

*It is going to stay the same as a 10.*0186

*9/10 plus 3/10 is going to equal 12/10.*0189

*Again 11/20 plus 9/20, same denominator.*0199

*I take 11, add it 9; I get 20 over... 20 plus 20?*0206

*No, you do not add them together; the denominators are 20 here.*0216

*The denominator for this one has to also be 20.*0221

*Let's look at this fraction right here; my answer is 20/20.*0225

*20/20, if the numerator and the denominator are the same, this is equal to 1.*0231

*My answer would just be 1.*0240

*The fourth example when we are adding and subtracting fractions, 23/95 plus 6/95.*0247

*Whenever I add or subtract fractions, I have to make sure that the denominators for both fractions are the same.*0258

*In this case, the denominator is 95; for this one, the denominator is 95.*0265

*Since they are the same, I can go ahead and add the fractions together.*0270

*I take the numerators which is 23 and 6; I am going to add them together.*0276

*23 plus 6 is 29.*0282

*For my denominator, denominator here is 95; here is 95.*0287

*For my answer, the denominator also has to be a 95.*0293

*You do not add the denominators together; the denominator stays the same as 95.*0298

*23/95 plus 6/95 is going to equal 29/95.*0304

*That is it for this lesson; thank you for watching Educator.com.*0313

*Welcome back to Educator.com.*0000

*This lesson, we are going to add and subtract fractions with different denominators.*0002

*Before we begin with that, let's review over the lesson on least common multiple, the LCM.*0013

*This I believe was a few lessons ago.*0022

*If you want, you can go back to that lesson.*0025

*We are just going to do a brief example here.*0028

*To find the LCM of 6 and 4, I am going to take the two numbers.*0031

*I am going to do the factor tree method on each of them.*0041

*For 6, a factor pair of 6 is going to be 2 and 3.*0046

*They are both prime; I am going to circle them both.*0053

*For 4, it is going to be 2 and 2; I am going to circle them.*0056

*Here, to find the LCM, I am going to look at what they have in common.*0072

*I know that I have a 2 here; I also have a 2 here.*0082

*I am going to write that 2 by itself.*0087

*This pair cancels out one of them; I am going to write a 2.*0093

*The other numbers, 3 and 2, the other remaining numbers are going to go tag along with it.*0098

*Again to find the LCM, I just find what they have in common.*0112

*Since there is a 2 here and a 2 here, one of those 2s get cancelled.*0119

*It is going to be 2 times 3 times 2.*0124

*2 times 3 is 6; 6 times 2 is 12; my LCM is 12.*0130

*The LCM of 6 and 4 is going to be 12.*0139

*LCM is the same thing as LCD.*0147

*LCM stands for least common multiple; LCD is least common denominator.*0154

*When I am using those two numbers as my denominators, then it is going to be called LCD.*0162

*But I am still going to find the LCM between those two denominators.*0170

*The reason for this, whenever I add fractions or subtract fractions,*0174

*I have to make sure that these denominators are the same.*0179

*In order to make them the same, I need to find the LCM or the LCD between the two numbers.*0184

*Here 6 and 4, just like what we did, the example, we know that the LCM is 12.*0193

*The LCM, 1/6, what I am going to do is I am going to make 1/6 the same fraction with the denominator becoming 12.*0204

*Same thing here, 3/4, the denominator is going to change to 12.*0221

*I want to figure out what these top numbers are going to be, my numerators.*0229

*How do I go from a 6 to a 12?--what do I multiply it by?*0235

*I multiplied this by 2; or I can do 12 divided by 6; I get 2.*0240

*Since I multiplied the 6 by 2 to get 12, I need to also multiply the top number by 2.*0250

*1 times 2 is 2; the fraction 1/6 became 2/12.*0258

*These are the same fractions; 1/6 is the same thing as 2/12.*0267

*Same thing here, 3/4; to go from 4 to 12, I have to multiply it by a 3.*0274

*Then I have to multiply the top number by 3; 3 times 3 is 9.*0285

*3/4, because I multiplied the top and the bottom by the same number, these fractions become the same.*0294

*They are the same fraction; 3/4 is the same thing as 9/12.*0304

*Now since I know that 2/12 is the same thing as 1/6 and 9/12 is the same thing as 3/4,*0310

*I can add these two fractions, this fraction and this fraction.*0318

*If I add these two, then my answer will be the same as if I add these two.*0330

*I have to do that because these denominators are different.*0337

*I have to make them the same by converting these fractions,*0341

*by changing these fractions so that the denominators will be the same.*0344

*Now that they are, I am going to take my numerators and add them together.*0351

*2 plus 9 is 11; here the denominator is 12; 12.*0357

*Then my denominator here has to stay the same as a 12.*0365

*2/12 plus 9/12 is 11/12; or I can say that 1/6 plus 3/4 is 11/12.*0369

*Let's do another example; here I am going to subtract.*0382

*But before I do that, I have to check my denominators.*0388

*This denominator is an 8; this one is a 4; they are different.*0391

*I have to find the LCD or the LCM between 8 and 4 so that I can make the denominators the same.*0397

*I am going to take 8 and 4; you could do the factor tree.*0407

*4 and 2; circle that one; that is a prime.*0419

*2 and 2; this is 2 and 2; look at this one.*0424

*There is a common one here; I am going to cancel out one of them.*0433

*I have another common between these two so I am going to cancel out one of them.*0438

*Then I just multiply all the remaining circled numbers.*0442

*It is 2 times 2 times 2 which is 8.*0447

*If I look at these, I can just look at them and figure out what the LCM is by looking at the multiples.*0455

*Multiples of 8 would be 8, 16, 24, and so on.*0464

*For 4, it would be 4, 8, 12, 16, and so on.*0469

*You are going to find the smallest common multiple between them which is 8.*0473

*Here I am going to change this fraction and this fraction so that their denominators will be the same.*0480

*For this fraction, 7/8, my LCM is already 8.*0489

*My LCM or my LCD, it is already 8.*0494

*For that one, I can just keep it the way it is.*0498

*For this one however, 1/4, I have to convert it; I have to change it.*0504

*I need a top number; 4; to get 8, I multiply it by 2.*0515

*Again I have to multiply the same number to the top which is 2.*0523

*Whenever you are converting fractions, as long as you multiply the top*0530

*and the bottom by the same number, then your fraction will stay the same.*0533

*Even if you change the numbers, it is still the same fraction; 1/4 became 2/8.*0538

*Now I am going to rewrite my problem, 7/8 minus 2/8.*0548

*Make sure the denominators are the same.*0556

*If they are not the same, then you did something wrong.*0558

*Go back and check your work.*0561

*Since they are the same, I can go ahead and subtract them.*0565

*7 minus 2 which is 5; then my denominator, 8.*0568

*8 here; it stays an 8 there; 7/8 minus 1/4 is going to equal 5/8.*0576

*Let's add this next problem, 9/10 plus 3/15.*0591

*Again I have to check my denominators; they are not the same.*0599

*I have to find the least common denominator with them.*0602

*I am going to take 10; do the factor tree which is 5 and 2.*0606

*Circle them if they are prime; only circle them if they are prime.*0612

*Then 15, this becomes 5 and 3.*0616

*If you are confused about how to find the LCD or LCM,*0625

*then you can go back and look at the lesson on that one before continuing.*0630

*My LCM or I am just going to call it the LCD since they are my denominators.*0636

*I look for any common numbers between them; they have a 5; 5 is common.*0642

*Whenever they have something in common, just cancel one of them out.*0650

*That is all they have in common.*0654

*Then for my LCD, I am just going to write out the remaining circled numbers.*0656

*Remember they can only be circled; 5 times 2 times 3.*0662

*5 times 2 is 10; 10 times 3 is 30; my LCD is going to be 30.*0667

*I have to change this fraction so that my denominator will become 30.*0676

*Same thing here, change this fraction so my denominator will be 30.*0681

*9/10, going to convert it; I can take 30 divided by 10; that is 3.*0688

*I know that I did 10 times 3 to get 30.*0702

*Again you have to do it to both the top and the bottom, the same number.*0707

*That is the only way you are going to have the same fraction because you don't want to change your fraction.*0711

*Even if you are changing the numbers, it is still the same fraction.*0715

*9 times 3 is 27.*0720

*I am going to do the same thing for the other fraction.*0727

*15 times 2 was 30; 3 times 2... again multiply it by the same number.*0733

*It is going to be 6.*0742

*Since 9/10 is the same thing as 27/30 and 3/15 is the same as 6/30, I need to add my new fractions.*0747

*Again double check your denominators; make sure they are the same.*0764

*It is going to be 27 plus 6.*0769

*27 plus 6 is 33 over... your denominator will stay the same.*0772

*It is 33/30; let's look at this fraction.*0785

*This is your answer; this is a solution to this problem.*0788

*But I have an improper fraction because the top number, the numerator, is bigger than the denominator.*0793

*You can either leave it like this; this is still the correct answer; or I can simplify it.*0800

*I know that a 3 goes into 33 and a 3 goes into 30.*0811

*I can take that number, the common number, the common factor between 33 and 30,*0819

*divide it to both the top and the bottom.*0828

*Remember as long as you are doing the same thing to the top and to the bottom of the fraction,*0831

*you are not changing it; you are just simplifying it.*0835

*33 divided by 3 is 11; 30 divided by 3 is 10.*0839

*This is your new improper fraction, 11/10.*0850

*Since it is an improper fraction, we can change it to a mixed number.*0854

*Or we can just leave it like that; that is fine too.*0858

*But if I do want to change it to a mixed number,*0861

*then this 10 fits into the top number 11 only one time.*0864

*10 fits into 11 only one time.*0873

*How many left over do I have?--only one.*0877

*My denominator always has to stay the same.*0881

*11/10 is the same thing as 1 and 1/10.*0884

*Another example, we are going to take 11/20 and subtract it to 11/30.*0896

*My denominators are different; I have to find the common denominator.*0903

*I can take 20; 5 is a prime number; I am going to circle it.*0910

*4, 2, and 2; I circle those; and then 30.*0918

*For this one, I can either do 3 and 10 or I can do 15 and 2, any factor pair.*0927

*Let's do 3 and 10; here 3 is a prime number; I am going to circle it.*0933

*10 is 5 and 2; they are both prime.*0939

*I am going to look for any numbers they have in common.*0945

*Here; I have a 2 here; and I have a 2 here.*0949

*I am going to cancel one of them out.*0954

*Here I have a 5; and I have a 5 here.*0957

*I am going to cancel just one of them out.*0960

*Any others?--nope, that is it.*0963

*My LCD or my LCM is going to be 2 times 2 times 5 times 3.*0967

*This is going to be 4 times 5 which is 20, times 3 which is 60; my LCD is 60.*0981

*Then my next step is going to be to change each fraction so that their denominator will become 60.*0993

*20, to figure out what you have to multiply to 20 to get 60,*1004

*I can just take 60 and divide it by 20.*1009

*This is going to be 3; 20 times 3 was 60.*1014

*Again you have to multiply the top number by the same number.*1019

*11 times 3 is 33.*1022

*For the second fraction, 11/30, 30 times 2 is 60.*1028

*Multiply the top number by that number; 22.*1040

*11/20 is the same thing as 33/60; I am going to subtract.*1047

*Then 11/30 is the same thing as 22/60.*1055

*Again double check your denominators; make sure that they are the same.*1061

*Since they are, now I can subtract; 33 minus 22 which is 11 over...*1068

*Keep your denominator the same; do not add or subtract your denominators.*1078

*11/20 minus 11/30 became 11/60.*1085

*Let's do another example; this example, 23/95 plus 4/5.*1093

*In order for me to add these two fractions, I have to make sure they have a common denominator.*1104

*In this case, they don't; 95 is this denominator; 5 is the other one.*1108

*I have to look for the common denominator.*1115

*For 95, I can either look for the LCM, the least common denominator or least common multiple, between 95 and 5.*1122

*Or I can list all the multiples out and see the smallest common multiple.*1134

*I know that 95 is divisible by 5 because any number that ends in a 5 or 0 is divisible by 5.*1143

*In this case, a 5, if this number is divisible by this number,*1156

*then this becomes the new common denominator, the least common denominator.*1162

*Or if you want to just do the factor tree to find the least common denominator, then you can do that too.*1167

*95 is going to be 5 times 19; these are both prime numbers.*1175

*I am going to circle them; 5 is just 5 and 1.*1189

*To find the LCD, I am going to look for any factors they have in common.*1197

*Here, there is a 5 here and a 5 here.*1207

*I am going to cancel only one of them out.*1210

*Whenever they have something in common, just cancel only one of them out.*1212

*Then I am going to write all the circled numbers again; 5 times 19.*1217

*This is just a 1 so I don't have to write that.*1225

*5 times 19 I know is 95.*1227

*My LCD, my least common denominator, is going to be 95.*1232

*For this fraction here, since the denominator is already 95, I don't have to change it.*1239

*This one can stay as it is.*1246

*This one however, I have to change that 5 to make it a 95 so they will have a common denominator*1250

*because that is the only way I can add these fractions, if their denominators are the same.*1256

*For this fraction right here, I need to change it so that the denominator will become 95.*1260

*I am going to take this 95, divide it by 5 to see what I have to multiply this by.*1271

*That is 19; here I am going to take this and multiply it by 19.*1281

*This will become 76; 4/5 became 76/95.*1296

*Make sure you multiply it by the same number.*1311

*You have to multiply the top and the bottom number by the same number.*1314

*That way you are not changing the fraction.*1318

*You are just changing the numbers; but they are still equal fractions.*1321

*Now I am going to do 23/95 plus 76/95.*1326

*Again I have to make sure the denominators are the same.*1340

*If they are not the same at this point, then there is something wrong.*1344

*Go back and check your work.*1347

*But since they are the same, I can go ahead and add the fractions.*1350

*23 plus 76, I am going to add the numerators together.*1354

*If I add them, it is going to be 99.*1358

*Here denominator stays the same; it is 95 here; 95 here.*1365

*My denominator is going to become 95; 23/95 plus 4/5 is 99/95.*1372

*That is it for this lesson; thank you for watching Educator.com.*1385

*This lesson, we are going to be adding and subtracting mixed numbers.*0000

*Remember a mixed number is a fraction with a whole number in the front.*0006

*If I have a whole number with a proper fraction, then I have a mixed number.*0012

*When I am adding mixed numbers together, the main thing here is the fraction.*0018

*We are look at fractions here.*0024

*Whole numbers, 3 and 1, we can just add them together.*0026

*For the wholes, we have 4 wholes.*0032

*Then we have the fractions that we have to worry about.*0036

*In this case, I am going to just add the whole numbers and I am going to add the fractions.*0040

*3 plus 1, the whole numbers, that is going to become my new whole number.*0047

*Then 3/5 plus 1/5.*0054

*Again from the last few lesson in adding fractions, we have to make sure that they have a common denominator.*0059

*In this case, this fraction and this fraction both have a denominator of 5.*0068

*We can go ahead and add those fractions.*0073

*It is going to be 3 plus 1; I am adding the numerators; 3 plus 1 is 4.*0075

*My denominator stays the same as always as a 5.*0081

*This mixed number plus this mixed number equals this mixed number.*0088

*When you have your answer, when you find the answer,*0095

*you have to make sure that this fraction here is a proper fraction.*0098

*Meaning the top number must be smaller than the bottom number.*0104

*If that is the case, then that is your answer, 4 and 4/5.*0110

*Let's do a few problems; we are adding 2 and 1/2 plus 2 and 2/3.*0118

*Again I am going to add the whole numbers together, this whole number and this whole number.*0129

*That is going to become 4.*0136

*Then I am going to add my fractions together, 1/2 plus 2/3.*0139

*But there is a problem; our denominators are different.*0145

*Whenever you have fractions with different denominators,*0150

*then you can't add them or subtract them until you make the denominators the same.*0155

*In order to make the denominators the same,*0162

*you have to look for the least common denominator or the least common multiple.*0165

*Between 2 and 3, the least common denominator will be 6.*0170

*The multiples of 2 would be 2, 4, 6, 8, so on.*0178

*For 3, 3, 6, 9, and so on; the least common multiple is 6.*0188

*I have to change these fractions.*0203

*I have to convert the fractions so that the denominators will become a 6.*0206

*1/2, I am going to multiply this 2 by 3 to get a 6.*0214

*2 times 3 is 6.*0226

*Whatever I do to this part, I have to do to the top.*0228

*1 times 3 is 3.*0233

*Again you have to multiply the top and the bottom by the same number.*0235

*If you don't, then it is going to be wrong because these fractions have to stay the same.*0240

*All you are doing is changing the numbers, but it is still the same fraction.*0246

*Then we have to do this next one, 2/3.*0253

*2/3, I have to change this so the denominator will become a 6.*0258

*3 times 2 became 6.*0264

*I have to multiply the top number by the same number.*0269

*It is going to be 4.*0272

*This mixed number could be the same thing as 2 and 3/6.*0276

*This one can change to 2 and 4/6.*0286

*They might look different; but they are the same thing.*0294

*This problem is the same problem as this one as long as you did everything correctly.*0297

*All you did was just change the fractions so that their denominators would be the same.*0301

*Again I am going to add the whole numbers.*0309

*It is 2 plus 2 which is 4.*0312

*Then since their denominators are the same for the fractions, I can add them.*0317

*It is going to be 3 plus 4 which is 7 over...*0322

*The denominator always stays the same; it is going to be 6.*0328

*Let's look at this answer right here; I have a problem.*0333

*Because I have 4 and 7/6, remember my mixed number, this is supposed to be a mixed number.*0339

*The mixed number has to be a whole number with a proper fraction.*0346

*But since my numerator is bigger than my denominator, this is actually an improper fraction.*0351

*I have to change this so that this will no longer be an improper fraction.*0359

*Let's just look at just this part right here, 7/6.*0366

*7/6, since it is an improper fraction, we can change this so that it becomes a mixed number.*0371

*Remember I ask myself how many times can 6 fit into the top number 7?*0381

*I know that 6 can only fit into 7 one time.*0389

*If 6 fits into 7 one time, how many do I have left over?*0395

*I only have 1 because I have 7; 7 minus 6 would be 1.*0400

*Again my denominator stays the same.*0408

*This right here, 7/6, became 1 and 1/6.*0412

*But then again I have a 4 right here; I have another whole number, 4.*0418

*I have a whole number 4; I have a whole number of 1.*0422

*Since this 4 and 7/6 is the same thing as 4 plus 7/6,*0429

*I can just take this whole number and add them together.*0442

*This will become 5 and 1/6.*0451

*Again if your answer, your mixed number, has an improper fraction, you have to take out the whole number,*0458

*change your improper fraction into a mixed number, and then add your whole number to it.*0469

*7/6 became 1 and 1/6.*0478

*I have a whole number 4 that I have to consider.*0481

*I am going to add that 4 to that mixed number.*0484

*It is going to be 5 and 1/6; that is my answer.*0487

*Another example here, I have 6 and 5/7 minus 2 and 1/5.*0495

*The main problem here is my fraction.*0505

*I have to look at my fractions, 5/7 and 1/5; my denominators are different.*0507

*I have to make sure to make a common denominator.*0516

*Normally you can do the factor tree to find the least common denominator.*0523

*But for 7 and for 5, they are both prime numbers.*0527

*If they are both prime numbers, then you can just list out all the multiples*0531

*or the first few until you find the common multiple--7, 14, 21, 28, 35.*0536

*For 5, the multiples are 5, 10, 15, 20, 25, let me continue right here, 30, 35.*0553

*I found one, 35.*0568

*You have to make sure when you find the least common denominator that it is the smallest one.*0570

*They are going to have more than one common denominator or common multiple.*0575

*You just have to make sure it is the least common multiple.*0580

*It is the smallest common multiple; in this case, it is 35.*0583

*I am going to change just this fraction.*0590

*I am going to ignore my whole number for now.*0592

*I am just going to change the fractions.*0596

*5/7, I want to make the denominator become 35.*0599

*7, what did I multiply by 7 to get 35?--multiplied 5.*0607

*Then I have to multiply this top number by 5; this becomes 25.*0614

*Then I am going to look at this fraction right here, 1/5.*0621

*5 times 7 became 35; multiply this top number by 7; get 7.*0630

*I am going to rewrite my mixed numbers, my problem, so that I will have common denominators for each of these.*0639

*This becomes 6 and 25/35 minus 2 and 7/35.*0648

*Now that I have common denominators, I can go ahead and subtract these two fractions.*0666

*Let's do 6 minus 2; the whole number is 6 minus 2 which is 4.*0673

*Then I can take my numerator here, subtract it by this numerator.*0680

*25 minus 7 is 18; my denominator stays the same as 35.*0685

*My answer here is 4 and 18/35.*0700

*Again you have to look at this fraction right here, this mixed number.*0707

*Make sure that this is a proper fraction.*0710

*Your top number, your numerator, is going to be smaller than the denominator.*0712

*This next example, 3 and 3/4 plus 4 and 1/10.*0722

*Again I have to look at my fractions because I can't add them until I have a common denominator.*0729

*In this case, you can either, just like the other examples, to find the LCD, you can list out their multiples.*0738

*Since these are not prime numbers, you can do factor trees.*0746

*I am just going to do the factor tree; this is 2 and 2.*0751

*For 10, it is going to be 5 and 2.*0759

*There is a common number of 2; cross one out.*0765

*My LCD, my least common denominator... I am going to write out all the remaining circled numbers.*0771

*2 times 2 is 4; times 5 is 20; my LCD is 20.*0783

*I am going to change these two fractions so that my denominators will become 20.*0792

*3/4; 4, I multiplied it by 5 to get a 20.*0797

*I am going to do the same thing to the top, 15.*0806

*The other fraction, 1/10; 10 times 2 became 20; 1 times 2 is 2.*0813

*Again this fraction 3/4 is the same thing as 15/20 and 1/10 is 2/20.*0825

*I am going to rewrite this problem, 3 and 15/20 plus 4 and 2/20.*0833

*I add my whole numbers together; it is going to be 7.*0851

*Then I have to add my fractions.*0856

*They have a common denominator so I can add them together.*0859

*I am going to take my numerators, 15 and 2; add them up.*0862

*I get 17 over... guess what my denominator is going to be?*0866

*20, your denominator has to stay the same.*0873

*I look at this answer.*0877

*Is my top number in my fraction smaller than my bottom number?*0879

*If it is, then it is a proper fraction.*0883

*This is going to be my answer, 7 and 17/20.*0887

*This next example, 12 and 9/11 minus 12 and 3/22.*0895

*Before I begin here, I have to make sure that the denominators for these fractions are the same.*0904

*I am looking at this fraction here and this fraction here.*0910

*Here I have an 11 as my denominator; this one, I have a 22.*0916

*I have to change these denominators so that they are the same in order for me to be able to subtract these fractions.*0922

*I need to find the least common denominator.*0930

*I am going to write out the multiples of 11--11, 22, 33, and so on.*0935

*For 22, it is going to be 22, 44, and so on.*0944

*My least common multiple is 22.*0952

*Since they are denominators, it becomes the least common denominator.*0961

*Make sure, if you are going to list out the multiples to find the LCD,*0965

*then you have to find the one that is smallest.*0969

*It has to be the smallest common multiple because these two numbers,*0973

*they are going to have more than one common multiple.*0976

*It has to be the smallest one.*0979

*Now that I have my LCD, I have to make sure that these fractions*0983

*will be converted so that I will have my denominator as 22.*0994

*This fraction right here, 9/11... I know that I have whole numbers here.*0999

*But I am just going to worry about my proper fractions first.*1004

*9/11, I want to make that denominator 22.*1009

*I take this number, 22, divide it by 11.*1017

*Or I can just figure out 11 times 2 gave me 22.*1020

*Whatever I do to that number, I have to do to the top number.*1026

*I have to multiply the top number by 2 as well.*1030

*9 times 2 is 18.*1034

*This next fraction, 3/22, the denominator is already 22.*1039

*We don't have to change it; we can just keep it the way it is.*1045

*I know that 9/11 is the same thing, is the same fraction as 18/22.*1050

*I can just rewrite this whole problem so that they will have common denominators.*1059

*12 and 18... let me erase that.*1069

*12 and 18... it has to change to this fraction right here.*1076

*12 and 18/22 minus 12 and 3/22; again double check your denominators.*1082

*Make sure that they are the same; then we can go ahead and subtract those.*1094

*For this, my whole numbers, 12 minus 12, is going to be 0.*1101

*Then I don't have to worry about my whole numbers for now.*1109

*I have to subtract my numerators; 18 minus 3 is 15.*1113

*My denominator again, it is the same as 22.*1124

*My denominator for my answer has to also stay the same; it is 15/22.*1129

*I don't have a whole number because 12 minus 12 gave me 0.*1136

*I don't have a whole number here.*1140

*Here I have to make sure when I have my fraction that this top number is smaller than the bottom number.*1144

*This is a proper fraction.*1150

*I can't simplify it because 15 and 22 do not have any common factors.*1154

*There is no number that can go into both the top number and the bottom number.*1160

*Once I ask myself all those questions and I can't simplify, then this would be my answer.*1165

*12 and 9/11 minus 12 and 3/22 became 15/22.*1173

*That is it for this lesson; thank you for watching Educator.com.*1181

*Welcome back to Educator.com.*0000

*This lesson is on multiplying fractions; that includes mixed numbers.*0002

*If you are multiplying fractions and you do have mixed numbers,*0009

*then make sure to change them to improper fractions first.*0013

*I am going to do a few examples here.*0018

*But if you don't remember how to do that, just go back*0020

*to the previous lesson on switching between mixed numbers and improper fractions.*0023

*Once all of your fractions are either proper fractions or improper fractions,*0029

*then you are going to take the numerators which are the top numbers, multiply them together.*0034

*If we have A/B, A/B which are variables, they are going to represent numbers.*0042

*If this is a fraction, a number over number, times another fraction, C/D,*0050

*then you are going to take this top number, multiply it to this top number.*0057

*You get AC.*0062

*Then you are going to take your denominator, B, and multiply it to the other denominator, D.*0064

*It becomes AC/BD.*0069

*Don't get confused between multiplying and adding and subtracting fractions.*0073

*Remember when we add or subtract fractions, you have to make sure that these denominators are the same.*0078

*For your answer, your denominator is going to be the same as well.*0086

*But in this case when you are multiplying fractions, you are just going to multiply the denominators together.*0090

*Here are some examples, 2/3 times 3/4.*0099

*2/3, that is not a mixed number; 3/4 is not a mixed number.*0107

*I can go ahead and multiply them.*0112

*2/3... I am just going to write it out again... 3/4.*0116

*You can do 2 times 3 which is 6 over... 3 times 4 which is 12.*0123

*6/12, they have a common factor; they have a common factor of 2.*0132

*Or the greatest common factor, the biggest factor that they have in common, is 6.*0141

*I can just divide the top number by 6 and then divide the bottom number by 6.*0146

*This is going to become 1/2; 1/2 is going to be the answer.*0153

*Another way when you have the problem like this,*0160

*I can go ahead and simplify straight from here and get my answer as 1/2.*0163

*I can do what is called cross cancelling.*0172

*If I have a 3 down here and a 3 up there, then I can cross this out because they are the same.*0175

*They can cancel each other out.*0186

*2/4, there is a common factor between 2/4 which is 2.*0187

*I can just divide this by 2; that becomes a 1.*0193

*4 divided by 2 becomes a 2; I can simplify it that way.*0199

*Make sure if you are going to simplify two numbers,*0204

*one of the numbers has to be on the top, one of the numerators.*0206

*Another one has to be on the denominator.*0211

*It doesn't matter if one is on the top up here and the other one is a denominator here*0213

*or one is numerator up here and then the other one is a denominator here.*0217

*As long as one number is the numerator and the other number is the denominator, you can cross cancel them.*0222

*If you multiply 1 times... this is just a 1.*0231

*Then 3 cancels out to make a 1; 1 times 1 is 1.*0234

*1 times 2 is 2; either way you get the same answer.*0240

*In the other example here, I have 1 and 2/5 times 2 and 3/5.*0246

*These numbers, these fractions, are mixed numbers; I have a whole number in the front.*0253

*I have to change this fraction from a mixed number to an improper fraction, 1 and 2/5.*0259

*If I want to change this to an improper fraction, I take my bottom number, my denominator.*0275

*Multiply it to my whole number; it is going to be 5 times 1.*0281

*Then add the numerator; 5 times 1 is 5; plus 2 is 7.*0285

*This becomes 7/5; I have another mixed number here, 2 and 3/5.*0291

*I multiply my denominator to my whole number and then add my numerator.*0304

*5 times 2 is 10; I add the 3; that is 13; 13/5.*0308

*Instead of multiplying this number, the mixed number, I am going to multiply 7/5 times 13/5.*0321

*This is a 5 right here; this is a 5 right here.*0337

*I can't cancel those out because those are both numerators.*0340

*Remember if I want to cancel or simplify numbers, one of them has to be the denominator.*0343

*Another one has to be a numerator; in this case, they are both denominators.*0349

*5 to 13, nothing can simplify; I can't cancel anything out.*0358

*I am just going to go ahead and multiply my numerators.*0364

*7 times 13 is... you can use your calculator if you want.*0367

*7 times 13 is 91 over... 5 times 5 is 25.*0376

*If you look at this, they are not going to have any common factors, 91/25.*0383

*That is an improper fraction, but you can just leave it like that.*0390

*This will be my answer, 91/25.*0393

*Another example, we have 6/5 times 3/4.*0400

*6/5... let me rewrite this problem.*0406

*6/5 is an improper fraction because we know that 6 is bigger than the denominator.*0413

*This is an improper fraction.*0422

*This is a proper fraction because the numerator is smaller than the denominator.*0423

*It doesn't matter; we can still multiply it the same way.*0429

*From here, I can either just multiply it out.*0434

*If you don't feel like checking to see if numbers can cancel or can reduce,*0439

*you can just multiply and then simplify your answer.*0447

*Or you can see if you can simplify any of these numbers.*0451

*5 and 3, they have no common factors; the greatest factor is 1.*0458

*We have to leave those numbers.*0463

*But 4 and 6; 4 and 6 have the common factor of 2.*0466

*I can divide each number by 2 and just simplify that way.*0472

*4 divided by 2 is 2; 6 divided by 2 is 3.*0477

*When you cross cancel numbers, you have to make sure you are going to divide by that same number.*0484

*4 and 6, I have to divide by 2 to both numbers.*0490

*4 divided by 2; it becomes 2; 6 divided by 2; that becomes 3.*0494

*3 times 3 is 9; 5 times 2 is 10.*0504

*9/10, there are no common factors besides 1; that is my answer.*0512

*The next problem is 4 and 1/4 times 5/3.*0521

*This is a mixed number; I need to change that.*0527

*4 times 4, 16; add the 1; that is 17; 17/4 times 5/3.*0530

*Again here I need to check to see if any numbers can cancel out.*0548

*I can't because 17 and 3 have no common factors.*0555

*5 and 4 have no common factors besides 1 of course.*0559

*I can just multiply 17 times 5.*0563

*If you want to do that on the side, let's just do a little multiplication right here.*0567

*7 times 5 is 35; 5 times 1 is 5; plus 3 is 8.*0573

*Again when you have a double digit times a single digit, you are just going to do 7 times 5.*0584

*You are going to put that number, the 3... because it is 35.*0590

*The 3 goes up there; the 5 goes down here.*0594

*Then 5 times this number, 5; you are going to add this number right here.*0597

*5 times 1 plus 3; it becomes 8.*0603

*17 times 5 is 85 over... 4 times 3 is 12; this is my answer.*0606

*Some more examples; I want you guys to try to do these problems on your own.*0620

*You can just pause the video; look at these problems, write them out, and try them.*0628

*After you are done, you can play it again and just check your answers that way.*0636

*This problem, 7/8 times 3 and 2/9.*0641

*7/8 is a proper fraction; and we have a mixed number, 3 and 2/9.*0646

*Times... I need to change this.*0655

*It is 9 times 3 which is 27; plus 2 is 29; 29/9.*0657

*Remember the denominator has to stay the same.*0666

*When you are changing it from a mixed number to an improper fraction, you are going to keep the denominator.*0669

*7 and 9, do they have any common factors?--no.*0680

*8 and 29?--no, they have no common factors.*0684

*You are going to do 7 times 29; let's do that right here.*0689

*29 times 7; 9 times 7 is 63; the 6 goes up here.*0692

*The 3 goes down here below lined up with the 9 and the 7.*0702

*7 times 2 is 14; plus 6 is 20.*0707

*There is no number to carry over.*0712

*You are just going to write both numbers down here, 20; it becomes 203.*0714

*203; 8 times 9 is 72; that is your answer.*0720

*This is an improper fraction; you can leave it like that.*0732

*Or you can change it to a mixed number if you would like.*0734

*If you would like to do that, you would just take the 72.*0738

*See how many times it will fit into 203; you can do that by dividing.*0743

*You could just do 203 divided by 72 and see how many times it will go into there.*0748

*Then you find how many leftovers you have.*0755

*The leftover becomes your numerator; 72 becomes your denominator.*0759

*But otherwise just leave it like this, improper fraction.*0764

*The next example, 5/4 times 9/11.*0768

*Since we are multiplying these fractions, they are both... *0781

*5/4 is improper; 9/11 is a proper fraction; we can go ahead and multiply that*0787

*Let's see; can we cancel anything?*0794

*5 and 11, they have no common factors besides 1.*0795

*4 and 9 have no common factors besides 1.*0799

*5 times 9 is 45; 4 times 11 is 44; that is an improper fraction.*0806

*But we can still leave it like that; that is our answer, 45/44.*0820

*The last couple of examples, we have 3 and 3/7 times 6 and 1/4.*0828

*These are both mixed numbers.*0836

*Remember when we multiply fractions, we have to make sure that the fractions are not mixed numbers.*0838

*We have to change these mixed numbers to make it an improper fraction.*0846

*3 and 3/7, to change this to a mixed number, again I take the denominator of 7.*0852

*I am going to multiply it to my whole number.*0861

*It is going to be 7 times 3 which is 21.*0864

*Then you are going to add the numerator, plus 3.*0867

*It is 7 times 3 is 21; plus 3 is 24.*0871

*You are going to keep the same denominator of 7.*0879

*3 and 3/7 becomes 24/7; they are the same fraction.*0883

*But you are changing it from a mixed number to improper fraction.*0888

*We have to do this one because that is also a mixed number.*0894

*6 and 1/4, I am going to multiply my denominator to my whole number.*0898

*4 times is 6 is 24; add the 1; this is 25/4.*0904

*Now that I converted this to an improper fraction and this one as well, I can now multiply those fractions.*0913

*It becomes 24/7 times 25/4.*0921

*From here, you can just multiply your numerators.*0934

*24 times 25; get your answer; then 7 times the 4; get your denominator.*0940

*Before you do that, you can just see if you can cross cancel any numbers.*0950

*7 with 25, one has to be denominator; the other number has to be the numerator.*0956

*7 and 25, do they have any common factors?*0964

*I know that all the factors of 7 are just 1 and 7.*0969

*It is a prime number; in that case, no.*0972

*They do not have any common factors besides 1.*0975

*4 and 24, 4 and 24, they are both even numbers.*0979

*I know that since the factors of 4 are 2 and 4 and 4 goes into 26, I can reduce these numbers by 4.*0988

*I am going to take these two numbers and divide both by 4.*1000

*4 divided by 4 is 1; 24 divided by 4 is 6.*1006

*Again when you simplify it, you have to make sure that one is on the top and another one is on the bottom.*1014

*Then you are going to see what common factor they have and divide both numbers by that same factor.*1022

*That is the most I can simplify; I need to multiply 6 times 25.*1034

*You are just going to do that on the side; 25 times 6.*1043

*5 times 6 is 30; you put the 3 up here; 0 down there.*1047

*6 times 2 is 12; plus the 3 is 15.*1052

*Since you have no numbers to bring it up, you write the whole number down there.*1057

*150 over... 7 times 1 is 7; this is an improper fraction.*1062

*But you can just leave it like that; that would be the answer.*1073

*The next example, we have 2 and 4/5 times 10/3.*1078

*2 and 4/5 is a mixed number; we have to change that.*1084

*I am going to take my denominator, 5; multiply it to my whole number.*1091

*It is 5 times 2 which is 10; then add the top number.*1095

*That is 14; it will be 14/5 times 10/3.*1101

*Again you can just do numerator times the numerator and get the numerator of your solution.*1113

*5 times 3, that becomes your denominator.*1121

*Or first you can just see if any of these numbers will cancel; let's see.*1125

*14 and 10 have a common factor because they are both even numbers.*1133

*But I can't cancel those out; I can't reduce those because they are both on the top.*1139

*Remember if you want to reduce the numbers, you have to make sure*1146

*that one is on the top and one is on the bottom.*1148

*14 and 3 have no common factors.*1153

*5 and 10 have a common factor of 5.*1156

*I can take both numbers and divide it by that factor of 5.*1161

*5 divided by 5 is 1; 10 divided by 5 is 2.*1166

*You have to make sure that you are going to divide both numbers by that same factor.*1171

*Then you are going to do 14 times 2.*1178

*Now that everything is simplified here, we are going to go ahead and multiply.*1180

*It is 14 times 2 is 28; 1 times 3 is 3.*1184

*This becomes your answer; this is an improper fraction.*1197

*If you want to change it to a mixed number, you can go back a few lessons*1201

*to the lesson where I talk about how to change improper fractions to mixed numbers.*1206

*Just to change this real quick, if you want to change this to a mixed number,*1214

*I have to see how many times the 3 is going to fit into 28.*1218

*How many times 3 fits into 28; I can do that by dividing.*1223

*I do 28... this bar right here means divide.*1228

*I can do 28 divided by 3; 28; I do 3.*1231

*How many times does 3 go into 28?*1240

*Let's see, 9; 9 times 3 is 27; I have 1 left over.*1245

*You can leave it like this; this can be your answer.*1255

*Or if you need to change it to a mixed number, you are going to see how many whole numbers.*1257

*Since 3 fits into 28 nine times, 9 becomes your whole number.*1264

*Then how many you have left over, I had 1 left over.*1270

*That becomes your numerator; you are going to keep the same denominator.*1274

*28/3 or 9 and 1/3, they are both your answers.*1281

*That is it for this lesson; thank you for watching Educator.com.*1290

*Welcome back to Educator.com.*0000

*The next lesson is on dividing fractions including fractions that are mixed numbers.*0002

*In the same way that we multiply fractions,*0013

*when we divide fractions, we have to make sure that we have no mixed numbers.*0015

*If you do have mixed numbers, make sure to change them to an improper fraction.*0020

*When you have a fraction A/B and you are dividing it by another fraction C/D,*0028

*what you are going to do is take the second fraction and you are going to flip it.*0035

*You are just going to make this number, the top number, the bottom number.*0044

*Then the bottom number becomes your top number.*0049

*When you do that, you are going to change the division to a multiplication sign.*0052

*Now your problem becomes A/B times D/C.*0058

*You are going to multiply the fractions the same way.*0065

*Again take the second fraction and flip it.*0069

*When you flip it, it is going to go from dividing to multiplying.*0074

*Let's do a few examples; 2/3 divided by 3/4; I have no mixed numbers.*0083

*I can go ahead and work with these fractions right here, 2/3.*0094

*I am going to switch this sign to a multiplication sign because I am going to take this fraction right here.*0101

*My 4, my denominator, becomes my numerator; my numerator becomes my denominator.*0110

*Don't forget; if you flip the first fraction, you are going to get it wrong.*0119

*Make it is not the first fraction; it is the second fraction that you flip.*0124

*From here, I am going to multiply my numerators.*0129

*2 times 4 is 8; 3 times 3 is 9.*0135

*2/3 divided by 3/4 is 8/9.*0143

*The next example, 1 and 2/5 divided by 2 and 3/5.*0151

*Both of my fractions are mixed numbers.*0156

*I have to make sure to change them to improper fractions before I go ahead and divide them.*0159

*When you change it from a mixed number to an improper fraction, you take the bottom number 5.*0167

*Multiply it to your whole number and add the top number.*0172

*You are going to go 5 times 1 is 5; plus 2 is 7.*0177

*It is 7/5 divided by... 5 times 2 is 10, plus 3 is 13; 13/5.*0182

*Look at how I kept the division sign.*0199

*I can only switch it to multiplication when I flip my top and bottom numbers.*0201

*For now, I didn't flip it yet.*0209

*All I did was just convert this mixed number to an improper fraction.*0210

*I had to keep this sign.*0214

*My next step, I am going to make it a multiplication problem by flipping these.*0219

*My 5 now goes on the top.*0226

*My 13 is going to go on the bottom.*0228

*Remember from last lesson, if I have this problem right here, I can see if a number from the top*0233

*and a number from the bottom can reduce, can simplify if they have common factors.*0242

*7 and 13 have no common factors because they are both prime.*0249

*7 is a prime number; 13 is a prime number.*0253

*The only common factor between them would be 1.*0256

*This 5 and the 5 up here, they have a common factor of 5.*0260

*They are the same numbers so their common factor would be itself.*0267

*I can simplify these numbers by dividing their common factor of 5.*0272

*I am going to take this 5; divide it by 5 which is 1.*0277

*Then this number divided by 5; that also becomes 1.*0282

*Now I can multiply the top numbers together and then multiply my bottom numbers together.*0289

*7 times 1 is 7; 1 times 13 is 13; that is my answer.*0295

*Some more examples, 6/5 divided by 3/4.*0307

*Since none of these are mixed numbers, I can go ahead and just work with those.*0315

*Since I am dividing, I can now multiply after switching these.*0324

*It becomes 4/3; again you can just multiply them.*0329

*Just do 6 times 4, 24, over 5 times 3 which is 15; then simplify that fraction.*0338

*Or you can simplify the problem--this is the problem--by seeing if any numbers have common factors.*0346

*Again 6 and 4, this number 6 and this number 4 have a common factor of 2 because they are both even numbers.*0356

*But I can't cancel those out because they are both on the top.*0362

*Whenever you cancel numbers out, make sure that one of them is on the top and one of them is on the bottom.*0368

*I can only do maybe 5 and 4.*0374

*But they don't have any common factors besides 1.*0377

*6 and 3 have a common factor of 3.*0381

*3 divided by 3 is 1; 6 divided by 3 is 2.*0388

*Make sure you divide both numbers by that same factor.*0394

*Now I am going to multiply; 2 times 4 is 8.*0401

*5 times 1 is 5; that is an improper fraction.*0407

*But I can leave it like; that is my answer.*0412

*The next example, 4 and 1/4 divided by 5/3.*0417

*I have a mixed number; I need to convert that.*0423

*4 times 4 is 16; plus 1 is 17; 17/4 divided by 5/3.*0427

*I didn't change it yet because I didn't flip this fraction.*0443

*17/4 times 3/5; make sure it is the second fraction that you flip.*0448

*From here, let's see, 17 with 5; no, they don't have any common factors.*0461

*4 with 3?--nope, no common factors besides 1.*0469

*I have to just solve it out; 17 times 3; and then 4 times 5.*0474

*17 times 3; let's do that problem.*0481

*7 times 3 is 21; the 2 up here; 1 down here.*0485

*Multiply these two and then add that number.*0492

*3, 5; 51 over... 4 times 5 is 20; this is my answer.*0494

*You can change it to a mixed number if you would like.*0509

*All you are going to do is see how many times 20 is going to fit into 51.*0512

*That is going to be your whole number.*0518

*How many you have left over is your numerator; keep the same denominator of 20.*0521

*Let's just do that; 20 is going to fit into 51 two times.*0527

*If you do it two times, it is 51 divided by 20.*0534

*It is going to fit in there two times; that becomes 40.*0541

*I have 11 left over; I can write it like this.*0545

*Or I can write it like... 2 is my whole number; I have 11 left over... over 20.*0551

*Either fraction, improper or mixed number, is going to be your answer.*0564

*You can leave it this way; or you can write it this way.*0568

*Few more examples, 7/8; you can divide and then change this fraction.*0575

*9 times 3 is 27; add the 2; that is 29/9.*0586

*I am going to do 7/8 times 9/29.*0595

*8 and 9 have no common factors; 7/9 have no common factors.*0606

*I can go 9 with 29; I can compare those two.*0611

*But nothing has any common factors for me to simplify or to reduce.*0617

*I am going to multiply; 9 times 7 is 63; 29 times 8.*0623

*29 times 8; 9 times 8 is 72; 8 times 2 is 16.*0632

*Plus 7 is 23; that is 232.*0641

*I know they have no common factors because none of these were able to reduce.*0653

*This will be my final answer.*0658

*The next example, 5/4 divided by 9/11; that is my problem.*0663

*Before I start cancelling numbers out, I have to make sure I change this to a multiplication problem first.*0674

*5/4 times 11/9; don't forget to flip this one.*0683

*Let's see; 11 with 9--no; 11 with 4--no; 5 with 9--no.*0692

*None of these have any common factors for me to cross cancel numbers out.*0700

*5 times 11 is 55; 4 times 9 is 36; that is my answer.*0710

*The next couple of examples, I have 3 and 3/7 divided by 6 and 1/4.*0726

*These are both mixed numbers; I have to convert them to improper fractions.*0734

*I take my denominator of 7, multiply it to my whole number, and then add the numerator.*0743

*7 times 3 is 21; plus 3 is 24;*0749

*Then 4 times 6 is 24; plus 1 is 25.*0758

*Make sure you keep the same denominator.*0767

*Now I can go ahead and divide these fractions.*0772

*For me to divide fractions, I have to take my second fraction and flip it.*0774

*My top number is going to become my bottom number.*0780

*My bottom number will become the top number; let's write it right here.*0782

*Once I flip it, I am going to change my sign to multiplication.*0790

*It is going to be 24/7 times 4/25.*0797

*Make sure you flip the second one and not the first one.*0804

*If you flip the first one, you are going to get the wrong answer.*0806

*It is 24/7 times 4/25.*0809

*You are going to see if you can cancel any of these numbers out.*0816

*Make sure that one number is the numerator and another number is in the denominator position.*0820

*7 with 4, they have no common factors because 7 is a prime number.*0826

*24 and 25 also have no common factors.*0834

*You are going to have to multiply the top numbers and then multiply the bottom numbers together to get your answer.*0839

*24 times 4; 24 times 4; 4 times 4 is 16.*0845

*I put the 1 right here; 6 down there.*0853

*4 times 2 is 8; add the 1; 9; this becomes 96 over... 25 times 7.*0856

*5 times 7 is 35; I put the 3 up here; 5 right there.*0869

*7 times 2 is 14; plus 3 is 17; this is 175.*0875

*Since none of these were able to reduce or cross cancel, I know that this is my answer.*0887

*The next example, 2 and 4/5 divided by 10/3.*0898

*10/3 is an improper fraction; I can leave that as it is.*0903

*This one I have to convert; 5 times 2 is 10.*0908

*Plus the 4 is 14; 14/5 divided by 10/3.*0915

*I am going to flip the second fraction so that the top number and my bottom number switch positions.*0930

*That changes this division to multiplication; 14/5 times 3/10.*0938

*From here, now that it is a multiplication problem, I can see if any numbers will cross cancel.*0953

*5 and 3 have no common factors because they are both prime numbers.*0960

*14 and 10 have a common factor.*0965

*Before we do that, let me just point out... 5 and 10, I have a common factor of 5.*0969

*5 goes into both of these numbers.*0977

*But remember you can't cancel those out because they are both denominators.*0979

*They are both on the bottom.*0985

*When you cancel numbers out, you have to make sure that*0987

*one of them is on the top and another one is on the bottom.*0990

*Since 14 is on the top and 10 is on the bottom, I can cross cancel those numbers out.*0996

*A common factor between 14 and 10 is 2 because they are both even numbers.*1003

*I am going to take both numbers and divide it by that common factor.*1011

*Since the common factor is 2, I am going to divide this number by 2.*1017

*That is 5; then take the 14, divide it by 2; that becomes 7.*1021

*You have to make sure that you are going to divide by the same factor.*1028

*There are no more; 7 and 5, they have no common factors.*1036

*I can go ahead and multiply them.*1041

*7 times 3 is 21; 5 times 5 is 25; that is your answer.*1042

*Make sure when you divide fractions, number one, convert all mixed numbers to improper fractions.*1058

*Then you are going to switch the second fraction.*1065

*The top number becomes the bottom number; the bottom number becomes the top number.*1069

*Then you are going to multiply those fractions together.*1073

*That is it for this lesson; thank you for watching Educator.com.*1077

*Welcome back to Educator.com; this lesson is on distributive property.*0000

*When you are using distributive property, you are multiplying everything inside the parentheses to the number on the outside.*0010

*You are going to distribute the outside number to everything inside the parentheses.*0022

*Here are just some examples with variables.*0028

*A and B, C and D, are all variables.*0032

*If I have the outside number is A, I am going to multiply it to the inside number of B.*0035

*This is just going to become A times B which is AB.*0042

*Here I have two numbers inside the parentheses.*0050

*You are going to take this outside number; multiply it to both numbers.*0055

*First I am going to multiply it to B.*0060

*It is going to be... A times B is AB.*0063

*Then I have the plus to separate them.*0068

*Then A, the outside number, to that number.*0072

*It is going to be A times C which is AC.*0075

*This one, there is three numbers on the inside.*0082

*I take my outside number, multiply it to that one.*0087

*It is going to be A times B; separate using that plus; A times that one; it is AC.*0092

*Then I have a third one; I separate it with that plus; A times D.*0102

*You are just multiplying the outside number to everything inside the parentheses.*0110

*Some examples; 4 times 2 plus 3.*0120

*Here you can go ahead and solve inside the parentheses first before multiplying the 4.*0126

*You can solve this out by doing 4 times... 2 plus 3 is 5.*0136

*Then this 4 times 5 is 20.*0142

*If you want to use distributive property because sometimes you do have to use distributive property,*0148

*you are going to take the outside number of 4.*0156

*Multiply it to the first number 2; it is... 4 times 2 is 8.*0159

*I have a second number; I am going to separate that by that plus sign.*0166

*4 times that number is 12; 8 plus 12 is 20.*0171

*You are going to get the same answer.*0181

*We notice that this one is a lot easier.*0183

*If you can solve inside the parentheses first, then go ahead and do it that way.*0186

*But sometimes you have to do it this way.*0192

*You have to use distributive property just like in the next example.*0195

*2 times A plus 6; here inside the parentheses, A plus 6.*0200

*I can't solve that out because this one A is a variable and this is a number.*0209

*I can't combine those; this is not 6A.*0214

*This A plus 6 is just A plus 6.*0217

*In this case, since I can't solve within the parentheses, I have to use distributive property.*0221

*You are going to take the outside number of 2.*0227

*Multiply it to the first number or letter of A.*0229

*That becomes 2A; 2 times A is 2A.*0235

*You still have a second number.*0242

*You are going to separate it; you are going to write your plus sign.*0243

*Then take your outside number; multiply it to that second number.*0248

*2 times 6 is 12; right here, 2A plus 12.*0253

*Again I can't solve that out because I have a variable here and this one does not have that same variable.*0261

*My answer will just be 2A plus 12.*0267

*You are going to leave it like this, 2A plus 12.*0271

*Few more examples; this one right here.*0276

*Again I can't solve within the parentheses; I have to use distributive property.*0283

*Take the outside which is C; multiply the first number inside.*0288

*C times 5 is 5C; 5C.*0296

*When you have a number multiplied to a variable, you are just going to write it together like this.*0302

*5C, with the number first; instead of C5, you are going to write 5C.*0308

*I have a minus to separate those two; then this one times the second one.*0316

*CA or you can do AC; it doesn't matter.*0323

*When you have a variable times a variable, you are just going to write them together.*0326

*It is CA or AC; we can't subtract these together because they have different variables.*0329

*So that is just my answer.*0339

*The next one, 10 times 8 minus 3; here I can solve within the parentheses.*0342

*I can do 10 times... 8 minus 3 is 5; 10 times 5 is 50.*0354

*Or if you have to use distributive property, take the outside number.*0365

*Multiply it to that first one; this is 80; write this sign to separate them.*0374

*Then I am going to do 10 times the second number which is 30.*0384

*Then 80 minus 30 is 50.*0391

*You have to make sure that these numbers are the same; that is your answer.*0394

*This example right here, 4 times B plus C.*0402

*Again you take the 4, the outside number.*0409

*You are going to multiply it to that first number inside or in this case letter.*0411

*4 times B is 4B; again number times a letter.*0417

*You are just going to write it together with the number in the front.*0425

*It is 4B plus... multiply it to that one, the second one.*0428

*4 times C is 4C; I can't combine them; they have different variables.*0435

*That is my answer.*0441

*Next example, 9 times 5 minus D; I can't solve within the parentheses.*0446

*Take the outside number; multiply it to that first one of 5.*0457

*9 times 5 is 45; you are going to write this sign to separate them.*0465

*Then 9 times D which is 9D; sorry... 9D.*0472

*I can't combine them; that is my answer.*0492

*Let's do a couple more.*0497

*7 times M minus 8; let me just write that again.*0501

*Here again I can't solve within the parentheses because I have a variable, M, minus 8, a number.*0509

*This one doesn't have the same variable.*0516

*In this case, I have to use distributive property.*0520

*I am going to take my outside number which is 7.*0523

*Multiply it to everything inside the parentheses.*0527

*I take the 7; multiply it to M, the first thing in there.*0531

*7 times M is 7M.*0538

*Again you have a number times a letter or a variable.*0541

*When you do that, you are going to write it 7M with the number first.*0546

*Then you are going to write this sign to separate them.*0551

*You are going to take the outside number and multiply it to that second number in there.*0554

*7 times 8 is 56; I look at this; I can't combine them.*0561

*I can't subtract them because this has a variable of M and this one doesn't.*0571

*That becomes my answer; I am just going to leave it like that.*0577

*My next example, X plus Y plus 3.*0581

*For this one, I have three different things I have to distribute the outside number to.*0590

*I am going to take the 2; multiply it to the X first.*0598

*2 times X is 2X; separate it with that sign, plus.*0602

*2 times the Y is 2Y; again I have to separate it with that sign.*0610

*Then 2 times the 3 which is 6; can I combine any of these?*0618

*No, none of these are like terms because this has a variable of X.*0629

*This one has a Y; this one doesn't have a variable.*0632

*You are just going to leave it like that as your answer.*0636

*Again when you are using distributive property, you are going to take the outside number,*0639

*multiply it to each thing inside the parentheses, separate them with a sign--with the plus or the minus sign.*0643

*After you distribute that, see if you can combine like terms together.*0654

*If not, then that is your answer.*0659

*That is it for this lesson; thank you for watching Educator.com.*0662

*Welcome back to Educator.com; this lesson is on units of measurement.*0000

*We are going to look at the different types of measurement, starting with length.*0005

*To see how long something is, we have different measurements.*0013

*1 foot we know is 12 inches; 1 yard is equal to 3 feet.*0021

*1 mile is equal to 5280 feet.*0029

*Using this, we can convert from foot to inches, yards to inches to feet, and so on.*0034

*Also for length, we have meters; 1 meter is equal to 100 centimeters.*0043

*It is also equal to 1000 millimeters; 1000 meters is equal to 1 kilometer.*0054

*The key here in converting is... you should know all of these.*0061

*But it is also to determine which measurements are larger, which ones are smaller than the others.*0069

*Right here, centi means 100; but it means small 100.*0081

*That means to make up 1 meter, it takes 100 centimeters.*0088

*If I said that this is 1 meter, then it takes a lot of centimeters, 100 centimeters, to make up 1 meter.*0097

*We know that the meter is bigger than the centimeter; centi means 100.*0117

*Anytime you have anything with centi, it just means 100, but in a small way.*0122

*They are very small.*0127

*Same thing, 1 meter, it also takes 1000 millimeters to make up 1 meter.*0131

*Milli means 1000, but in a small way also.*0140

*It takes 1000 of these very small units to make up 1 meter.*0143

*Kilo also means 1000, but in a big way.*0152

*That means it takes 1000 meters to make up 1 kilometer.*0157

*We know kilometer is bigger than the meters because it takes 1000 of these to make up 1 of those.*0164

*Let's go over the next one, mass, which is looking at weigh, how heavy something is.*0177

*1 pound is the same as 16 ounces; 1 ton is equal to 2000 pounds.*0184

*Depending on what you are weighing, you are going to use these different measurements.*0194

*If you are going to weigh something that is really, really light, then you are probably going to use ounces.*0200

*Pounds we know.*0206

*Tons would be something very heavy; like maybe a car or something that is pretty large, pretty heavy.*0209

*Kilogram which is kg; remember kilo; kilo means 1000, but in a big way.*0219

*It takes 1000 grams to make 1 kilogram.*0231

*A kilogram, again, anything with kilo means 1000; that is also used for mass.*0238

*For liquid, 1 gallon is the same as 4 quarts.*0253

*1 quart is the same as 2 pints; 1 of those is the same as 2 cups.*0260

*For this, there is... let's look at this.*0269

*I am going to show you an easy way to remember these.*0276

*I am going to write a G for gallon; let me write this part over.*0281

*Let's see; I have a G like that; G for gallon; this is 1 gallon.*0294

*Then since 1 gallon is equal to 4 quarts, I am going to write 4 Qs in here.*0300

*1, 2, 3, 4; that means 4 quarts; 4 Qs is equal to 1 gallon.*0306

*Within the Qs, I am going to write 2 Ps because I know that 2 pints is equal to 1 quart.*0320

*Here is P, P, P, P, P, and then there we go.*0327

*I have 2 Ps within 1 quart, 1 Q.*0341

*Then 1 pint is equal to 2 cups.*0346

*Within each P, I am going to write 2 Cs.*0349

*We know G is for gallon, Q is for quarts, P is for pints, C is for cups.*0366

*If I ask you how many pints are in 1 gallon,*0373

*you can just count how many Ps there are within the G for the gallon.*0379

*1, 2, 3, 4, 5, 6, 7, 8; there are 8 pints in 1 gallon.*0386

*What if I ask you how many pints are in 2 quarts?*0392

*You are going to count all the Ps within 2 Qs.*0396

*It is 1, 2, 3, 4.*0400

*How many cups are in 1 quart?*0402

*1, 2, 3, 4; there is 4 Cs in 1 Q.*0406

*You can just use this to help you when it comes to the liquid measurements.*0411

*When we convert these, if I want to convert meters into centimeters,*0423

*first of all I have to know the measurement of 1 meter equals how many centimeters?*0430

*Centi meant 100, but in a small way.*0439

*I know that 100 centimeters equals 1 meter.*0444

*I am just going to write that below here; 1 meter equals 100 centimeters.*0448

*This is just an easy way to do this.*0455

*We are going to probably cover this again when we go over proportions and that chapter.*0457

*But for now, you can just write this below here just to help you.*0463

*From meter, 1 meter was the same as 100 centimeters.*0471

*To get from 1 to this, you multiplied by what?--by 10; multiplied by 10.*0478

*In the same way, I have to take 100.*0487

*Then multiply it by 10 to get how many centimeters is going to be equal to 10 meters.*0491

*Whenever I take this number and multiply it by 10,*0499

*there is just one 0 for 10 so I can just add a 0 there at the end of it.*0505

*It is going to be 1000; 100 times 10 is 1000.*0510

*I know that 10 meters is the same as 1000 centimeters.*0514

*Quarts to cups; if you don't remember how many cups is equal to the pints and the quarts,*0523

*you can just draw your G and then use that as a reference.*0531

*Let me just write that out just so you guys can see it.*0538

*Remember I had 4 Qs; within each Q, I had 2 Ps.*0542

*Within each P, I had 2 Cs.*0550

*1 quart, 1 Q, how many cups do I have?*0555

*I have 4Cs; I have 4 cups; 1 quart is the same as 4 cups.*0560

*Yards to inches; I know that 1 yard is equal to 3 feet.*0575

*1 yard is equal to 3 feet; 1 feet is equal to 12 inches.*0592

*If I want to go from yards to inches, since I know that 1 yard is 3 feet*0605

*and only 1 of these is equal to 12 inches,*0614

*to get from here to here or from here to here, you multiplied by 3.*0618

*Then if I want to go from inches to yards, I have to multiply by 3.*0625

*12 times 3 is 36; 1 yard is equal to 36 inches.*0635

*I know this seems a little confusing.*0645

*But I just know that I have the top and the bottom.*0646

*To get from 1 to 3, you have to multiply by 3.*0653

*You have to multiply this side also by 3 to get back to the yards.*0659

*1 feet is equal to 12 inches; 3 feet is going to be 36 inches.*0666

*3 feet is the same thing as 1 yard.*0672

*So you can look at it that way.*0674

*Ounces to pounds; I know that 16 ounces is equal to 1 pound.*0678

*How did you get from 16 to 8?*0694

*Once these are the same, you can just convert it back over.*0701

*To get from 16 to 8, I divided by 2.*0706

*I have to also take this number and divide by 2.*0714

*I have to do the same thing to both sides.*0719

*1 divided by 2 is 1/2; this means 1 divided by 2.*0722

*1/2 is the same thing as 1 divided by 2.*0732

*8 ounces is half a pound.*0734

*Few more; 2 yards equals how many feet?*0741

*I know that 1 yard equals 3 feet.*0747

*Again I have yard to yard, feet to feet.*0756

*To go from 1 to 2, I multiplied by 2.*0762

*Then I have to do the same thing on this side.*0771

*Multiply by 2; take this number; times 2 is 6.*0774

*2 yards is the same as 6 feet.*0781

*Now 8 quarts to how many pints?*0790

*1 quart equal to 2 pints; 1 quart is equal to 2 pints.*0794

*Again I have quarts to quarts and pints to pints.*0807

*How do I go from 1 to 8?--you multiply.*0812

*It is always going to be multiplication or division.*0817

*You are not going to add 7 to get 8.*0819

*You don't do plus and minus; you are going to do times and divide.*0822

*1 times 8 is 8; then I need to multiply this by 8 which is 16.*0827

*8 quarts is equal to 16 pints.*0839

*Let's do a couple more; 1 gallon equals how many pints?*0844

*You can draw your G again.*0853

*Let's draw a big G so you guys can look at it, G for gallon.*0855

*I know that 4 quarts equals a gallon; you just draw this over.*0862

*Within each quart, there are 2 pints.*0870

*Since I am only going up to pints, I am not using cups.*0883

*I don't have to draw the Cs in there.*0886

*Within 1 G, how many Ps do I see?*0890

*I see 1, 2, 3, 4, 5, 6, 7, 8.*0894

*In 1 gallon, there are 8 pints.*0899

*Then 2 kilometers equals how many meters?*0908

*Remember kilo means 1000 in a big way.*0912

*I know that there is going to be a lot more meters than kilometers*0918

*because since kilometers are big, it means 1000 in a big way,*0922

*there has to be more meters to make up for the kilometer.*0928

*1 kilometer equals 1000 meters.*0934

*Again this is kilometer to kilometer and meters to meters.*0943

*I can just go ahead and see what I multiplied this by.*0947

*1 times 2 equals 2.*0953

*You have to multiply this by 2 to get the 2 kilometers.*0957

*You are going to do the same thing on this side.*0961

*You have to also multiply by 2; take this number; 1000 times 2 is 2000.*0962

*2 kilometers is 2000 meters.*0972

*If you need to, go back to the beginning of this lesson and review over the different units of measurement.*0978

*Try to get yourself familiar with all of them.*0984

*That is it for this lesson on units of measurement.*0989

*Thank you for watching Educator.com.*0993

*Welcome back to Educator.com.*0001

*We are going to talk about integers and the number line.*0003

*What are integers?--integers are all positive and negative whole numbers.*0008

*Whole numbers are all numbers that are whole; no decimals, no fractions.*0016

*Numbers like 4, 5, 10, 20, 100, 0, those are all whole numbers.*0024

*All whole numbers and their opposites...*0034

*Opposites just mean that if I have a whole number like 5, the opposite would be -5.*0037

*It is basically all whole numbers because whole numbers are only positive numbers including 0.*0045

*When you say whole numbers and their opposites, integers include whole numbers*0053

*and the negative of the whole numbers; positive and negative whole numbers.*0060

*-3 is an integer; -5 is an integer; and so on.*0068

*If I have a number line right here, this is a number line.*0076

*I am going to make that 0 right in the middle.*0083

*All of my numbers to the right of it are going to become positive.*0089

*If this is 1, 2, 3, 4, 5, 6, 7, 8, this right here then on the other side of 0 are negative numbers.*0095

*This one is going to start off at -1, -2, and so on, -3, -4, -5, -6, -7, -8.*0111

*It is going to keep going; this is 1, 2, 3, 4, 5, 6, 7, 8.*0123

*This is -2, -3, -4, -5, -6, -7, -8, right there.*0129

*This is a number line that includes positive numbers and negative numbers.*0136

*The opposite of 8 is going to be -8.*0143

*But all of these together are called integers.*0149

*Absolute value is the number's distance from 0; distance from 0.*0156

*If I ask you what is the absolute value of 3?*0165

*How many units or how far away is 3 from 0?*0169

*If it is 0 right here and 3 is right here, the absolute value of 3 will be 3.*0177

*It is 3 units away from 0.*0182

*If I ask you the absolute value of -3, the distance away from 0 is going to be 3.*0185

*Distance we know cannot be a negative number.*0195

*Whenever they are asking for a distance, make sure that the number is a positive number.*0199

*In this case, when I ask for absolute value of a number,*0203

*whether it is positive or negative, it is going to have to be positive.*0206

*This is how you would write it; absolute value of x.*0213

*You are going to write a 1, but make sure it is a little bit longer.*0216

*Then you are going to write the number inside.*0221

*Two bars, in between two bars, and then the number inside; this is the absolute value of x.*0224

*If I ask you what the absolute value of 5 is, then my answer is 5 because it is 5 units away from 0.*0234

*If I ask you what the absolute value of -5 is, again -5 is right here.*0246

*It is 5 units away from 0; this is also 5.*0254

*It is just asking how far is it from 0; that is absolute value.*0261

*Whenever you have an absolute value of a number, you are just taking the positive of it.*0267

*You are making it positive.*0272

*Let's go over a few examples; let's compare the integers, 3 and -5.*0276

*I just want to know which one is bigger than the other.*0282

*3 and -5; negatives are very small.*0289

*Positive numbers I know are bigger than negative numbers.*0296

*If you have the number line again, as the numbers go to the right, they become bigger.*0300

*This way is bigger; if the numbers go this way, then it gets smaller.*0311

*Positive numbers are big; negative numbers are small.*0321

*If one number is positive and one number is negative, I know...*0324

*By the way, this doesn't have a plus sign in front of it to show that it is positive.*0328

*But if there is no negative sign, then that means it is positive.*0333

*+3 and -5, I know that +3 is greater no matter what these numbers are because this is negative.*0338

*This is positive; this is negative.*0349

*Positive numbers are always greater than negative numbers.*0351

*The next one, -2 and -4, they are both negative.*0356

*But I want to see which one is greater, which one is bigger, which one is smaller.*0361

*If I make this 0, then here is -2.*0370

*Here is -4 because it goes -1, -2, -3, -4.*0375

*Remember the numbers that go this way, that are on the right side of it, are bigger.*0382

*The numbers that are on the left side are smaller.*0388

*-2 is on the right side.*0391

*It is closer to 0 or to the positive side than -4 is.*0395

*So I know -2 is bigger than -4.*0400

*In this case, -2 is greater than -4.*0405

*You can also think of this as money.*0410

*If you want to think of it that way, you can.*0416

*-2 means that maybe you owe money.*0420

*A negative, you either owe 2 dollars or you can owe 4 dollars.*0427

*You know that owing 4 dollars is worse than owing 2 dollars.*0435

*If you owe 2 dollars, then that means that you have more value or that it is bigger*0441

*because you owe less, because the more you owe, the worse it is.*0449

*So -2 is going to be bigger than -4.*0452

*10 and -10; 10 is +10; this is -10.*0457

*Automatically 10 is greater because this is positive and this is negative.*0466

*I would rather have 10 dollars than owe 10 dollars.*0475

*The next one, this is positive; it is +8.5 and -8.9.*0479

*Don't get confused with these numbers because... just look at this line.*0487

*This is a positive number; this is a negative number.*0490

*I know that this number is bigger.*0494

*What if I make this number negative, -8.5 and -8.9?*0496

*Let's see; let me draw this out a little bit bigger.*0509

*If I have a number line, if I make the 0 right here, here is -8 and here is -9.*0511

*I know that both numbers are going to be in between -8 and -9.*0522

*Which one is closer to -9?--which one is closer to -8?*0530

*-8.5 would be right in the middle of -8 and -9.*0535

*-8.9 is actually very close to -9, right here.*0542

*This number of -8.5 would be greater because it is closer to the right.*0548

*It is on the bigger side; this one is going to be greater than -8.9.*0556

*Write an integer to represent each; 200 feet below sea level.*0566

*This is the keyword here, below.*0573

*If it is below, then I know it is going to go negative.*0575

*If it goes above, then it is positive.*0578

*200 feet below sea level is going to become -200 feet or -200.*0582

*I don't have to write the feet; it is just write an integer.*0590

*A gain of 30 yards, if you are gaining something, then it is a positive value.*0594

*Here is the keyword, gain; that is +30.*0601

*The next one, a decrease of 100 points.*0606

*Decrease, the keyword, means you are losing it; increase means you are gaining.*0610

*Decrease would be negative; -100.*0619

*12 degrees below 0; below so it is going to become -12 degrees.*0625

*Write the opposite integer; remember we talked about opposites.*0640

*How if you have a number line, then it is going to be on the other side of 0.*0644

*The opposite of -3 is going to become +3; or I can just write 3.*0650

*+94, the opposite is going to become -94; -50, +50 is the opposite.*0659

*Again this doesn't have a plus sign in front of it.*0670

*But I know it is a positive because numbers can only be positive or negative.*0672

*There is no negative sign; it has to be positive; this is -48.*0678

*The fourth example, find the absolute value of these numbers.*0689

*Absolute value remember again was the distance from 0.*0697

*How far away is that number from 0?*0700

*Distance, if I am asking you how far away something is, we know we can't have a negative number.*0705

*We have to have a positive number; distance can only be positive.*0710

*The absolute value of 45 is 45; 45 units away from 0.*0715

*+98, 98; absolute value of 10 is going to become 10.*0728

*Here the absolute value of 726 is 726.*0741

*If I have the absolute value of -10, on the number line, -10 is to the left of 0.*0748

*It is on the negative side.*0759

*Again if I am asking you how far away -10 is from 0,*0762

*then it is just 10 because it is 10 spaces away from 0.*0766

*Whether it is to the right or to the left, it is still 10 units, 10 spaces.*0771

*If I have an absolute value of a negative number again, then it becomes positive.*0775

*We can never have a negative distance.*0780

*Even if you find the absolute value of a negative number, then it is still going to be positive*0785

*because you are just finding the distance; that is it.*0791

*That is it for this lesson on absolute value and introducing integers.*0796

*Thank you for watching Educator.com.*0803

*Welcome back to Educator.com; this lesson is on adding integers.*0000

*To use a number line, we can add integers using a number line here.*0007

*What we are going to do is start with the first number.*0012

*If I give you a problem, 4 plus -2.*0014

*I am going to start off at this number right here, 4.*0023

*Let me just write out the numbers on this number line.*0026

*Here is 0, 1, 2, 3, 4; I can just do a few more on this side.*0028

*Here are my negatives; this is -4.*0037

*I start at this first number 4; 4 is right here.*0045

*I am going to start right here and then use the second number to move spaces.*0051

*This number right here is how many spaces I am going to move.*0061

*If it is positive, I am going to move to the right because this is the positive direction.*0064

*If it is negative, I am going to move to the left because left is always negative.*0073

*Since I have a negative number, -2, I am going to be moving to the left two spaces.*0078

*The negative and the positive is to just see which direction you are going to go, to the right or to the left.*0085

*I am going to move this many number of spaces.*0092

*We are starting at 4, moving to the left two spaces.*0095

*I am going to go 1, 2; his is where I land.*0099

*This number right here is 2; my answer is 2.*0104

*Let's do another example; if I have let's say 5 plus -8.*0111

*Again I am going to start off at 5; here is 5 right here; I start right here.*0122

*I am going to move 8 spaces to the left because it is negative.*0129

*Negative is going to make me go left.*0134

*If this was a positive, I would move 8 spaces to the right.*0136

*8 spaces to the left is going to be 1, 2, 3, 4, 5, 6, 7, 8.*0142

*This is where I am going to land.*0151

*That is -3 because it is -1, -2, -3.*0154

*The number I land on is the answer; my answer here is -3.*0157

*That is how you use a number line to add integers together.*0164

*Again integers are all positive and negative whole numbers.*0168

*No decimals, no fractions, just whole numbers and their opposites.*0174

*Another way to add integers without using the number line... first let's remember that opposites add to 0.*0183

*If I have +5 and I have a -5, then they are going to become 0.*0194

*A 5 plus a -5 equals 0; 5 and -5 are opposites.*0203

*As long as they are opposites, if I add opposites together, then my answer becomes 0.*0210

*If both numbers are the same sign, then you are going to add the numbers and keep that sign.*0219

*If they are opposite signs, we will talk about that in a second.*0225

*But let's do an example of this one--if I have -2 plus -3.*0229

*They are both negative; they have the same sign.*0238

*This is negative and this is negative.*0240

*I am just going to add up the numbers and I am going to keep that sign.*0245

*My answer is -5.*0250

*If I have -1 plus -10, then my answer is -11.*0253

*As long as I have the same sign, I am adding two numbers to the same sign,*0260

*then I am just going to add it and keep that sign; that is it.*0265

*Same thing goes with positive numbers.*0269

*If I have 3 plus 6, this is a positive number and this is a positive number.*0271

*We know that 3 plus 6 is 9; +9; you kept the same sign.*0278

*If the signs are opposite, the two numbers that you are adding together have opposite signs, then that becomes a little tricky.*0285

*If I have 2 plus a -3, this one I know is a positive number because integers can only be positive or negative.*0294

*There is no negative sign here so I know it is positive.*0306

*It is +2 plus a -3.*0308

*Since they are opposites, what I am going to do is take the absolute value of both numbers.*0313

*The absolute value, if you don't remember absolute value, then take a look at the lesson right before.*0321

*Absolute value asks me for the distance from 0.*0326

*On the number line, how far away is 2 from 0?*0333

*I know it is 2; the absolute value of 2 is 2.*0336

*The absolute value of this is 3.*0340

*I am actually going to take their absolute values... or I am going to subtract them.*0348

*Basically I am going to just do 3... because the absolute value of -3 is 3... and -2.*0357

*I am going to find the difference; that becomes 1; 3 minus 2 is 1.*0364

*You are going to keep the sign of the number that is greater, with the greater absolute value.*0372

*For this one, this has a negative sign.*0380

*I am going to keep that negative sign; this is a little confusing.*0384

*But just keep in mind that when you have the same sign,*0387

*then it is like you are adding the two numbers and you are keeping that sign.*0392

*When you have opposite signs, it is like you subtract the numbers.*0395

*Then you take on the sign with the greater absolute value.*0400

*Another way you can do this is think of positive numbers as dogs.*0409

*I like to use dogs and cats; you can use anything.*0416

*You can use stars; you can use different colors.*0419

*Let's say positive numbers are like dogs and negative numbers are cats.*0429

*If I am adding dogs and cats together, I am going to add a D for dog and C for cat.*0436

*It is going to be dog and a dog because there is 2 of them, and then 3 cats.*0442

*That is cat, cat, and cat; because it is -3, it is 3 cats.*0447

*Each dog and cat cancels out; they are like opposites, add to 0.*0453

*Dog and cat cancels out; these cancel out; what do I have left?*0460

*I have 1 cat; remember a cat is a negative number; it is a -1.*0465

*I can also use colors.*0472

*If I have a positive, then let's say I am going to use blue.*0475

*Blue circle, blue circle; -3 is going to be red circles.*0478

*Each time I have a blue and a red, I am going to cancel it out; cancel it out.*0488

*What do I have left?--1 red; that makes a negative.*0494

*Let's do a few more examples; add the integers.*0501

*If I have -4 plus -2, I am adding two negative numbers together.*0507

*You can just add up the two numbers and keep the same sign.*0516

*-4 plus -2 is going to be -6.*0520

*If I want to draw it out, -4... red circles because it is negative; -4 plus a -2.*0529

*I don't cancel this out because I only cancel out a blue and a red or opposites.*0541

*These are not opposites; they are the same.*0547

*I can't cancel them; instead I have to add them all together.*0549

*1, 2, 3, 4, 5, 6; 6 and it is red; it is -6.*0553

*Next one, I have -3 plus 8; 3, negative, that is red.*0559

*Then +8; +8 is blue; 1, 2, 3, 4, 5, 6, 7, 8.*0571

*The blue and the red cancels out; cancels out; cancels out.*0582

*Then I have 1, 2, 3, 4, 5 left; blue is positive; it is +5.*0589

*I can leave it like this; or I can write +5.*0598

*Again another way to think of this, you can take the absolute value of each number.*0603

*The absolute value of this is going to be 3.*0608

*The absolute value of this is 8; I just find the difference.*0611

*I subtract them; I can just do 8 minus 3 is 5.*0616

*I take on the sign of the greater number which is 8.*0623

*That is a positive so this becomes a positive.*0629

*The next couple of examples; find the sum; that just means to add them up.*0634

*-15 plus -10; we have a negative number and a negative number.*0639

*I just add them up; my answer is also a negative number, -25.*0650

*The next one, I have a positive number and a negative number.*0660

*Again I take the absolute value.*0666

*I am just going to think of both of these as positive numbers; 37 minus 25 which is 12.*0667

*I take the sign of this number because 37 is bigger than 25.*0680

*That has a sign of negative.*0690

*That means I am going to give this that same sign.*0691

*It is going to be -12.*0694

*Find the value; this one right here, I am looking for x.*0699

*To find x here, -2 plus something is going to give me -7.*0703

*-2 plus what is going to give me -7?*0711

*If I give myself 5 circles or 5 cats, what do I have to add to this?*0722

*Plus what is going to give me 7?--how many more do I need?*0730

*I have 5 here; I have 7 here; how many more do I need?*0738

*I need 2 more, 2 more red.*0744

*That is -2 because red I know is negative; I have to add -2 more.*0748

*The next example, the absolute value of 15 plus the absolute value of -9.*0761

*The absolute value of 15 is 15; the absolute value of -9 is 9.*0767

*My answer is 24; that is a positive.*0778

*Again if it is a positive number, you don't have to write the plus sign, the positive sign.*0784

*You can just leave it as 24.*0788

*The next couple, add the integers.*0791

*Again we have a negative number here plus a negative number here.*0796

*As long as they have the same sign... in this case, they both have a negative sign.*0800

*Then we can just add the numbers together, 12 and 20.*0807

*Or you can add the absolute values together.*0813

*It is 12 plus 20 which is 32.*0815

*You are going to give it the same sign.*0820

*Negative plus a negative equals a negative; -12 plus -20 equals -32.*0823

*The next one, 8 plus -6.*0831

*Again 8 is a positive number even though there is no sign written there*0834

*because numbers can only be positive or negative.*0838

*We know it is not negative so it has to be positive... plus a negative number.*0842

*Again if I want to draw a visual representation of this, this is going to be 2, 3, 4, 5, 6, 7, 8.*0851

*The blue represents the positive number; that is +8.*0862

*My red is going to represent the negative number, -6; 3, 4, 5, 6, *0867

*Each time I have a blue and a red, an opposite, I am going to cancel it out.*0875

*All of these cancel out; what do I have left?--2.*0880

*That is positive because blue is positive; it is +2.*0887

*Or you can just take the absolute value of these numbers.*0893

*The absolute value of 8 is 8; the absolute value of -6 is 6.*0897

*We are going to subtract those numbers; 8 minus 6 is going to be 2.*0903

*This is only when the signs are opposite; so 2.*0909

*Then this was an absolute value of 8 and this was 6.*0914

*The bigger number is 8; that has a sign of positive.*0920

*You are going to give that same sign to the answer which is 2.*0925

*It is going to become +2.*0932

*Again you take the absolute value of each number; subtract them.*0935

*It is 8 minus 6 because absolute value of 8 is 8.*0939

*Absolute value of -6 is 6; 8 minus 6 is 2.*0944

*Then you are going to give it the same sign as the bigger number.*0948

*That is a positive; your answer becomes +2.*0955

*That is it for this lesson on adding integers.*0960

*We will see you next time; thank you for watching Educator.com.*0963

*Welcome back to Educator.com; this next lesson is on subtracting integers.*0000

*The previous lesson was on adding integers.*0011

*What we are going to do first in order to subtract integers is *0015

*use a two-dash rule to change the subtraction problem to an addition problem*0019

*because adding integers is always a lot easier than subtracting integers.*0026

*We are going to use the two-dash rule.*0033

*What that is is in order to change the minus sign to a plus sign, you are going to add a dash to it.*0036

*For example, if I have 3 minus 5, first step, I have to make two dashes.*0045

*The first dash is to make this minus sign a plus sign.*0053

*That is the whole point--to change the minus to a plus.*0057

*I am going to use the first dash to make that a plus.*0060

*Then I have to make two of them.*0063

*My second one will be to make this a negative.*0065

*My subtraction problem is now an addition problem.*0069

*Another example, if I have 3 minus -5.*0072

*This is a minus; this is a negative.*0078

*The first dash is going to make this minus a plus.*0081

*Since I have to make two dashes, this is already a negative.*0085

*My second dash would be to make that a positive.*0089

*3 minus -5 would be the same as 3 plus 5.*0092

*If I have -3 minus 5, this right here would be just a -3.*0100

*That is not a minus problem.*0108

*The minus is right here because it is -3 minus the 5.*0109

*When you use a two-dash rule, you are not going to be using it for negative signs.*0114

*All you are doing is changing the subtraction problem to an addition problem.*0118

*You are not going to use two-dash rule for this one up here.*0127

*But you are just using it for the minus sign; it will be one, two.*0130

*That is your subtraction problem to an addition problem; then you just add the integers.*0136

*If you remember from last lesson, when we add integers,*0141

*if we have the same sign here, if it is the same sign,*0145

*if this is a negative and this is a negative, then we can combine these numbers.*0149

*We can add the numbers which is 8; or take the absolute value.*0154

*Remember absolute value takes the distance from 0.*0158

*-3 is 3 from 0.*0163

*Absolute value must be a positive number because distance...*0166

*if you are looking at how far away -3 is from 0... here is 0, 1, 2... here is -3.*0170

*How many units away from 0 is it?--it is 3.*0179

*Whenever you measure distance, it cannot be a negative; it has to be a positive.*0184

*The absolute value of -3 is 3; the absolute value of -5 is 5.*0190

*If you add those, it is going to be 8.*0196

*But then be careful because you have to give it the same sign.*0200

*This is a negative; this is a negative.*0205

*Then this is going to be a negative; -3 plus -5 is -8.*0206

*If you look at these problems right here or this one, let's do this one first.*0212

*This one doesn't have a sign in front it.*0217

*But it is a positive because numbers are going to be either positive or negative.*0219

*If there is no negative sign in front of it, then it has to be a positive.*0224

*This is a +3 plus a +5.*0228

*That is just the same thing as 3 plus 5 which is 8 or +8.*0232

*For this one, their signs are different; this is a +3 plus a -5.*0238

*Since their signs are different, again you take the absolute value.*0245

*This is 3; absolute value of this is 5.*0248

*Instead of adding them, you are going to subtract them.*0251

*You are going to find the difference; that is 2.*0253

*Then you take the number with the greater absolute value which is this one right here.*0256

*Get that sign; give it to this one; the answer is -2.*0262

*If you don't remember how to do this or if you want to look at a few more examples,*0270

*we are going to do a few more here for this lesson.*0275

*But you can also go back to the lesson on adding integers.*0278

*The first set of examples is to rewrite the subtraction problem to an addition problem.*0286

*This one, the first one is 5 minus 9.*0291

*We want to use the two-dash rule.*0296

*The two-dash rule is strictly just used to make a subtraction problem to an addition problem.*0299

*Later when you start getting really comfortable with these kind of problems, you won't have to use the two-dash rule.*0306

*The two-dash rule is just to make the problems a lot easier.*0311

*For here, the first dash I am going to make is to make that minus into a plus.*0314

*That is the first one.*0323

*My second one that I have to make will be to make that one a negative.*0325

*Now we have 5 plus -9.*0329

*Now that it is a plus problem, we have to add these two integers.*0335

*But they have different signs; this is +5; this is a -9.*0340

*We take the absolute value; the absolute value of 5 is 5.*0344

*The absolute value of -9 is 9; remember their signs are different.*0348

*So we find their difference.*0353

*Absolute value of 5 plus absolute value of -9 is going to be...*0358

*I am sorry... this is a minus; let me just make that a minus.*0365

*This is going to be 4; it is a 4.*0372

*But remember again you have to figure out which one...*0378

*If this is a 5 and this is a 9, this is the bigger number.*0382

*You look at the sign that goes with that number; it is a negative.*0387

*You are going to give that negative sign to the answer; it is -4.*0391

*This next problem, we have a minus negative; it is -10 minus a -4.*0396

*Again the first dash will be to make this one into a plus.*0404

*The second dash... for this problem, we made it into a negative.*0409

*But it is already a negative; we have to make that into a plus.*0414

*Again you are not going to use the two-dash rule for this negative sign up here.*0419

*It is only to make this a plus.*0423

*Whenever you do the two dash rule, it has to be right in there.*0426

*-10 plus 4; this is -10 plus 4; that is the same thing.*0432

*Plus positive is the same thing as just plus.*0439

*I don't have to put my positive sign.*0442

*Again they have different signs.*0445

*The absolute value of 10 minus the absolute value of 4 is 6.*0448

*Which one has the greater number?*0456

*The absolute value is going to be this one right here.*0459

*You are going to take that same sign and give it to this.*0463

*Be careful with the signs because you can get this number right.*0466

*But if the sign is wrong, then the answer will be wrong.*0470

*You have to make sure you have the correct sign along with the correct number.*0473

*The next one, you are just finding the difference.*0480

*Again we are going to use the two-dash rule to make that a plus and then there.*0483

*They are different signs, a positive and negative sign.*0489

*We are going to find the difference of their absolute values which is 7.*0494

*You are going to give that a negative sign because the 11 is the greater absolute value.*0500

*You are going to give that a negative.*0508

*The next one, again you are going to make this a plus and then a plus.*0511

*-5 plus 5; this is -5 plus 5;*0517

*-5 plus 5, again they are different signs; you take the absolute value.*0521

*That is 5; minus the 5 is 0; this is just 0.*0530

*If you have +2 minus 2, that is 0.*0536

*A -5 plus a 5, if you have opposites, then it will just be 0.*0540

*Next one, 32 minus 9, make this a plus negative; they have different signs.*0551

*I take the difference of the absolute values.*0561

*This is 32, the absolute value; this is 39.*0565

*If I find the difference of that, it is 7.*0569

*With that negative sign belonging to the 39, to the greater value, that becomes a negative.*0573

*For this, absolute value of -15 minus absolute value of -9.*0582

*The absolute value of -15, how far is -15 from 0?--it is 15.*0589

*You can also think of it, whenever you take the absolute value of something, it is just the positive of that number.*0597

*If it is -15, then the positive of -15 is 15.*0602

*Minus... this is not inside the absolute value sign.*0608

*This doesn't change; this has to stay the same; absolute value of -9 is 9.*0612

*15 minus 9, that is 6; we don't have to change this.*0619

*If you want to, you can change this minus to a plus using the two-dash rule.*0626

*But 15 minus 9, we know that that is just 6.*0632

*The last couple examples; here we have -3 minus -8 plus 5.*0640

*The first thing I want to do is -3 minus -8;*0648

*I want to solve this first; but I have a minus.*0653

*Remember we want to change our subtraction problems to additions problems.*0656

*We do that by using the two-dash rule.*0662

*For the two-dash rule, the first step will be to make that minus into a plus*0667

*because that is the whole point of using it.*0672

*The minus will change to a plus.*0675

*Then I have to make one more little dash either to make this a negative or make it a plus*0677

*because it already is a negative so I have to make that a plus.*0683

*I change -3 minus -8 to -3 plus 8; -3 plus 8.*0686

*Plus positive is the same thing as just plus; -3 plus 8.*0695

*-3 plus 8, again they have different signs.*0701

*This is a negative; this is a positive.*0708

*I am going to take the absolute value.*0711

*If it is different signs, then you are going to take the difference of the absolute values.*0714

*-3, the absolute value of that, which is the distance from 0, how far away is -3 from 0?*0718

*That is 3; this absolute value is 8.*0725

*When you find how far apart they are from 8 and 3, the absolute values, you get 5.*0730

*This is 5.*0737

*You look at which number has the greater absolute value; that is 8.*0740

*You are going to give it the same sign as that number.*0747

*Since 8 is a greater number, it has the sign of a positive.*0751

*You are going to give that 5 a positive sign.*0755

*We don't have to write the positive sign.*0758

*If you just don't write anything, then that is the same thing as giving it a positive sign.*0760

*This right here became 5; then I have to do 5 plus this 5.*0766

*My answer is going to be 10.*0773

*The last example, this is 4 minus 10 minus a -9.*0779

*I am going to solve that first.*0785

*Again since it is a subtraction problem, 4 minus 10,*0789

*I am going to change that subtraction to an addition problem by using the two-dash rule.*0793

*The first dash will be to change this minus to a plus.*0800

*My next dash is going to make that into a negative.*0805

*Remember both dashes have to be within those two numbers.*0809

*You can't make this number negative instead of this number.*0813

*Both dashes you make are going to be within the two numbers.*0818

*It is 4 plus -10; again their signs are different.*0822

*This has no sign which means it is a positive; +4 plus -10.*0827

*They have difference signs which means you are going to take the difference of the absolute values.*0834

*This is 4; this is 10; their difference is going to be 6.*0839

*This has the greater value; it has the sign of a negative.*0851

*That is going to go there too.*0855

*Then I am going to do that; -6 minus a -9.*0859

*Again we have a minus problem.*0866

*I am going to change this to addition by doing that; that is 1.*0867

*It is already negative; I have to make that a +2.*0874

*-6 plus +9, opposite signs; they are different signs so you find the difference.*0878

*This is 6; this is 9; they are 3 units apart.*0887

*Which one has the greater value?--the 9; that has a positive sign.*0896

*This is going to be a positive sign; the answer is +3.*0901

*That is it for this lesson on subtracting integers.*0908

*If you want to go back and review over some more problems, the previous lesson on adding integers,*0912

*that will probably help you freshen up a little bit for the next lesson.*0919

*Thank you for watching; we will see you soon.*0923

*Welcome back to Educator.com; this next lesson is on multiplying integers.*0000

*When you multiply integers, it is very important to keep in mind that*0007

*if one of the two numbers is negative, then your answer is going to be negative.*0014

*If both numbers are negative, then your answer will be positive.*0019

*Very different than when you add or subtract integers.*0023

*Make sure you keep that rule in mind; this is very important.*0026

*If only one number is negative, if you have one negative sign in the problem, then the answer will be negative.*0029

*If you have two numbers, think of it as those two numbers cancel the negatives out.*0036

*The product will be a positive; one number, then the answer will be negative.*0041

*If it is two numbers, then the answer will be positive.*0049

*If I am going to multiply let's say A and B together,*0054

*if I multiply A times a ?B, then my answer is going to be... this is a positive.*0057

*Remember there is no sign in front, then it is a positive.*0065

*Positive times negative is going to be negative; this is ?AB.*0068

*Or if I have ?A times ?B, we have two negatives signs.*0073

*My answer will be a +AB.*0081

*Let's do a few examples; the first one, -5 times 7; -5 times 7.*0088

*We are going to multiply these numbers the same way; 5 times 7 is 35.*0096

*I only have one negative sign.*0104

*From the two numbers that I am multiplying, only one is negative.*0107

*My answer will be a negative.*0110

*This one, -8 times -4, I know that is a 32.*0116

*I have two negative signs; two negatives signs gives me a positive.*0123

*Another way to think of it, if you have an odd number of negatives in the problem,*0131

*then your answer, your product will be a negative.*0137

*If you have an even number of negatives, like 2, 4, 6, 8, then your answer would be a positive.*0140

*Every two negatives cancel each other out to make a positive.*0146

*Even if you are multiplying four numbers together and they are all negatives, you have four negative signs.*0153

*That is going to give you a positive answer, a positive product.*0159

*3 times -10; 30; I have only one negative sign; that is a negative product.*0165

*What about this one?--this one, the answer is 60.*0178

*If I multiply these two numbers, a positive and a positive.*0183

*This doesn't change; this is just 12 times 5 is 60.*0186

*There is no negative signs involved.*0191

*20 times -12; 20 times 12 is 240.*0195

*We have one negative sign right here; positive times a negative is a negative.*0205

*This next one, 11 times 10 is 110.*0214

*A negative sign times a negative sign, we have two negatives.*0223

*That makes a positive; +110.*0226

*Again when we are multiplying two numbers, we have only one negative sign.*0234

*If only one of the numbers is negative, then the product will be a negative.*0241

*If you have two negative signs, it is going to become a positive.*0248

*In this problem, 7 times 9 is 63.*0253

*I have a positive here; I have a negative here.*0260

*I have one negative sign; that is going to make my answer a negative.*0262

*This next problem, I have a few things I have to solve out.*0269

*But the first thing I always solve out is parentheses.*0276

*I must solve my parentheses first; this is order of operations.*0279

*Order of operations says parentheses; I have parentheses right here.*0284

*I am going to solve this out; that is -7 minus a 3.*0287

*From the previous lesson on subtracting integers,*0292

*I want to use the two-dash rule to make this subtraction problem into an addition problem.*0294

*I am going to do that; keep this in parentheses.*0300

*-7; make this a plus; then a negative; this becomes -7 plus a -3.*0304

*They have the same sign; I add the numbers and keep that same sign.*0316

*The absolute value of -7 is 7; plus the absolute value of -3 is 3.*0324

*If I add those two numbers together, I get 10.*0331

*But then since they are both negative, I have to keep that same sign.*0335

*This rule is very different than multiplying integers.*0338

*Always keep in mind what you are doing.*0343

*Are you adding integers?--are you subtracting?--are you multiplying?*0345

*The next lesson is going to be dividing integers.*0349

*For each of them, think of the different rules; just keep practicing the problems too.*0353

*This was -10; I still have another parentheses I have to solve out.*0360

*This one right here is already an addition problem.*0369

*We don't have to change this problem like we did this one because it is already a plus right here.*0373

*The whole point of changing this was to make it a plus.*0380

*A -15 plus 8, they have different signs.*0385

*We are going to take the difference of their absolute values.*0390

*-15, the absolute value of that would be 15.*0393

*The absolute value of 8 is 8.*0397

*If you find the difference of 15 and 8, that is 7.*0399

*The sign, we take from this one, the 15; that sign is a negative.*0406

*I give this a negative; just going to write this out again right here.*0413

*-10 times a -7; 10 times 7 is 70.*0425

*Then I have a negative times a negative; I have two negatives.*0432

*That is going to make my answer, my product, a positive.*0435

*Think of two negatives cancelling each other out; that is going to be +70.*0440

*That is it for this lesson on multiplying integers; thank you for watching Educator.com.*0448

*Welcome back to Educator.com; this next lesson is on dividing integers.*0000

*If you remember the previous lesson on multiplying integers, *0006

*the rule was to multiply the numbers together to find the product *0011

*and determine if the product is going to be a positive or negative.*0016

*Remember that if only one of the numbers were negative, then the product was going to be negative.*0022

*If you have two negative numbers, then the product was going to be positive.*0029

*The same rule applies when you divide integers.*0033

*The only difference is that you have to divide the numbers instead of multiply them.*0038

*When you divide integers, you are going to divide the numbers.*0043

*You are going to apply the same rule as when you multiply integers.*0046

*If only one number is negative, then the quotient is going to be negative.*0051

*If both numbers are negative, then the quotient is going to be positive.*0055

*Let's just do a few examples.*0063

*The first problem, -9 divided by 3; 9 divided by 3 is 3.*0066

*Again if we have one negative sign, then my answer, my quotient, becomes a negative.*0075

*If I have two, positive.*0080

*This one right here, 14 divided by 2, ignoring the negative, is going to be a 7.*0084

*Don't forget to go back to the negatives though.*0091

*You can look at the negative signs as you do the problem too.*0094

*Or you can just divide it and then go back and look at how many negative signs do I have?*0098

*In this case, I only have one; that is going to make that a negative.*0104

*It is a -7; 14 divided by -2 is a -7.*0107

*49 divided by 7 is 7.*0113

*I don't have any negative signs; it is just positive.*0117

*-40 divided by -8.*0125

*I have two negatives which means that my answer, my quotient, is going to be a positive.*0128

*40 divided by 8 is 5; my answer is +5.*0136

*-90 divided by -10; again I have two negatives.*0143

*That is going to make my answer a positive; 90 divided by 10 is 9.*0149

*You can also look at these problems this way.*0155

*If I have it written like this, -90 divided by -10, *0159

*even though it looks like a fraction, this can also be divided by.*0165

*It is -90 divided by -10; you are going to get the same answer, +9.*0169

*If the problem is written like this or like this, it is still going to be +9.*0175

*55 divided by -5.*0183

*I have one negative sign which gives me a negative answer of 11.*0185

*55 divided by -5 is -11.*0195

*The next couple of examples, -99 divided by 11; again we have one negative.*0198

*If I have one negative here, then that is going to make my answer a negative.*0209

*99 divided by 11 is 9; -99 divided by 11 is -9.*0214

*This next problem, I have a few things to solve out.*0223

*Remember order of operations, I must solve out my parentheses first.*0228

*I am going to solve this out first; -40 minus 4.*0234

*Since I am subtracting integers here, I want to change my minus sign to a plus sign.*0241

*My subtraction problem to an addition problem.*0245

*I do that by using the two-dash rule.*0248

*If you don't remember how to do this, you can go back and review over that lesson.*0252

*-40 minus 4, the first dash will be to make that a plus because that is the whole point.*0257

*Then make that a negative; this will be -40 plus -4.*0262

*They have the same sign; this is a negative; this is a negative.*0269

*I just add the two numbers together, just 44, and give it the same signs.*0273

*-40 plus -4 is going to be -44.*0277

*Then I have another parentheses that I have to solve out before I divide.*0285

*-85 plus 96, their signs are different.*0290

*I don't have to apply the two-dash rule for this one because this problem right here was a minus problem.*0295

*That is the only time you are going to use that rule.*0302

*This is already a plus; I can just add them straight from here.*0304

*This is a negative number; this is a positive number; they have different signs.*0308

*You are going to find the difference of their absolute values.*0314

*The absolute value of -85 is 85; the absolute value of 96 is 96.*0317

*To find the difference between the two numbers, you are going to get 11.*0325

*96 is the number with the greater absolute value.*0331

*You are going to look at that sign which is a positive and then give that sign to the answer.*0336

*-44 divided by +11; now I can go ahead and divide.*0343

*-44 divided by +11; I have a negative here; I have a positive here.*0351

*I have one negative; my answer, my quotient, is going to be a negative.*0358

*Then 44 divided by 11 is 4.*0363

*Make sure when you solve a problem like this, you have to solve within the parentheses first.*0370

*Then you divide; one negative is going to make your answer negative.*0375

*Two negatives is going to make your answer positive.*0384

*This rule applies for when you multiply integers and when you divide integers.*0386

*Not when you add and subtract; the rules are very different.*0391

*That is it for this lesson on dividing integers; thank you for watching Educator.com.*0397

*Welcome back to Educator.com.*0000

*For this next lesson on integers and order of operation,*0002

*now that we went over all the different ways we can solve integers,*0007

*we are going to put them together into the same problem and solve it by using order of operations.*0012

*Remember that when you solve, you have to solve in this order.*0023

*Parentheses is always, always first; then it is exponents.*0029

*Then after that, it is multiplication and division.*0035

*This is the same; you don't multiply before you divide.*0039

*You just do it in order from left to right.*0045

*If you have multiplication and division, then you would just solve it in that order.*0049

*You don't solve one over the other; same thing for addition and subtraction.*0053

*If you have something to subtract before you add, then you would just do that*0059

*in order from left to right instead of adding first and then subtracting.*0064

*Again parentheses; then exponents; then multiplication and division; then addition and subtraction.*0070

*The first problem is going to be -4 plus 8 times -3 divided by 6 minus 2.*0080

*Within all these operations, I have to multiply and divide first.*0091

*Multiplication and division, there is no order between those two.*0099

*I just have to solve it out whichever comes left to right.*0103

*I am going to solve this first right here; 8 times -3.*0107

*When you multiply integers, you have to look at how many negative signs there are.*0114

*There is only one negative sign; only this one has a negative.*0122

*My answer is going to be -24.*0126

*After you solve one thing out, write out the rest.*0133

*-4 plus the 8 divided by 6 minus 2.*0138

*I rewrote the whole problem with only this solved.*0146

*From here, my next step will be to divide these.*0150

*This is going to be -24 divided by 6.*0155

*Again the same rule applies for the negatives; negative sign here; positive here.*0159

*My answer is going to become a negative; that is going to be a -4.*0166

*This is not a divide; don't get confused with that.*0175

*-4; rewrite the whole problem again; plus that, minus that.*0178

*Don't forget to include these signs; this sign was included when we solved it.*0184

*This was part of this answer; but this wasn't.*0190

*You are going to write out that sign here when you rewrite the whole problem.*0194

*-4 plus a -4, I have the same sign.*0200

*When you add integers, if they have the same sign, then you are just going to add up the numbers.*0207

*Then you are going to give the same sign.*0213

*For this one, -4 plus -4 is -8.*0216

*I am going to write out this right here, -8 minus 2.*0223

*I have a minus; I am going to use the two-dash rule; one, two.*0227

*-8 plus -2 is -10; all of this became -10.*0234

*I know it looks like a lot of work.*0243

*But it is only a lot of work because you are doing one little step at a time.*0245

*But just remember, you are multiplying integers, then dividing, then adding, and then subtracting integers.*0249

*The next one, 3*^{2} plus 2 minus 5 times 6.0261

*I have parentheses; parentheses always, always, always comes first.*0270

*I am going to solve this first; I have a minus problem, 2 minus 5.*0276

*I am going to use the two-dash rule to make this minus to a plus, the subtraction problem to an addition problem.*0282

*It would be one, two.*0289

*They have different signs; I am going to find the difference of their absolute values.*0293

*This is a 2; the absolute value of -5 is 5.*0298

*The difference will be 3.*0303

*I am going to give it the same sign as the 5.*0306

*That is a negative so this is a negative.*0308

*Then I am going to rewrite the whole problem.*0312

*This is 3*^{2}; don't forget this plus; then times 6.0316

*The next step on the order of operations is exponents.*0325

*I have to solve this exponent next; 3*^{2} is 9; plus -3 times 6.0330

*Then after that, I have adding and I have multiplying; multiplying comes first.*0339

*That will be again one negative sign here; my answer will be -18.*0346

*Write everything else out; you don't have to write everything out.*0355

*But that is the best way to do it because that prevents any mistakes that you might make.*0358

*9 plus -18; again we are adding integers.*0368

*They have different signs; you find the difference.*0372

*This is a 9; absolute value of this is 18.*0376

*If you find the difference of those, you get 9.*0380

*Give it the sign of the 18; that will be your answer, -9.*0383

*This next problem, I have parentheses that I have to solve out first, -15 plus 4.*0397

*Again we are adding integers; they have different signs.*0406

*Find the difference of their absolute values.*0410

*Absolute value of -15 is 15; absolute value of 4 is 4.*0413

*If you find the difference of those, it is 11 with a negative sign.*0419

*Then times another parentheses that I have to solve out.*0429

*Here is a minus problem; use the two-dash rule; one, two.*0433

*Again two-dash rule, do not apply it to any other negative signs.*0438

*This negative sign, you are not going to touch that.*0441

*The two-dash rule only changes the minus to a plus.*0444

*Then this number to either a negative or a positive.*0449

*-2 plus -8; they have the same sign; -2 and -8.*0454

*You are going to add the numbers, their absolute values; then give it that same sign.*0459

*-2 plus -8 is going to be -10.*0464

*-11 times -10; I have two negative signs.*0472

*That is going to make my answer a positive; 11 times 10 is 110.*0479

*That is it for this problem; that is the answer.*0488

*The fourth example, let's see, I have a negative.*0494

*This, divided by absolute value of that minus a -7.*0501

*The first thing I want to solve out is the parentheses which is right here.*0507

*2*^{3}; 2^{3}, be careful, 2^{3} is not 2 times 3.0513

*This is 2 times 2 times 2; I can just rewrite the whole thing.*0519

*Absolute value of -8 minus a -7; I have to solve this out first.*0530

*2 times 2 is 4; 4 times 2 is 8.*0536

*I am going to make this an 8.*0541

*-8 divided by absolute value of -8 minus a -7.*0544

*I want to solve this out too because absolute value is like parentheses.*0555

*Just solve that out first before you divide and do anything else.*0560

*Absolute value of -8 is 8 because the distance from 0 is 8.*0567

*I am going to write everything else out; don't forget these signs.*0580

*Minus a -7; write everything else out exactly the way it is.*0584

*-8 divided by 8 minus -7; I have to divide before I subtract.*0590

*Do this next; -8 divided by 8 is -1.*0597

*Because I have one negative sign, that is going to make my quotient a negative.*0602

*Then I am going to write all this out again; minus a -7.*0608

*I am subtracting integers; I am going to apply the two-dash rule.*0612

*Make this a plus; make that a plus; -1 plus a 7 here.*0617

*Again I have different signs so I am going to find the difference of their absolute values.*0625

*The absolute value of -1 is 1; absolute value of 7 is 7.*0630

*That is 6; then I am going to give it the sign of the 7.*0635

*That is it; that is your answer.*0642

*If any of these problems were confusing to you, if you forgot how to do some of these*0646

*like either dividing or multiplying, maybe adding and subtracting integers,*0653

*then just go back to the previous lessons and review over it.*0657

*Then try to come back here and figure out these problems too.*0660

*That is it for this lesson; thank you for watching Educator.com.*0665

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over writing expressions.*0002

*Expressions again are math statements without an equal sign.*0007

*We just have statements; we can use words.*0014

*We can use different operations using numbers and variables to express some kind of statement.*0018

*We are not going to use any equal signs.*0026

*Some words that we can use to express adding--addition, plus, and more than.*0030

*The minus, we have subtraction, minus, and less than.*0039

*For this, we have multiplication, times, and product.*0045

*You can also say multiplied or just multiplication.*0052

*This is division, divided by, quotient.*0058

*Remember this is probably going to be the hardest words to remember, product and quotient.*0062

*Product, just remember that you are multiplying; quotient, you are going to be dividing.*0068

*For these here, when you see the word than being used, you are going to actually switch the order.*0075

*For example, if I say 2 more than X, we know that this right here means plus.*0087

*But instead of saying 2 plus X, we are going to do a switch.*0101

*That is every time you see the word than, you are going to switch the order.*0107

*This is not 2 plus X; instead it is going to be X plus 2.*0112

*Same thing for less than; if it is 2 less than X, it is going to be X minus 2.*0119

*We are going to do some examples.*0131

*We are going to write each as an expression; here it says 7 plus X.*0134

*We know that plus means addition; this will be 7 plus X.*0140

*That is it; that would be your answer; that is how you write an expression.*0148

*The product of 4 and 3; when it says product, you are talking about multiplication.*0153

*The product of something and something, you are going to be multiplying two things, this one and this one.*0164

*That is going to be 4 times 3.*0171

*That would be how you are going to write it as an expression.*0175

*You can also simplify it if you can.*0179

*We know that we can't simplify these because this is a number and this is a variable.*0181

*So we are going to leave it like that; this is 4 times 3.*0186

*If you want, you can also write this as that, simplified.*0190

*Next one is C divided by 5 which is going to be C divided by 5.*0198

*Also with division, you can write it like this.*0206

*C divided by 5; this is also C divided by 5.*0213

*This is a fraction but fractions are also division.*0217

*They are also the top number divided by the bottom number.*0221

*C divided by 5; you can write it either like this or like this.*0225

*2 less than A; we know that less than means minus.*0233

*But then here we see the word THAN.*0239

*Whenever we see that, we are going to switch the order.*0242

*It is not going to be 2 minus A; it is going to be A minus 2.*0246

*This one comes first; this one is going to come last.*0253

*If you write 2 minus A, that is actually going to be wrong.*0258

*You have to make sure that you switch them; that becomes A minus 2.*0261

*Let's do a few more; the sum of 10 and K.*0269

*The sum you know is addition.*0274

*We are going to be adding two things, this one and that.*0278

*It is going to be 10 plus K.*0283

*9 minus 10; this is... who knows?... minus.*0291

*You are just writing it using the actual operations like that.*0297

*This one again you can simplify because you are just subtracting two numbers.*0302

*This is 9 minus 10; if you do 9 minus 10, it is not 1.*0307

*It is actually going to be -1.*0311

*Here we have a positive number.*0316

*This minus is actually going to be part of this number.*0318

*The sign actually goes with whatever is behind it.*0322

*This becomes a -10; I can write a plus here.*0325

*9 minus 10 is the same thing as 9 plus -10.*0330

*This minus is the same thing as negative.*0337

*If you want, you can change this to a plus problem and make that a negative right there.*0341

*Here if you have 9 of something, say you have $9 but you need 10.*0348

*Let's say you borrow 10; how much do you have left?--you have -1.*0356

*12 more than Z; more than we know is plus.*0365

*It is not going to be 12 plus Z because we see that word right there.*0371

*We are going to switch them; it becomes Z plus 12.*0375

*That we can't simplify; that is the answer for that.*0381

*The quotient... quotient means divide... of P and 4.*0385

*It is going to be P divided by 4.*0393

*Or you can say P divided by 4 as a fraction.*0398

*These we are going to write them using words.*0409

*For the minus, for that operation, we can just say minus.*0415

*We can say less than; we can say subtracted.*0420

*Here I can say 10 minus 4; or if I want to use less than.*0425

*Since you are using this word, remember you have to switch them; don't forget.*0439

*You can say 4 less than 10.*0443

*This one, 6 divided by 2; just write it out like that.*0448

*Or you can say the quotient of something and something.*0459

*That will be 6 and 2.*0475

*Here we can use the word plus; we can use more than; or we can use sum.*0480

*If I am going to use plus, then I just say B plus 3.*0496

*I am going to use more than.*0500

*Since you have to switch them, I am going to say 3 more than B.*0502

*Or for the sum, I can say the sum of something and something.*0508

*That will be, in order, B and 3.*0516

*This one I can use times; I can say multiply; I can say product.*0522

*It will be just be 5 times 5.*0535

*If I am going to use product, then I have to say the product of something and something*0538

*which will be 5 and 5, the product of 5 and 5.*0545

*That is it for this lesson; thank you for watching Educator.com.*0552

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to be writing equations.*0002

*An equation is a math statement with two expressions equaling each other.*0007

*The previous lesson, we went over expressions, how to write expressions.*0014

*Remember an expression is a math statement that also uses numbers and variables.*0017

*An equation is when two of those expressions are equal to each other.*0025

*The main difference between expression and equation is that equation has an equal sign.*0030

*If you look at the first four letters of equation, it is almost the full word equal.*0041

*Think of equation as equal or having an equal sign whereas expressions do not.*0051

*If I just said A plus 3, A plus 3 without equaling 5, that would be considered an expression.*0059

*But if you see A plus 3 equal to 5, then that becomes an equation.*0070

*Here we are going to use words and operations into words to write this two ways,*0079

*as an equation using numbers and variables and using just words only.*0091

*To go over our different operations, this for addition, we can use plus.*0101

*We can use more than.*0108

*For the subtraction, we can use minus; we can use less than.*0111

*For multiplication, we can use times or product.*0116

*For division, we can say divided by; we can say quotient.*0122

*This one, when you see the word is, that means equals.*0129

*You can say A plus 3 equals 5; then you can say A plus 3 is 5.*0136

*Also to go over this, more than and less than, when you see the word than,*0146

*just like when we were writing expressions, you have to switch.*0150

*For example, if I say A more than B, more than means plus.*0155

*But instead of saying A plus B in that order, we have to switch those two.*0167

*Instead of A plus B, you are going to write B plus A.*0174

*It is for this one and this one.*0180

*Whenever you see the word than, you are going to switch them.*0182

*Let's do a few examples; write each as an equation.*0188

*For the first one, 10 minus A is 5.*0192

*We know that... we can write that...minus means minus; A.*0197

*Then is you know is equal; and 5.*0205

*That is it; that would be our equation; 10 minus A equals 5.*0211

*The second one, the product of 7 and 8 is X.*0217

*Product means times; 7 times 8 is X.*0223

*I know that this means times; this means product.*0239

*But when I start writing equations, I don't want to use this anymore.*0244

*I don't want to use the X to represent multiplication because, X, we use that as a variable.*0248

*Since we are using it as a variable, as an unknown number, unknown value,*0257

*I don't want to write it here because it looks like I have a variable.*0261

*Instead of using the X to represent times, you can either for now write the little dot.*0269

*But even this later on, you are not going to be able to do that anymore.*0279

*The best way to show two numbers being multiplied together is in parentheses.*0284

*To write each of them in parentheses.*0290

*That is the best way to represent multiplication, two numbers being multiplied together.*0292

*If you see two numbers written in parentheses like this, then that means times.*0299

*It means 7 times 8.*0305

*The next one, a number plus 4 is 10; here it just says a number.*0310

*A number plus 4 is 10; we know this is plus 4 equals 10.*0317

*But we don't know that that is; anything unknown, we write as a variable.*0325

*You can just pick whatever variable; you can say A.*0333

*You can say B; you can say D; whatever your favorite letter is.*0336

*You are going to write that as a number.*0340

*The next one, 6 is P divided by 3; 6 equals P divided by 3.*0346

*Or divided by can also be written as a fraction.*0359

*This can be 6 equals P divided by 3.*0365

*You can write it like this; or you can write it like that.*0373

*Let's do a few more; 12 is 2 more than A.*0381

*12 is 2 more than A; we know more than is plus.*0387

*But you see that word right there.*0393

*Instead of writing 2 plus A, I have to write A plus 2.*0396

*You are going to leave it like that.*0405

*A number multiplied by 2 is 20; a number; again we see that, a number.*0409

*I can say M multiplied, times, 2 is 20; M times 2 equals 20.*0417

*When it comes to multiplying letters and numbers together, you don't have to write it like this.*0433

*We can just stick them together with the number in front.*0442

*I can write M times 2 as 2M.*0445

*When you have a number in front of a variable like that, it means multiply them.*0450

*This is 2 times M; you can write it in that way.*0456

*Here they said a number first; a number multiplied by 2.*0461

*You would think you write it like that.*0467

*But when you have a number times a variable, you always write the number in front.*0469

*You don't have to write anything between them.*0477

*You don't have to write the dot; you don't have to write parentheses.*0478

*Only when it comes to numbers with variables.*0481

*Or you can do it with variable to variable also.*0485

*If it is M times N, then you can just write them together like that.*0487

*Just be careful, you cannot write it with a number being multiplied to a number.*0493

*If I want to say 2 times 3, if I put them together like that, then that just becomes 23.*0497

*That doesn't say 2 times 3 so you can't write it like that.*0505

*Only when you are multiplying a number with a letter, a variable, or variable with variable.*0509

*Only when a variable is involved, you can write them together.*0517

*M times 2 equals 20; this can be written as 2M equals 20.*0522

*The quotient of a number and 4 is 40; quotient is divide.*0532

*Quotient of... what are the two things that we are dividing?--a number and 4.*0541

*A number again is just a variable; you can say Z.*0548

*Z divided by 4 equals 40; again we can write this as a fraction.*0556

*This can be Z over 4 equals 40; Z divided by 4 equals 40.*0569

*The last one, 2 added to a number is 21.*0581

*2 added to a number P equals 21.*0588

*Again you can use whatever variable you want; 2 plus P equals 21.*0598

*Write the equations using words.*0610

*You can write this several ways because we know that addition, we can use plus.*0614

*We can use added; we can use the sum.*0619

*Here we can say 5 plus Z is 9; I am going to use sum.*0628

*I am going to say the sum of 5 and Z.*0636

*If you use sum, then you have to write the two things that you are going to be adding together.*0648

*5 and Z, those are the two things; equals translates to is; 9.*0654

*The sum of 5 and Z is 9.*0666

*Here 3; this is is; 18; 18 minus A.*0671

*You can also say 3 is A less than 18.*0691

*Remember that here, I had to switch these because of this.*0706

*The next one, the product of 2 and B; or you can say 2 times B.*0714

*The product of 2... be careful, you are not going to say 2 times B because you already said product.*0726

*You already used that word to show operation; here you are going to write and.*0739

*You are saying between this and B; this is... and then 10.*0748

*The product of 2 and B is 10.*0762

*The fourth one, 36 divided by 3 is 12.*0768

*Or I can say the quotient of 36 and 3 is 12.*0772

*It is okay if yours is a little bit different than what I wrote as long as you use the words correctly.*0797

*Make sure if you are going to use less than or more than here, then you are going to switch.*0804

*You can say Z more than 5 is 9.*0810

*We are going to determine if the equation is true or false.*0820

*The first one, 12 is the sum of 10 and 2.*0824

*I am going to write it as an equation first.*0831

*12 equals... we know is is equals... the sum... we are going to add.*0832

*We are going to be adding what and what?--10 and 2.*0840

*10 plus 2; is this true?--12 equals 10 plus 2?--yes it is true.*0844

*Number two, the product of 6 and 3 is 15.*0857

*Product means we are going to be multiplying.*0861

*6 and 3, remember if you are going to multiply two numbers,*0866

*the best way for you to write that is to write them each in parentheses.*0869

*6 and 3, is, equals, 15; 6 times 3 we know is not 15.*0875

*This one is false.*0888

*The next one, 8 minus 10 is 2; equals 2.*0895

*This may look right; this may look true.*0906

*But it is actually not because here this is 8 minus 10.*0909

*If I had 10 minus 8, this equals 2.*0916

*But that is not the equation; it is 8 minus 10.*0923

*Remember this number is smaller than this number.*0926

*If I have a number line, this is 0; say this is 8.*0935

*If I am going to start at 8 and then go backwards 10 because that is what minus says,*0947

*then I am going to be... all the way to here is... that is 8.*0953

*I moved 8 spaces; but then I have to move 2 more.*0959

*Where am I going to land?--if I move 10 spaces, then I am going to land at -2.*0964

*Then this is not true; this is false because 8 minus 10 is -2.*0972

*Again you are starting at +8 and then you are going to move backwards 10 spaces on the number line.*0979

*You are going to land at -2; so this is false.*0986

*You can also think of this as +8 and -10.*0997

*Remember minus and negative is the same thing.*1002

*If you want, you can circle this sign with that number.*1006

*This is +8 and -10; we know that this is the bigger number.*1010

*We have to give this the sign of that number if you remember from going over integers; +8 minus 10.*1018

*Another way you can think of this for integers, if you have 8 apples.*1028

*It is a +8; you have something; you have 8 apples.*1035

*But let's say you need 10 apples.*1039

*Say you are going to bake an apple pie or something.*1042

*You need 10 apples; negative means that you need it or you borrowed it.*1044

*You have 8; you need 10; are you short?--do you have 2 left?*1052

*No, you don't have 2 apples left.*1059

*You used up all of your apples and you need 2 more.*1060

*That would be a -2 because you are short 2; that is a -2.*1065

*It would be 8 minus 10 would be -2.*1072

*That is it for this lesson; thank you for watching Educator.com.*1080

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to be solving equations that involve adding and subtracting.*0002

*To solve an equation means to solve for the unknown variable.*0011

*Whenever you have an equation and you have one variable only, you can solve for it.*0018

*Once you get the variable by itself, you are going to solve the equation.*0026

*In order to get the variable by itself, you are actually going to use inverse operations.*0032

*If you have A plus 1 equals 2, here A this is what you are solving for.*0037

*That means I want to get rid of this number.*0050

*I want to get rid of this number so that I have A by itself.*0058

*Some equations are easy to do; you can do it in your head.*0063

*I know something plus 1 equals 2; that something has to be 1.*0066

*A has to be 1 because 1 plus 1 equals 2.*0073

*But when you write it, you are going to write it like that.*0076

*I'm sorry... like this because you know that A is 1.*0081

*This is how you are going to have to write the answer.*0089

*When the equation gets a little bit tough, a little bit harder,*0092

*you want to use inverse operations to solve for this variable.*0096

*because once you have the variable by itself like this here, you have solved the equation.*0102

*If I want to solve for this variable, I have to get rid of this 1.*0111

*The inverse operation of addition is subtraction; it is like the opposite.*0117

*What is the opposite operation?--it is subtraction.*0124

*What is the opposite of subtraction?--it is addition.*0128

*Here this is A plus 1; this involves addition.*0133

*That means the inverse operation would be subtraction.*0138

*We would have to subtract this to get rid of it because 1 minus 1 equals 0.*0143

*That is how you get rid; that is how you make it go away.*0151

*But whatever you do to one side of an equal sign for an equation, you have to do to the other.*0155

*If I am going to subtract 1 from this, this is the left side.*0162

*If I subtract one here, then I have to subtract one over here too.*0168

*Whatever you do to one side, you must do to the other side.*0172

*This goes away.*0178

*We want it to go away because we want to get the variable by itself.*0180

*We want to isolate the variable; that is why we had to subtract the 1.*0183

*This becomes A equal to... 2 minus 1 is 1.*0189

*Again in order for you to solve an equation, if you can't do it in your head,*0197

*if you can't use mental math, you want to get the variable by itself*0204

*by getting rid of whatever is next to it on that side of the equal sign.*0211

*In this equation here, it is a +1.*0216

*To get rid of a +1, I need to subtract it; do the inverse operation.*0220

*Subtract it; because it is +1, I need to subtract 1.*0226

*Whatever I do to one side, I have to do to the other side.*0230

*This will cancel out because it will be a 0.*0234

*You are just going to bring this down.*0237

*This is the only thing left on the left side.*0239

*What is left on the right side of the equal sign?*0242

*2 minus 1, you have to solve that out; that becomes 1.*0244

*Once the variable is by itself, you have solved the equation.*0248

*Let's do a few examples; let's try it.*0255

*For these problems, let's just use mental math.*0258

*That means we are just going to do it in our head.*0260

*4 minus a number is 3; what do you subtract from 4 to get 3?*0263

*Isn't that 1?--4 minus 1 equals 3.*0271

*Instead of just writing 1, I want to write that my variable A is 1.*0275

*You would have to write it like this.*0282

*You are writing that the variable, the unknown value, is 1.*0285

*X minus 3 is 4; what number subtract 3 will give you 4?--7.*0292

*Instead of just writing 7, I am going to write X is 7; X equals 7.*0303

*Here 5 plus something equals 11; 5 plus 6.*0311

*Again V, the variable, equals 6.*0316

*26 equals 8 plus C; 8 plus what equals 26?*0322

*If you subtract it from here, that will be 18.*0331

*C equals 18 because 8 plus 18 is 26.*0338

*For these, we want to use the inverse operation so that we can solve.*0346

*That means if it is plus, I want to subtract; that is the inverse operation.*0352

*Inverse operation of plus is minus; inverse operation of minus is plus; the opposite.*0363

*I am just going to draw a line just to split my two sides, my left side and my right side.*0374

*I want to solve for this variable A.*0382

*I want to keep going until I have A all by itself on the left side.*0387

*Since I want this to be by itself, I have to get rid of this number here.*0392

*That means in order to get rid of it, since it is +3, how do I get rid of a +3?*0399

*I have to subtract 3; the inverse operation of plus is minus.*0407

*I am going to subtract it so that I can make it go away.*0412

*Whatever I do to one side, remember I have to do to the other side.*0416

*On my left side, since this is no longer there, all I have left is this A.*0426

*I can just write A down there.*0432

*Because this becomes 0, I don't have to write A plus 0.*0435

*A plus 0 is just A.*0438

*I am going to write A; then bring down the equal sign.*0440

*10 minus 3, I need to solve this out on the right side.*0445

*10 take away 3 is 7.*0448

*This one is an easy equation; you can just do it in your head.*0452

*You know that 7 plus 3 is 10.*0454

*But we want to be able to know how to solve equations in this way using inverse operations.*0457

*My answer is A equals 7. *0463

*The next one, -2 plus C equals 11.*0468

*Again I am going to draw a line through just to separate my two sides--left side, right side of the equal sign.*0475

*Again what am I solving for?--always know what you are solving for.*0484

*Here is my variable; I want it to be by itself on the left side.*0488

*It is not by itself; it has this number right here next to it.*0493

*I want to get rid of it; how do I get rid of a -2?*0497

*This is negative; again it is the same thing as minus.*0502

*Minus, negative, they are the exact same thing.*0506

*If this is a -2, if I owe $2, how do I get rid of that?*0511

*How do I make that into 0?--I have to give a +2.*0517

*-2 plus 2; that makes 0.*0523

*Whatever I do to this side, I have to do to the other side.*0527

*If I add 2 here, I have to add 2 here.*0532

*What is left on my left side?*0538

*This went away; this is just a +C.*0539

*I don't have to write a plus in front of it because plus C is the same thing as +C.*0544

*Even if I don't write the positive sign, I know that the C is positive still.*0550

*This is C equals... 11 plus 2 is 13; C equals 13.*0558

*For this next one, a common mistake here would be to subtract this 4 to the other side.*0568

*But again you are solving for the variable.*0580

*This time, the variable is on the right side; that is OK.*0584

*Just make sure that you identify the variable.*0588

*You want to get the variable by itself on whichever side it is on.*0591

*That means I want to get rid of this number here because that is on the same side of the variable.*0595

*The whole point is to get the variable by itself.*0604

*To get this by itself, let's get rid of that number; this is a +5.*0609

*If there is no negative sign in front of it, then it is always a positive.*0614

*This is 5 minus 5 to get 0.*0619

*Whatever I do to one side, I have to do to the other side.*0624

*This is a positive.*0632

*Again if there is no sign in front of it, it is a positive.*0633

*+4 minus 5; +4 minus 5.*0636

*With these integers, it still might be a little bit difficult to do 4 minus 5.*0643

*If it is, think of this as having 4 apples.*0654

*But again you need 5 apples.*0661

*If you have 4, you need 5, you are short.*0665

*If you need 5 apples but you only have 4, then you need 1 more.*0671

*Whenever you need something, whenever you don't have it, it is a negative number.*0675

*You can also think of this as having 4 dogs.*0681

*Let's say dogs are positive numbers; D is for dog; we have 4 dogs.*0690

*Whenever you have a negative number, that is going to be a cat.*0697

*We are going to have 5 cats because we have a -5.*0701

*A dog and a cat cancels out; those cancel out; those cancel out; and cancels out.*0707

*What do you have left?--you have 1; is this a positive or negative?*0715

*Cats are negative; that would be a -1; here this becomes -1.*0721

*Be careful, do not put 1; this has to be a -1.*0729

*If you want, you can also use a number line.*0734

*This is again only if you are having trouble with integers.*0738

*Here is a 0; I am going to start off at my first number.*0742

*That is 4; I am going to start here; then minus 5.*0747

*If I subtract, then I have to go this way; how many?--5.*0753

*I am going to go 1, 2, 3, 4; then I have to go one more, 5.*0759

*What is 1 more?--this is a -1.*0766

*This is not 1; 1 is right here; -1 is right there.*0770

*You can do it that way too.*0777

*Continuing, this I bring down; that went away.*0784

*That was the whole point--to make that go away so that this variable M will be by itself like that.*0788

*Now that I have the variable by itself, I have my answer.*0798

*This is the same thing as... if you want, you can rewrite it like this.*0801

*If you don't like that variable on that side,*0810

*since you are probably used to having variables on this left side, you can rewrite it.*0812

*-1 equals M; or you can say M equals -1.*0819

*For this next one, be careful here because the variable is here.*0826

*The variable is already by itself.*0834

*There is no need to move anything over.*0837

*There is no need to add, subtract, do any inverse operations.*0839

*We can just go ahead and solve this.*0843

*2 plus -2, if I have 2 of something, I take away 2.*0847

*It is like having 2 apples and then eating 2 apples.*0854

*You have 0 apples left.*0856

*Equals; then D; this is my answer.*0860

*0 is still a number; see how we have 0 right there?*0870

*It is still a number; D equals 0; that would be your answer.*0875

*Or again if you want to write the variable first, you can write D equals 0 like that.*0881

*They are both correct.*0888

*Let's do a few more; solve each equation; we are going to use inverse operations.*0890

*Again we are solving for B.*0898

*If you want, you can circle it to help you know what you are solving for and help you to see what to get rid of.*0901

*I am going to separate my sides with the line.*0910

*You don't have to draw the line; but it helps to see it.*0912

*This is the left side; this is the right side; you are moving this.*0915

*You are getting rid of it by doing the inverse operation to this side and to that side.*0921

*Whatever you do to one side, you must do to the other side.*0928

*The inverse operation of minus is plus.*0933

*I have to add 7 so that this will go away; add 7 here.*0938

*This is nothing; what is left on my left side?--on that side?*0948

*B equals -1 plus 7.*0952

*Again if the numbers are small, then you can just use the dog-cat example.*0959

*You can also say you need 1 apple; you have 7.*0964

*After you use that 1 that you need, how many do you have left?*0974

*You have 6 left.*0978

*To show you the dog and cat example again, a negative number is a cat.*0981

*That is 1 cat; then 7 dogs; 1, 2, 3, 4, 5, 6, 7.*0988

*7 Ds represents a +7; this cancels; how many dogs do you have left?*0998

*1, 2, 3, 4, 5, 6; this is 6; is that positive or negative?*1006

*Dogs are positive; it is +6; that is my answer.*1011

*Then here again you are solving for E; you can separate the sides.*1020

*I have to get rid of whatever is next to it which is the 4.*1026

*You don't have to get rid of pluses because plus is the same thing as positive.*1030

*Just like minus is the same thing as negative, a plus is the same thing as a positive.*1035

*Here this is a +4; there is no negative sign in front of it.*1042

*It automatically gets a positive; if that helps, you can write that in.*1047

*To get rid of a +4, you have to subtract 4; you have to take 4 away.*1052

*Whatever you do to one side, you have to do to the other side.*1057

*This becomes nothing; E is left only on that side which is what we want.*1063

*Equals; this is dog, dog, cat, cat, cat, cat; cancel, cancel.*1069

*This becomes 2; cats are negative; E is -2.*1081

*The next one, circle the variable.*1092

*Here this is what I have to get rid of because it is next to the variable on that side of the equal sign.*1101

*To get rid of a -6, I have to add 6; that is the inverse operation.*1108

*Whatever I do to one side, I have to do to the other side.*1113

*This goes away; P by itself now equals -5 plus 6; 1.*1118

*If you borrowed 5, you have 6, so after paying that 5 back, you have 1 left over.*1134

*For this one, you can circle and draw the line.*1147

*You are getting rid of the 3, not the 8, because the 3 is on the same side of the variable.*1153

*The whole point is to get the variable by itself.*1159

*Again this is a +3; I need to subtract 3 to make this 0.*1163

*Whatever I do to one side, I have to do to the other side which is over here.*1169

*Be careful, you don't do it to the same side.*1173

*It has to be to the other side.*1176

*That is why you draw this line just to make it easier to see this side and that side.*1179

*Went away; what is left?--my variable equals 5.*1187

*For these, we are going to translate to an equation first.*1200

*Then we are going to solve them.*1203

*The sum of A... this is the variable A... and 8 is -1.*1206

*Sum we know is plus of A and 8.*1216

*That means we are going to add A and 8 together.*1222

*A plus 8, is means equals, -1.*1226

*To solve this, again we are going to solve for A.*1235

*Separate the two sides to get rid of this because that is still on the same side.*1243

*Subtract it; that is the inverse operation.*1251

*Whatever I do to this side, I have to do to the other side.*1257

*Went away; A is left by itself; equals -1 minus 8.*1263

*Here is a negative; here is a negative.*1271

*You have 1 cat; you have 8 more cats.*1273

*How many cats do you have total?--9 cats.*1276

*Cats we know are negative so it has to be a negative number.*1281

*There is my answer; A is -9.*1286

*The next one, 9 minus A is 7; 9 minus A equals 7.*1293

*I am solving for this.*1309

*For this one, we don't have to use the inverse operation.*1315

*If you can solve it in your head, if it is easy, then you can just go ahead and do that.*1319

*We know 9 minus what equals 7?--that is 2.*1324

*Just be careful that you are not going to just write 2 as your answer.*1329

*If you just write 2, you have to identify it as your variable.*1333

*That is what you found A to be.*1338

*You want to write A equals 2 because 9 minus 2 equals 7.*1344

*Next one, K more than 14 is 20.*1354

*We know more than is plus; this is plus.*1359

*But then I see this word here.*1365

*Remember whenever you see that word, you have to switch them.*1367

*You have to write it in the opposite order.*1371

*Instead of K plus 14, you are going to say 14 plus K is 20.*1375

*Again you are solving for K; you can just do this in your head.*1386

*You know that 14 plus 6 more is going to give you 20.*1392

*You can say K equals 6.*1396

*Or if you want, you can just practice doing the inverse operation.*1401

*You are going to subtract 14 because this is a +14.*1405

*Whatever you do to one side, you have to do to the other side.*1409

*Goes away; K is left by itself which is what you want.*1415

*Equals, this is 6.*1419

*The last one, 5 less than Q is 11; 5 less than Q is 11.*1429

*Less than we know is minus; but then again here is that word.*1438

*It is going to be Q minus 5 equals 11.*1445

*A number Q, if you take 5 away, is going to be 11.*1453

*We know Q is going to be 16.*1459

*Again just to show you the inverse operation, I am solving for Q so get rid of everything on that side.*1463

*You are going to add 5 to get rid of it.*1471

*Whatever you do to one side, you have to do to the other side.*1474

*You are going to have Q equal to 16.*1480

*That is it for this lesson; thank you for watching Educator.com.*1490

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to continue solving equations.*0002

*We are going to solve multiplication equations.*0005

*Equations that involve multiplication, we are going to continue to solve using inverse operations.*0010

*Again whenever you solve equations, you always have to try to get the variable by itself.*0023

*The inverse operation of multiplication is division.*0030

*If I have a number being multiplied to a variable, 2 times A...*0037

*Remember if you have a number times a variable, you can write it together like that.*0044

*Equals 10; that is my equation; this would be a multiplication equation.*0050

*This 2 times A, 2 times the variable; we can do this in our head.*0056

*We know that 2 times 5 equals 10; we know A is 5.*0062

*But if I were to solve this using inverse operations, again I want to get the variable by itself.*0066

*I have to get rid of whatever is next to the variable on that side.*0078

*On the left side, I have to get rid of everything except for the variable and get the variable by itself.*0082

*That means I have to get rid of this 2.*0091

*Since this is 2 times A, the inverse operation would be to divide.*0094

*To get rid of the 2, we have to divide the 2; divide this 2.*0101

*We know that 2 over 2 is going to go away.*0107

*It is going to become 1.*0109

*Whatever you do to this side, remember you have to do to the other side.*0112

*If I divide 2 from here, then I have to go to the right side and then divide 2 there.*0116

*This then becomes 1A; 1A is the same thing as A.*0126

*Whenever you have a variable with no number in front of it like this one does, there is an invisible 1 here.*0139

*It is just saying that you have 1 A.*0148

*How many As do you see?--you see 1 of them.*0150

*If I say I have an apple, you know I have only 1 apple.*0154

*I didn't say I have 1 apple.*0159

*But just because I said I have an apple and I made it singular, you know that I have 1.*0161

*In the same way, if I have an A, you know that I have 1 of them.*0167

*That just means that there is an invisible 1 in front of it.*0172

*When this number cancels out like that, you don't have to write 1A.*0176

*You can just write A which is the same thing as getting the variable by itself.*0180

*Again whether you write the 1 in front of the A or just leave it as A, it is the exact same thing.*0186

*By itself now, that is the whole point; you want to get it by itself.*0196

*We got rid of the 2; equals... on the right side, I have to actually solve that out.*0199

*That becomes 10 divided by 2.*0207

*Remember this line right here like a fraction; that represents divide.*0211

*This would be 10 divided by 2.*0216

*We know that 10 divided by 2 is 5; that would be my answer.*0220

*That is how you would solve multiplication equations using inverse operation.*0227

*Let's go ahead and do our examples.*0234

*The first set of examples, we are going to use mental math*0237

*meaning we are just going to solve it in our head.*0240

*We don't have to divide or use inverse operations.*0241

*Here again this means 3 times F; 3F means 3 times F.*0247

*3 times what is 9?--3 times what equals 9?*0257

*I know 3 times 3 equals 9; F has to be 3.*0263

*Again when you are solving equations, you don't want to just write 3.*0268

*You don't want to just write the number.*0271

*You have to write what that number represents; you are saying that F is 3.*0273

*Once you write it like that, variable by itself equaling the number, then that is your answer.*0283

*10 times what equals 100?--10 times 10 equals 100.*0290

*Then I have to say A is equal to 10.*0296

*18 equals C times 6; this C times 6, same thing.*0305

*Whether you see it like this or whether you see it like that, they both mean multiplication.*0314

*What times 6 is 18?--I know 3 times 6 is 18.*0321

*That means C has to be 3.*0327

*The next one, 21 equals 3 times a number; 3 times what equals 21?*0335

*3 times 7 equals 21; that means P has to be 7.*0344

*Now let's use the inverse operation to solve the equation.*0357

*We are going to use that method that we did earlier.*0360

*We are going to use that same method to go out and solve for our variable.*0364

*This is -10 times S equals 100.*0370

*I am solving for S; I am solving for my variable.*0377

*I can circle it just to see that that is what I want.*0381

*That is my goal; that is what I am solving for.*0385

*I am separating my sides.*0388

*Since this is -10, that number times S, inverse operation of times is divide.*0393

*To get rid of this -10, I have to divide it.*0404

*I am going to use the inverse operation to get rid of the number and get the variable by itself.*0407

*I need to divide.*0415

*Remember this line right here, writing it as a fraction; that means divide.*0415

*Whatever I do to one side, remember I have to do to the other side.*0423

*I have to divide this side by -10 also; what is left on this side?*0428

*On my left side of the equal sign, I got rid of that number.*0435

*I want to get the variable S by itself.*0440

*Now that I got rid of -10, I have S by itself now.*0444

*I am going to write S; that is what is left on my left side.*0448

*Equals... what became of my right side?--100 divided by -10.*0452

*Remember when you multiply or divide integers, meaning positive and negative numbers like this, you still get the same number.*0459

*Let's say I don't see that negative sign.*0474

*Then I am still going to do 100 divided by 10.*0476

*Whether or not this number is negative or positive, you are still going to divide 100 to 10.*0480

*But if you have one negative number, if only one is negative, then your answer becomes negative.*0487

*It is the same number when you divide.*0496

*You are just going to do 100 divided by 10.*0498

*But you only see one negative sign in that problem.*0502

*Then my answer becomes a negative.*0506

*Same number; just a negative sign in front of it.*0509

*If I have two negatives signs, whether it is a negative times a negative*0513

*or a negative number divided by a negative number, whenever you see two of them,*0520

*those two negative signs will pair up and become a positive.*0526

*This is only when you multiply or divide; two negatives make a positive.*0532

*One negative, it remains a negative.*0541

*If you have two negatives, it becomes a positive number.*0544

*This is 100 divided by 10; that is still 10.*0549

*But because there is only one negative sign here within this problem,*0553

*there is only one, so then my answer becomes a negative.*0558

*S equals -10; that is my answer.*0562

*The next one; again you are solving for T; separate the sides.*0571

*I have to get rid of the 2.*0579

*To get T by itself, I have to get rid of the 2.*0581

*This is 2 times T; my inverse operation, I have to divide; divide the 2.*0584

*Remember this line means divide also; that goes away.*0591

*Then I have to divide this side by 2.*0595

*On my left side, I only have T left which is what I want.*0602

*Equals -16 divided by 2.*0606

*A +16 divided by 2 is 8; I know 2 times 8 is 16.*0611

*But since I have only one negative sign within this problem, my answer, it stays a negative.*0619

*Be careful not to confuse multiplying and dividing positive and negative numbers and adding and subtracting positive and negative numbers.*0628

*When you add two negative numbers, that doesn't become a positive.*0635

*Only when you multiply or divide two negative numbers; so that is my answer.*0641

*For the third one, I am solving for my variable R.*0653

*It is on the right side of my equal sign.*0659

*This is my left side; this is my right side; I am solving for R.*0661

*I have to get rid of the 5 because R, the variable, has to be by itself.*0667

*This is 5 times R.*0673

*I have to get rid of the 5 using the inverse operation, dividing.*0675

*That is my way of making that go away.*0681

*Since I did it to this side, I have to do to the other side.*0685

*Now I am going to simplify; I am going to solve everything out now.*0691

*25 divided by 5 is 5.*0694

*We don't have to worry about any positive or negative numbers because there is no negatives.*0697

*25 divided by 5 is just 5; bring down the equal sign; this is R.*0700

*R by itself because we got rid of the 5; that is my answer.*0707

*Again if you want, you can leave it like that as your answer.*0712

*Or you can say R equals 5.*0715

*You can rewrite it 5 equals R or R equals 5.*0718

*It is the same exact thing.*0722

*This last one, again I am solving for D, the variable.*0727

*Separate my two sides; this is 8 times D.*0733

*To get rid of it, inverse operation would be to divide.*0736

*Whatever you do to one side, you have to do to the other side.*0742

*What is left here?--D only, equals... 64 divided by 8 is 8.*0747

*But because I have one negative sign when I am dividing numbers, my answer becomes a negative.*0755

*It is -8; that is my answer.*0761

*Here, is -2 a solution of each equation?*0771

*They are asking is -2 the answer for the variable?*0777

*That means I can plug this in.*0786

*I can substitute a -2 for the variable to see if my equation is going to be true or false.*0788

*4 times S; instead of writing S, I can write -2 to see if -2 is what S is going to equal.*0797

*4 times -2; then remember the best way to show two numbers being multiplied together*0810

*is to write each of them in parentheses like that; equals -8.*0818

*You are just seeing if 4 times -2 equals -8.*0827

*4 times -2 is -8; 4 times 2 is 8.*0832

*You only have one negative sign; that makes that negative.*0837

*They do equal each other; this does equal -8; so this one is yes.*0841

*Is this -2 a solution for this equation?--this one is yes.*0847

*Next one, 10 equals 5 times -2; I want to write it out.*0855

*Again I can write both of these in parentheses to show that those are two numbers being multiplied together.*0866

*What is 5 times -2?--isn't this -10?*0873

*because again 5 times 2 is 10 but then you have only negative number.*0880

*So this is not true; this one is no or false.*0884

*This one, -6 times -2 equals +12.*0892

*Again I am going to write that in parentheses to show that I am going to multiply them.*0900

*6 times 2 we know is 12; here I have two negatives signs.*0909

*For this, the two numbers that I am multiplying, they are both negative.*0915

*That means I have two negatives which makes a positive.*0919

*When you multiply or divide, two negatives make a positive.*0923

*This becomes +12 or just 12; I don't even have to write the positive.*0927

*12 equals 12; this one is yes; this one works.*0933

*Let's do a few more; we are going to solve this using inverse operation.*0944

*Here again we are solving for the variable.*0952

*Circle it; draw a line to separate the sides.*0954

*This is -4 times W; I am going to divide -4.*0959

*Whatever I do to one side, I have to do to the other side.*0965

*That was my way of making that number go away and make the variable by itself.*0970

*Equals... 28 divided by 4; we have a negative divided by a negative.*0975

*Does that make it a positive when you divide two negatives?*0987

*Yes; that becomes a +7 because 4 times 7 equals 28.*0992

*Be careful; this is a +7.*1000

*Or if you just write 7 without the plus sign, that is okay.*1003

*Number two, you are solving for N; I am going to separate my sides.*1012

*Here again I need to get rid of this number.*1022

*Do not divide this number; do not try to move this number; don't use this.*1025

*We are going to try to get rid of this number because that is what is next to the variable.*1032

*Again we are trying to get the variable by itself.*1037

*Divide -8; again I divided because that was -8 times N.*1041

*Dividing is the inverse operation.*1046

*Whatever you do to one side, you have to do to the other side.*1049

*Negative divided by a negative is a positive.*1054

*This becomes 10 because 8 times 10 is 80.*1057

*Equals; went away; left with N; same thing here.*1063

*I know some of you guys can still do this in your head.*1080

*If you can, that is fine.*1083

*But it is important to know inverse operations and know how to do these steps*1084

*because later on, the equations are going to get a lot harder.*1090

*If you know how to do it this way, then solving equations becomes really easy.*1094

*Just try to practice it a few times; just keep practicing.*1102

*Circle the variable because that is what you are solving for.*1105

*To get the variable by itself, I have to get rid of this number.*1109

*Divide; divide; this goes away; this becomes -2.*1113

*Again positive divided by negative; I only have one negative sign so my answer is a negative.*1122

*11 times 2 is 22; that is why it is a 2.*1129

*Equals K; there is my answer.*1133

*If you want, you can flip this and make it K equal to -2.*1140

*Or not flip but switch the sides; you can write it like that.*1146

*The last one, going to circle the A; this is 9 times A.*1151

*Divide the 9; inverse operation to get rid of it.*1158

*45 divided by 9 is 5; I only have one negative.*1163

*The answer stays a negative; equals A; that is my answer.*1168

*That is it for these multiplication equations; thank you for watching Educator.com.*1180

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to be solving division equations.*0002

*Just like the other equations that we were solving in the previous lessons,*0009

*we are going to use inverse operations to get the variable by itself.*0014

*The whole point of solving equations is to solve for the variable.*0018

*What does the variable equal?--what is the value of the variable?*0022

*For this one, the inverse operation of division is multiplication.*0027

*The opposite operation, the opposite of division is multiplication.*0032

*Again we are just going to solve for the variable just using mental math.*0042

*That means 3 equals a number divided by 2.*0047

*Don't forget, right here, this fraction also means divide.*0054

*It is the top number divided by the bottom number.*0059

*A divided by 2 is 3; what is A?*0064

*What number if you divide it by 2 is going to give you 3?*0067

*I know that 6 divided by 2 is 3.*0071

*Again instead of writing just 6, you have to write that the variable A is equal to 6.*0076

*Same thing here; a number divided by 3 is going to give you 5.*0086

*15 divided by 3 is 5; that means the variable D is equal to 15.*0095

*A number divided by 4 is 4; 16 divided by 4 is 4.*0104

*That means I have to make B equal to 16.*0114

*Then 100 divided by a number is going to give you 10.*0121

*What is K?--K has to be 10 because 100 divided by 10 is 10.*0127

*Now for these examples, we are going to solve for the variable using the inverse operation.*0135

*I am going to circle the variable just like I did for my other lessons and then separate the sides.*0145

*Again this is M divided by 4.*0153

*If I have a number on the bottom and if I multiply this by 4, a +4 is the same thing as 4/1.*0158

*4 is the same thing as 4/1; I could put 1 under any number.*0181

*If I need to turn this into a fraction, I can put a 1 under it.*0189

*4 divided by 1; what is 4 divided by 1?--isn't that 4?*0194

*Remember this also means divide; 4 divided by 1 is 4.*0200

*If you want, if it helps, you can put it over 1.*0206

*Then if you remember, to multiply fractions, this is M times 4 which is M times 4 or you can write 4M.*0210

*Remember if you have a number times a variable, you can just write it together like that.*0222

*Over... this number times 1 which is 4.*0227

*If you have the same number on top as the bottom, they will become 1.*0234

*4 divided by 4 is 1.*0240

*That is why the inverse operation, if this is M divided by 4, the opposite of divide is multiply.*0246

*If I want to get rid of this 4 down here, then I have to multiply the 4.*0257

*Again I am using the inverse operation.*0264

*Just in the same way, this 4 is on this side this time.*0267

*But it is the same thing; 4/1.*0274

*Or you can leave it as 4; or you can put it over a 1.*0277

*Times M/4; remember this becomes that same exact thing, 4M/4 because this times this is that.*0281

*1 times 4 is 4; 4/4 makes a 1; this is 1M.*0295

*If you remember, 1M is the same thing as M.*0303

*Here inverse operation of divide is to multiply.*0312

*If you multiply when it is divided from the variable, they will cancel out.*0317

*It will go away.*0323

*If you don't understand this, what I did here, just know that if you have to divide*0324

*or if this is M divided by 4, then you have to multiply it so that it will go away.*0330

*Again whatever I do to one side of the equal sign, I have to do to the other side.*0340

*Since I multiplied 4 to this side, I have to multiply 4 to this side.*0345

*You can write it like this for now.*0351

*Or the best way to show that you are going to multiply two numbers is to write them in parentheses.*0354

*What is left on the left side?*0361

*This side, all I have left is the M which is what I want.*0364

*I wanted to get rid of this 4; that is why I multiplied it.*0368

*Equals -3 times 4; remember... I have a negative times a positive.*0371

*Whenever you are multiplying or dividing, if you only have one negative sign, your answer becomes a negative.*0379

*You are going to multiply these two numbers just the same way.*0387

*3 times 4 is 12.*0390

*It is a -12 because I only see one negative sign.*0396

*Remember one negative, your answer becomes a negative.*0399

*If you are multiplying two negative numbers, then those negative numbers pair up to become a positive; a plus.*0403

*Then this is M equals -12; I have the variable by itself.*0414

*So I am done; I solved the equation.*0419

*Again next one, I am going to circle the variable.*0426

*That is my goal, to make the variable by itself.*0429

*I am going to separate the two sides like that.*0433

*This is A divided by 7.*0436

*If it is A divided by 7, the inverse operation is multiply.*0439

*I have to multiply this whole side by 7.*0445

*But these will cross cancel; they will become a 1.*0450

*Whatever I do to one side, I have to do to the other side.*0455

*7 times 6 is 42; on my left side, I get 42; then my equals.*0460

*On my right side, I have only A left because remember I got rid of that 7.*0472

*Once I have the variable by itself and I have the number, I am done.*0478

*That is my answer.*0482

*The third one, I am going to use a different color; solve for B.*0486

*Again this is B divided by -3; it is divided by -3.*0494

*I still have to get rid of this whole number right here.*0499

*Inverse operation, multiply this by -3; that way these will cancel out.*0502

*Whatever I do to one side, I have to do to the other side.*0509

*I have to multiply this by -3.*0513

*Then on this side, I only have a B left; equals... my right side.*0518

*I have to solve this out; 10 times -3; 10 times 3 is 30.*0526

*How many negative signs do I see?--I only see one.*0533

*That means my answer is going to be negative; B equals -30.*0537

*This last one, again solve for X; I am going to circle it.*0548

*Separate my sides; X divided by 5; inverse operation is to multiply.*0553

*Multiply this side; that way that number will go away.*0559

*Whatever I do to one side, I have to do to the other side.*0565

*What is left on this side?--X is left; equals... 8 times 5 is 40.*0570

*I only have one negative sign; that means my answer is going to be negative.*0581

*That is it; that is my answer.*0587

*For these, we are going to determine if -6 is the solution of the equation.*0595

*Here is my variable.*0604

*I want to know if -6 can be M or if M can be -6.*0605

*-6 divided by 2; is that 3?-- -6, I am going to replace it.*0613

*I am going to write it instead of M because I am trying to see if M is going to be that number.*0619

*I am just going to substitute it in; equals 3; is this true?*0626

*I know that 6 divided by 2 is 3; but is -6 divided by 2, 3?*0632

*No, because when you have a negative divided by a positive, *0637

*you only see one negative, that makes the answer a negative.*0641

*In this case, this is no or false.*0646

*This one, I am just going to write no.*0650

*The next one, here is my variable; 6 over... in place of the variable, -6.*0653

*I know that 6 divided by 6 is 1; is 6 divided by -6, -1?*0665

*Only one negative sign when you are dividing numbers; that makes this negative.*0672

*So this one is yes; this one is true.*0677

*Here -18 equals -6/4; does -6 divided by 4 equal -18?*0682

*No, so this one is no.*0703

*Let's solve each equation.*0714

*We are going to use the inverse operations to solve each of the equations.*0716

*The first one, I am going to circle my variable.*0722

*I am going to separate my sides; be careful here, this is A plus 11.*0728

*What is being used?--a plus; what is the inverse operation of plus again?*0739

*Inverse operation of plus is minus; the inverse operation of minus is plus.*0748

*What about times?--what is the inverse operation of times?--divide.*0756

*And the inverse operation of divide is times.*0762

*Since this is A plus 11, and I know I have to get the variable by itself,*0767

*I have to get rid of everything that is next to the variable.*0773

*I have to use the inverse operation of plus which is minus.*0777

*That means I have to subtract this number 11 because +10 minus 11, that makes 0; that goes away.*0781

*Whatever I do to one side, I do to the other; -4 minus 11.*0789

*Be careful here; remember we are not multiplying; we are not dividing.*0799

*Even though you see two negative signs, it doesn't make it go away; only when you multiply or divide.*0803

*When it comes to adding and subtracting, you can think of dogs and cats.*0809

*You can think of money; it is like saying you borrowed $4.*0815

*Then you borrowed another $11; whenever you see a negative, you are borrowing.*0821

*You borrow 4; you borrow 11; how much do you owe?*0826

*You owe 15; that is a negative because you still owe.*0830

*Whenever you don't have something, it is a negative.*0837

*Bring down that equal sign; then on this side, what is left?*0842

*Only the variable because whatever was here, we got rid of; that number that was there.*0847

*So this is it; this is the answer.*0854

*The whole point, the goal is to get the variable by itself.*0858

*The next one; circle the variable; separate my sides; this is N minus 8.*0864

*I know I have to get rid of this number because it is next to the variable.*0874

*Inverse operation of minus is plus.*0879

*That means to get rid of this ?A, I have to add A to make it go away so the variable can be by itself.*0882

*Then whatever you do to one side, you have to do to the other side.*0892

*It is the right side; only the variable is left.*0895

*Equals... this side, the right side... 28; 20 plus 8 is 28.*0902

*That is my answer; N equals 28.*0909

*Solving for my variable of E; separate my sides; here this is 9 times V.*0915

*If they are stuck together like that, that is a times; 9 times V equals -81.*0924

*Since it is multiplied together, I have to use my inverse operation which is divide.*0932

*I have to divide the 9 to get rid of it so that the variable can be by itself.*0938

*Then same thing; whatever you do to one side, you have to do to the other side.*0943

*If you did it to the left side, then you have to do it to the right side.*0947

*From here, this is left with V; bring down the equal sign.*0953

*Then 81 divided by 9; don't forget that this is divide.*0960

*81 divided by 9 is 9; but then you have a negative.*0964

*Negative divided by a positive, you only see one negative.*0971

*In this case, only when you multiply or divide, remember.*0974

*You are dividing now; one negative gives you a negative answer.*0977

*That is my answer; that is it.*0984

*The last one, D divided by -12 equals 3.*0989

*I am going to circle my variable; separate my sides.*0997

*This is D divided by -12; the inverse operation of divide is multiply.*1001

*To get rid of this number here, because again we need D to be by itself.*1008

*I need to multiply this number; make sure you divide the -12.*1014

*The number is -12; you are multiplying -12; that cross cancels out.*1021

*Whatever you do to one side, remember you have to do to the other side; 3 times -12.*1027

*Again the best way to write two numbers being multiplied is to write it in parentheses.*1032

*Can write that in parentheses too.*1042

*On this side, on my left side, I have D left by itself.*1045

*Equals... the right side, it is 3 times -12; first 3 times 12 is 36.*1049

*Remember the number stays the same; I have only one negative sign.*1060

*That makes this because it is multiplied; D is -36.*1066

*That is it for this lesson; thank you for watching Educator.com.*1075

*Welcome back to Educator.com; for the next lesson, we are going to go over ratio.*0000

*A ratio is when you compare two things.*0007

*You are making a comparison between two quantities; it is also same as division.*0011

*If you look here, there is three ways to express a ratio.*0020

*There is three ways to write a ratio.*0023

*If A is 1 and B is something else, you can say A to B.*0028

*You can write it out, A to B.*0033

*All this, whether you write it like this, like this, or like this, they are all read as A to B.*0036

*But you can write it like this, like this using a colon or as a fraction A to B.*0045

*You are still comparing A and B.*0053

*For example, if I said what is the ratio, you are comparing boys to girls.*0056

*Because I said boys first, boys is going to be written as A, the first one.*0064

*Then girls has to be the second one; boys to girls.*0071

*If I ask for the ratio of boys to girls, then I can't give you the number of girls to boys.*0077

*You can't do this. You have to write out the ratio in the order that was asked for; boys to girls.*0086

*If I say there are 5 boys and there are 3 girls, then the ratio of boys to girls would be 5 to 3.*0095

*You can also write it as 5 to 3 like that.*0108

*If I ask you what is the ratio of girls to boys, then you would have to*0114

*give me this number first, the number of girls to the number of boys.*0117

*You always have to write out the ratio in the order that it was asked for.*0124

*A to B, A:B, and A/B as a fraction; this is called ratio.*0132

*A rate is a ratio; you are still comparing A to B.*0141

*But you are given different rates; for example, if I say miles per hour.*0148

*Miles per hour would be... you have the number of miles and you have however many number of hours.*0158

*You are comparing, you are making a ratio between the number of miles and the number of hours.*0169

*A rate would be a ratio, same thing, A to B, but using different rates.*0175

*If I say $5 for 5 candies, then that is a ratio.*0183

*You are making the comparison between the amount over the number of candy.*0194

*If you make a comparison between two things, it is called a ratio.*0204

*When those two things have some kind of unit, then it is called a rate.*0209

*A unit rate is a rate with a denominator of 1.*0219

*That means that if I say I traveled 2 miles in 2 hours.*0225

*Here is my ratio, 2 miles every 2 hours; this is my ratio.*0239

*A unit rate would mean to make this, the denominator, the bottom number, a 1.*0247

*That means I need to change this to become 1 hour.*0255

*That would be a unit rate.*0259

*That means in order to turn this 2 into a 1, I have to divide the 2.*0261

*I am going to divide this by 2 which means I have to divide the top number by 2.*0267

*This would be 1 mile per hour because it is 1 hour.*0274

*1 mile per hour would be the unit rate.*0282

*This alone would just be a rate.*0286

*But when you make the denominator a 1, a unit of 1, then this is a unit rate.*0290

*This here is a unit rate because the denominator is 1.*0298

*Here is an example; $10 per 20 pieces; that is like the candy example.*0306

*If it is $10 for every 20 pieces,*0310

*in order to give me a unit rate, I want to find out how much it is per piece.*0323

*One piece, I am turning this denominator into a 1.*0328

*That means in order to turn this denominator into 1, I have to divide it by this number, divide it by itself.*0333

*That means I have to divide this top number.*0340

*Because this is money, I want to change it to a decimal.*0346

*I know that 10 divided by 20 or 10 divided by 20 is going to give me 0.5.*0350

*0.5 in money is the same thing as 50 cents.*0368

*If I add a 0 here, that becomes 50 cents.*0373

*Not 5 cents, be careful; this is 50 cents.*0375

*The unit rate would be 50 cents per piece.*0380

*I can put 1 in front of it if I want.*0391

*But if I just say per piece, then I am talking about 1 piece.*0393

*You can leave it like this; this would be your unit rate.*0400

*When you are converting rates, rate remember it is a ratio of two different rates.*0408

*You have different units on the top and the bottom; that is a rate.*0416

*To convert rates means you are going to go from whatever rates they give you,*0422

*whatever units they give you, and you are changing it to something else, changing it to different rates.*0429

*You are converting them.*0433

*For example, if I have miles per hour, let's say I want miles per hour.*0435

*I am going to put miles on top; I am going to put... 1 mile per hour.*0444

*This is the rate that I am starting off with.*0451

*I want to convert it to feet per minute.*0455

*This is miles; mi is miles; min is for minutes.*0467

*I am going to convert this number here, this ratio, this rate, to this rate, this ratio.*0471

*Remember rates are ratios; I am going to convert this to this.*0482

*That means miles I need to change to feet and hours I am going to change to minutes.*0486

*Miles and feet, they are both measurements of distance.*0493

*Mile and feet, they are both measuring the distance of something.*0498

*Hour and minutes, they are both measuring time.*0503

*I can convert miles to feet and hours to minutes.*0507

*In order for you to be able to convert rates, this to this, you have to know the equivalent units.*0512

*How many feet equals a mile?--1 mile equals 5280 feet.*0525

*This and this are the same; 1 mile is equal to 5280 feet.*0540

*Same thing for hours and minutes; I know that 60 minutes equals 1 hour.*0545

*If it helps, you need to just write this on the side.*0556

*You are going to use this to help you convert rates.*0559

*First thing I do, I am going to start here and I am going to end here.*0565

*I am going to change all these into these.*0571

*I am going to start from 1 mile to 1 hour.*0574

*I am going to multiply it to different units because I can cross cancel things out.*0582

*If I say that 5280 feet is the same thing as 1 mile, if they equal each other,*0594

*then I can say 5280 feet over 1 mile is going to equal 1 because this number and this are the same thing.*0603

*We said they are equal; this over this is equal to 1.*0623

*Anything over itself is 1.*0629

*If I said, for example, 5/5, isn't that 1?--because it is the same number over itself.*0631

*Same thing here; this equals this.*0638

*If I say 5280 feet over 5280 feet, isn't that equal to 1?*0642

*This does equal 5280 feet.*0649

*If I write it like this, you have to understand that this is the same thing as 1.*0653

*If I multiply this by 1, I am not changing this.*0659

*I can multiply this by 1 if I want because it doesn't change.*0665

*Instead of multiplying it by 1, I want to multiply it by this.*0670

*This is the same thing as 1.*0674

*I am going to multiply all this to this.*0682

*I want the miles to go away because the miles is going to have to change to feet.*0689

*I need to the miles to go away.*0696

*In order for me to cross cancel the miles, I have to have one on the top and one on the bottom.*0697

*This miles is going to go down here.*0702

*On the top, it is going to go 5280 feet.*0705

*That way this and this will cancel.*0718

*Again this whole thing is just equal to 1.*0724

*I can just multiply it to this if I want.*0730

*It is not going to change my answer because I am just multiplying it by 1.*0734

*Same thing for hours.*0740

*I also know that since this 60 minutes is equal to 1 hour,*0743

*if I put 60 minutes and I divide it by 1 hour,*0749

*since this whole thing equals this whole thing, this is also equal to 1, isn't it?*0755

*They equal each other.*0760

*Whenever the top and the bottom equal each other, that always equals 1.*0762

*I want to multiply this whole thing to this whole thing because again this is equal to 1.*0769

*I want the hours to go away.*0777

*That means if this is already in the bottom, then I need to write this on the top.*0779

*This is going to go 1 hour over 60 minutes.*0783

*I just flip this; this went to the top; this went to the bottom.*0790

*Because again if this is the same thing as this, then isn't this over this the same thing?*0795

*I am writing it on the top and the bottom, depending on where I have to cancel it.*0803

*If this is already in the bottom, then I need to cancel this.*0808

*That is going to go like that.*0811

*If I look on the top, what units am I left with?--feet.*0816

*For my answer, if I multiply all this out, then I am going to be left with feet which is what I want.*0823

*On the bottom, what am I left with?--minutes.*0828

*That is what I want left on the bottom.*0834

*I know that all I have to do is now solve this out.*0836

*I cancelled out everything that I need to cancel out.*0840

*If I just multiply this out and then multiply that out, solve for it, I will get my answer.*0844

*Here my top is going to be 1 times 5280 times 1 which is 5280 feet over 60 minutes.*0852

*1 mile per hour is the same thing as 5280 feet over 60 minutes.*0873

*This would be my answer; I can simply this if I want.*0885

*This is a ratio, is a rate.*0890

*But if I want to change it to a unit rate, I can divide this by... let me use a different color.*0893

*I can divide this by 60 and then divide this by 60.*0901

*5280 divided by 60 is going to give me... I am going to cross out these 0s.*0910

*Remember you can cross out the 0s if you want.*0921

*It is going to give me... 8, 4, 8; it is going to give me 88.*0926

*Again to multiply this, you are going to do 6 times 8.*0949

*Write it out, 48; my remainder is 4.*0952

*I am going to bring down this 8; then 6 times 8 is 48.*0954

*My answer is 88 feet per minute; this would be my unit rate.*0966

*Let's do a few more examples; the first example, write in simplest form.*0981

*Here these are just ratios; it is comparing this number to this number.*0991

*They look like fractions.*0997

*But you can also think of these numbers, the top number and the bottom number as ratios.*0998

*It is like division.*1005

*To write this in simplest form, 12/36, I can look for a common factor.*1008

*The greatest common factor is 12 because 12 goes into 12 here and 12 goes into 36.*1019

*If you don't see that 12 is the biggest factor,*1030

*you can just look for any factor because 12 and 36 have a lot of common factors.*1032

*If you want, you can just divide the 2 first and then just keep making the numbers smaller.*1038

*You can divide this by 4; divide it by 3.*1043

*Since I know that 12 is my biggest factor, I am going to divide this by 12 and then divide this by 12.*1050

*Whenever I am going to simplify, then I need to divide both*1057

*the top number and the bottom number by the same number, the same factor.*1060

*This is 1 over... 36 divided by 12 is 3.*1066

*This is saying that the ratio of 12 to 36 is the same as 1 to 3.*1074

*They are the same ratio; they are equivalent; they are the same.*1081

*This next one, I know because this ends in a 0 and this ends in a 5, that they are both divisible by 5.*1088

*I am going to take 30 divided by 5; 35 divided by 5.*1100

*30 divided by 5 is 6; 35 divided by 5 is 7.*1108

*This is simplest form.*1118

*That means the ratio of 6 to 7 is the same as 30 to 35.*1119

*Same thing here; let's divide this by...*1125

*Again if you just see any common factor, you can just keep dividing until you get simplest form.*1128

*Or if you find the greatest factor, that would be the fastest way.*1137

*But let's say that we wanted to just divide this by 2 because I just noticed that they are both even.*1141

*That is not the greatest factor; but let's just do that first.*1147

*I am going to divide both the top and the bottom by 2.*1152

*This is going to be 8/12.*1155

*This is still not simplest form because they are both even still.*1161

*4. or maybe 2 if you just notice that they are both even numbers.*1167

*But from these two, the greatest factor is 4; let's just divide them by 4.*1174

*8 divided by 4 is 2; over 3; that would be simplest form.*1180

*For this, let's see, this is not an even number so I know that 2 is not going to go into them.*1187

*If I add these two together, 5 plus 1, that is 6.*1195

*6 is a multiple of 3.*1200

*This bottom one, 1 plus 9, 9 is a multiple of 3.*1204

*So I know that 3 can go into both of these.*1209

*Divide this one by 3; divide this one by 3; 51 divided 3.*1213

*If you don't know what that is, you can always just divide it.*1222

*51 divided by 3; 1; subtract the number; bring this down.*1226

*That will be 17 over... 18 divided by 3; 3 times what equals 18?*1236

*That is 6; that would be simplest form.*1246

*Next example, Tommy has 4 blue marbles, 3 green marbles, and 7 red marbles in a bag.*1254

*Find the ratio of red to blue marbles; 4 blue, 3 green, 7 red.*1264

*We want to find the ratio of red to blue.*1274

*The ratio is going to be red to blue.*1278

*How many red do we have?--7 to 4 blue.*1286

*Make sure you have to write it as 7 to 4 and not 4 to 7.*1292

*Because they ask for red first before the blue, you have to write out the red first.*1298

*It is 7 to 4; you can say 7 to 4.*1304

*Or you can say 7 to 4 like that as a fraction.*1309

*Next, out of 27 students in classroom, 15 are boys.*1317

*Find the ratio of boys to girls; ratio is boys to girls.*1323

*They don't give us a number of girls.*1333

*They just tell us that there is 15 boys.*1334

*But I know that if there is 27 students total and 15 are boys, then the rest of the students have to be girls.*1339

*I have to subtract 27 students minus 15; I am going to get 12.*1350

*That means I know that 15 are boys and 12 are girls.*1358

*The ratio of boys to girls would be 15 to 12 or 15/12.*1364

*Find the unit rate.*1379

*Remember unit rate is when you have a ratio and the bottom number, the denominator, has to be 1.*1380

*Here the ratio is 250 for every 2 dozen.*1389

*I want to find how much it is going to be for 1 dozen or how much per dozen.*1403

*Think of it as per.*1411

*Every time you see unit rate, you are going to think of per.*1413

*Per whatever the unit is on the bottom; how much per dozen?*1418

*That means I need to turn this into a 1.*1424

*I divide this by 2 then to turn 2 divided by 2 into 1.*1428

*Then I have to multiply the top by 2.*1433

*250 divided by 2.... remember to bring out the decimal.*1436

*It is going to be 1; 2; bring down the 5.*1444

*2; 4; 1; bring down the 0; 5.*1455

*Going to be $1.25 per dozen; there is my unit rate.*1460

*A car goes 300 miles... mi means miles... on 10 gallons of gas.*1479

*300 miles on 10 gallons of gas; find the unit rate.*1488

*That means I want to turn this into 1.*1498

*It is going to be how many miles per gallon.*1501

*Then again divide this by 10; divide the top number by 10.*1504

*300 divided by 10... every time you divide by a number with the 0 at the end of it,*1512

*and they both have 0s at the end, you can just cross out one of the 0s.*1519

*If I cross out this 0 and cross out this 0, then I am going to be left with 30.*1524

*30 miles per gallon; this is the unit rate.*1529

*A skydiver falls 240 feet in 5 seconds; 240 feet every 5 seconds.*1541

*How many feet per second?--1 second; divided by 5; divided by 5.*1556

*240... let's do it over here; 240 divided by 5 is going to be 4*1564

*because that is going to be 20; 4, 0; that will be 8.*1571

*48 feet per second; here is my unit rate.*1581

*The fourth example, we are going to convert the units.*1595

*This is going to be the most difficult part of this lesson.*1599

*But just make sure you are going to...*1603

*Just try to cancel out the units so that you end up with the units that you want for your answer.*1606

*A car is moving at 8 miles per hour.*1616

*I am going to write that as a fraction; 8 miles per hour.*1620

*I want to convert this to feet on the top with what units on the bottom?--minutes.*1627

*I know it is 10 minutes.*1638

*But then I just want to focus on converting these units first--miles per hour to feet per minute.*1639

*I am going to start off here; I am going to write that over 1 hour.*1657

*Again I am going to multiply; I want to turn the miles into feet.*1667

*On the side, let's find out... 5280 feet equals 1 mile; this equals this.*1673

*That means if I put this over that as a numerator and denominator, it is going to equal 1.*1685

*Why don't I just do that right now; I am going to do times... *1698

*From the feet and the miles, which one do I want to go as my numerator?*1701

*I want my miles to go on the denominator because I want them to cross cancel out.*1707

*From these two, I am going to put this on the bottom.*1714

*I am going to put 1 mile on the bottom and then 5280 feet as my numerator.*1717

*That way my miles cancel out.*1727

*Hours I am going to change to minutes; 1 hour equals 60 minutes.*1733

*Write that out first so you can see it; it is a lot easier.*1744

*Then from this and here, one is going to go on as my numerator.*1747

*One is going to go on my denominator.*1752

*Which one do I want to go on the top?*1755

*The hours because here the hours is on the bottom.*1758

*I want it to cancel out so I have to put it on the top.*1760

*This is going to go on the top; this is going to go on the bottom.*1763

*1 hour over 60 minutes; cross cancel that out.*1767

*Here I want to now solve this out because if I look at the top, what units are left on the top?*1779

*Feet is left which is what I want.*1789

*This is what I want my answer in; I am on the right track.*1791

*On the bottom, what do I want left?--minutes.*1797

*That is where I am at; so I am good.*1800

*Now I know I just have to multiply this out and solve these numbers out.*1803

*Before we start multiplying this times this and get a big number and then*1809

*have to divide by a big number, let's try to cross cancel some stuff out.*1813

*Anytime you are multiplying numbers and you have numbers on top and you have numbers on the bottom,*1820

*you can start cross cancelling things out if they have common factors.*1825

*First thing I see is I see a 0 here and I see a 0 here.*1829

*I can cross cancel those out.*1834

*This is going to change to 6; this is going to change to 528.*1840

*Cross out that 0; cross out that 0.*1846

*8 and 6, I know that they have a common denominator of 2.*1852

*I can cross this out, divide this by 2; I get 3; that changes to a 3.*1858

*I am going to change that because that common factor was a 2 so that changes to a 4.*1865

*That means I divided this by 2 and I divided this by 2.*1871

*This became 4; this became 3.*1875

*Here, does 528, is it divisible by 3?*1879

*If you add this, it becomes 5 plus 2 is 7; plus 8 is 15.*1886

*Is 15 a multiple of 3?--it is.*1892

*Therefore I know that this is divisible by 3.*1896

*If you are wondering what I just did, I used the divisibility rule.*1899

*The divisibility rule of 3 is you add up all the digits.*1905

*You are going to do 5 plus 2 plus 8 which gives you 15.*1911

*You are going to see if that number is a multiple of 3.*1918

*Does 3 go into that number?*1922

*5 plus 2 plus 8 is 15; 3 does go into 15.*1925

*I know that 3 will go into this number; 528 divided by 3.*1929

*This is 1; this becomes 3; subtract it; you get 2; bring this 2 down.*1940

*3 goes into 22 seven times; that is 21; 1; bring down the 8.*1948

*3 times 6 is 18; this goes away; this became 176.*1957

*Now all I have to do to find my answer is just...*1979

*Since the bottom number is 1 times 1 and these all canceled out, then it is just 1 times 1.*1983

*That is just 1 minute; I just multiplied all the numbers; I get this left.*1991

*On my top, my numerator, it is just 4 times 176.*1997

*That is 24; 7 times 4 is 28; add 2; this becomes feet.*2005

*This is in a unit rate; 704 feet per minute.*2037

*I want to know how many feet it will move in 10 minutes.*2045

*What does that mean?--this is my unit rate; I am converting the units.*2051

*This would be the correct answer.*2059

*But then it is asking me how many feet it will move in 10 minutes, not per minute.*2061

*If they asked how many feet it will move per minute or in one minute,*2069

*this would be my answer, 704 feet per minute, for one minute.*2074

*But since they are asking for 10 minutes, I need to change my denominator to a 10.*2079

*They are not asking for a unit rate.*2086

*They want to know how many feet for 10 minutes.*2088

*I need to change this 1 to a 10.*2093

*In order to do that, I have to multiply by 10.*2094

*Same thing here; I need to multiply by 10.*2098

*If I need to multiply this by 10, I just have to add a 0 at the end of it.*2102

*That is pretty easy.*2106

*It just becomes 7 thousand, 0, 4, add the 0, feet per 10 minutes.*2106

*My answer, how many feet?--it is 7040 feet.*2118

*I know that problem seemed a little bit complicated.*2128

*But all I had to do was convert the units at miles per hour to feet per minute which is what I did.*2131

*Multiply your top numbers across; multiply your bottom numbers across.*2139

*If you want, you didn't have to cross cancel all this stuff out.*2143

*That is why it looks so complicated, because we ended up cross cancelling numbers out.*2146

*But if you want, forget about the cross cancelling.*2150

*Just multiply all the numbers straight across; get this number.*2153

*Multiply all the bottom numbers straight across and get this number.*2158

*Then simplify if you want; you can do it that way.*2161

*Once we get this, this is per minute; denominator is 1.*2165

*This is our unit rate.*2170

*But then because they are asking for 10 minutes,*2173

*I need to change this denominator to 10 by multiplying by 10.*2174

*We multiply the top by 10; you get 7040 feet per 10 minutes.*2179

*Let's try one more problem; the sprinklers used 2 gallons per minute.*2187

*How many quarts will it use in 30 seconds?--again we have to convert units.*2194

*This is 2 gallons per minutes.*2202

*I want to convert this to quarts... this is quarts... per seconds.*2209

*I am going to put just 30 seconds here.*2219

*Question mark, how many quarters per 30 seconds?*2223

*I am going to start off with this again; 2 gallons per minute.*2226

*Since I need to convert gallons to quarts, I know that 1 gallon is equal to 4 quarts.*2237

*Remember if this is equal to this, I can change this to a fraction, 4 quarts over 1 gallon.*2254

*That is going to equal 1.*2262

*1 gallon over 4 quarts, that is also going to be the same.*2263

*I can multiply this by... what do I want to get rid of?*2269

*I want to get rid of the gallons first by using this.*2275

*The gallons is going to go on the bottom.*2280

*This one is going to go on the bottom; that way this will cancel like this.*2282

*This one will go on the top like that.*2287

*See how one goes on the bottom and one goes on the top?*2293

*Or one goes on the top and one goes on the bottom?*2295

*Just depends on what you have to cancel.*2299

*Then I need to convert minutes to seconds.*2303

*The minutes to seconds is going to be 1 minute is equal to 60 seconds.*2310

*I need to write the minute one on the top so that it will cancel.*2322

*This is going to go on the top; this is top, bottom.*2326

*1 minute over 60 seconds; minutes will cancel.*2330

*What units do I have left on the top?*2341

*I have quarts which is what I want.*2343

*And I have seconds on the bottom which is what I want.*2346

*Now I just have to solve it out.*2350

*If I want, I can cross cancel out this 2 and this 60.*2354

*2 goes into 2; this changes to a 1.*2359

*2 goes into 60; cut it in half; that is 30.*2362

*You can cross cancel out again.*2369

*But then otherwise you can just write it out; 4 quarts over 30 seconds.*2371

*The reason why I decided to leave it...*2384

*You could have cross cancelled it out; that is fine.*2386

*Here they ask for 30 seconds.*2390

*They want to know how many quarts will it use in 30 seconds.*2393

*Here it will be 4 quarts every 30 seconds.*2399

*I know that my answer will be 4; 4 quarts.*2404

*It is 4 quarts per 30 seconds; that is my answer.*2410

*That is it for this lesson; thank you for watching Educator.com.*2417

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to be solving proportions.*0002

*A proportion is when we have two equal ratios.*0008

*A ratio is a comparison between two parts, A and B.*0014

*Here this is a ratio comparing A and B together.*0022

*You read it as A to B.*0028

*You can also write ratios A to B like that.*0030

*But when you are taking two ratios and you are comparing them*0036

*to each other and you are saying that they are equal,*0039

*the ratio of A to B is equal to the ratio of C to D, then you have a proportion.*0042

*You are actually going to have to write it A to B like a fraction.*0049

*Proportion is when you have ratio equaling another ratio.*0056

*To solve proportions, let's say you are missing one of these.*0062

*You are missing A; or you are missing B; one of these.*0065

*To solve a proportion, you are going to use what is called cross products.*0069

*Cross products is when you multiply across.*0075

*You are going to go A times D equal to B times C.*0078

*You have the proportion A over B equal to C over D.*0089

*Then you are going to find the products AD.*0096

*Cross products, that is A times D.*0100

*When you write two variables next to each other like that, that means multiply.*0103

*A times D equal to B times C.*0107

*Be careful, this is not the same thing as cross cancelling.*0113

*Cross cancelling is when you are multiplying fractions*0117

*and you can cancel out numbers if they have common factors.*0120

*But this is cross products; this is for proportions.*0126

*This is only when they are equal to each other.*0129

*Then you can multiply across and make it equal to this across.*0132

*AD, A times D is equal to B times C.*0140

*Let's do an example; if I have let's say 1/2 equal to X/ 4.*0145

*This one is easy; we know we can do this in our head.*0156

*One half, 1/2, is the same thing as 2/4.*0158

*I know that X has to be 2.*0163

*But to solve it, just to use cross products, it will be 2 times X.*0167

*2 times X is 2X; same thing as 2 times X.*0174

*It is equal to 1 times 4 which is 4.*0178

*Then to solve this out, we are going to... remember one-step equation.*0183

*This is 2 times X or 2 times what equals 4?*0188

*2 times 2 equals 4; X equals 2.*0193

*Let's do a few examples; the first example, find two equivalent ratios for each.*0200

*We know that proportions are when we have two equal ratios.*0207

*Let's find two other ratios that are equal to this.*0214

*3/4, I can say that if I multiply this by 2, then this is 6/8.*0218

*This ratio is equal to this ratio.*0228

*To find another one, how about if I multiply it by 3?*0232

*3 times 3 is 9; 4 times 3 is 12.*0235

*Here are my two equivalent ratios.*0241

*Here I can also divide.*0248

*If I divide by 10, divide this by 10, because I know 10 goes into both, this becomes 1/2.*0250

*That is one equivalent ratio.*0261

*I can multiply this by 2; multiplied by 2; multiplied by 2.*0266

*This can be 2/4.*0272

*This and this, they don't look like they are equivalent.*0276

*But they actually are because if you simplify this, this is 1/2.*0279

*If you simplify this, this is 1/2; it is the same; it is equivalent ratios.*0284

*This one, you can multiply it by 2; you can divide.*0294

*I know that 11 and 33 have factors of 11.*0301

*11 divided by 11 is 1 over... 33 divided by 11 is 3.*0308

*Again you can just base it on this for the next one.*0315

*Multiply it by 2; it will be 2.*0318

*Multiply this by 2; it will be 6.*0321

*There are actually many, many different ratios or fractions that you can write out to make them equivalent.*0327

*There is going to be many, many; these are not the only answers.*0336

*These are not the only fractions that are equal to this fraction.*0339

*If you want, you can multiply this by 10, multiply it by 20.*0344

*As long as you do it to both numbers, you are going to have equivalent ratios.*0347

*Let's solve these proportions; but we are going to use mental math.*0354

*Meaning we are going to try to solve these out in our head.*0358

*For the first one, I want to solve for X; 3/4 equals X/12.*0362

*If this fraction is going to have to equal this fraction, 4 times what is 12?*0371

*4 times 3 is 12; that means I have to multiply the top number by 3.*0376

*X has to be 9; X equals 9.*0382

*Same thing here; this is 1 times 5 which gave me 5.*0388

*Then I have to multiply 5 to this; A is going to be 25.*0394

*Same thing here.*0404

*If you look at the bottom numbers, this is 6 times 1 equals 6.*0406

*Something times 1 is going to equal 4; isn't this 4 times 1?*0414

*This is going to be the same fraction; 4/6 has to equal 4/6.*0418

*D is going to be 4.*0424

*The next one, this one is a little bit different*0428

*because I can't divide and multiply a number 15 to give me 12.*0433

*What I can do is I can just simplify this ratio because I have both the top and bottom number.*0441

*I want to simplify this ratio to help me solve for this ratio.*0449

*If I simplify this, I know that 4 goes into both numbers.*0454

*Divide this by 4; this is 3/2; this ratio is equivalent to that ratio.*0459

*I just have to base this one on this then.*0473

*3 times 5 is 15; to go from here to here, it is times 5.*0480

*To go from here to here then, it will be times 5.*0489

*Z has to be 10.*0493

*Tell whether the two ratios form a proportion.*0503

*That just means that they have to be equal.*0508

*It is just a yes or no.*0511

*Are they equal or are they not equal?--because proportions have to be two equal ratios.*0512

*Is this ratio equal to this ratio?*0519

*2/3, multiply this by 10 to get 20; multiply this to 10 to get 30.*0525

*Is it equal?--yes, this one is equal.*0533

*The next one, are these equal ratios?*0539

*This one, you had to multiply this by 7 to get 35.*0543

*How about this one?*0548

*If you multiply this by 7, you have to multiply it by the same number.*0549

*Does it give you that?--this one is yes.*0552

*This one here, again I can't multiply or divide this number to get this number.*0560

*I can find another equivalent ratio to base both of these on.*0566

*I know that 5 goes into both of these.*0572

*5 goes into 25 five times; I am dividing by 5.*0577

*70 divided by 5 is going to be...*0584

*Again if you want to just divide it out, it is going to be 70 divided by 5.*0589

*Otherwise it is going to be 14.*0594

*I know that because... I will just solve it out.*0598

*Let's do it right here; 70 divided by 5.*0602

*1 is going to give you 5; subtract it; 2; 5 times 4 is 20.*0608

*That means to get from 5 to 35, I have to multiply this by 7.*0617

*What is 14 times 5?--it is 70; I know it is not 70.*0626

*This has to be 70; so I am going to say no.*0636

*The next one, here 42/21, this also is going to be equivalent fraction.*0651

*This will be 2/1 because 42/21... 21 is half of 42.*0666

*Again I can just divide this by 21 to get 2; and then 21 to get 1.*0680

*To get from here to here... or see if this one equals the same thing.*0687

*This one, divide this by common factor; is this 12?*0693

*This becomes 2; this becomes 1.*0703

*See how this was equal to this?--and then this also equal to this?*0707

*That means these are the same; so this one is yes.*0712

*You can do the same thing for this one.*0716

*I know that this simplified to get that.*0720

*Here 35 divided by 5 because a common factor between this one and this one is 5.*0724

*This one is 7 over... 75 divided by 5.*0740

*75 divided by 5 is going to be 15.*0751

*That will just be 5, 2; bring down the 5; this is 15.*0757

*Automatically because these are different, I know that it is a no.*0765

*Again if you want to figure out if two ratios are equal, you can either multiply, see if it is the same factor.*0772

*Or can just simplify each one of them and see if those simplified fractions are the same.*0783

*Like the bottom one right here, this last one, you simplify this; it became 2/1.*0791

*You simplify this; it became 2/1.*0796

*Since they simplify to become the same fraction, you know that these are the same.*0798

*So that is yes.*0804

*For the next example, we are going to solve the proportion using cross products.*0807

*Just practice using these cross products.*0814

*You are going to multiply these across.*0816

*You are going to make it equal to those two multiplied.*0822

*This becomes 2 times X; I am just going to write that as 2X.*0828

*Remember whenever you multiply a number with a variable, you can write it together like that.*0832

*Then equals 5 times 10 which is 50.*0838

*Again be careful, cross products is not the same thing as cross cancel.*0843

*You are not cancelling anything out.*0849

*This 5 and this 10, they have a common factor of 5.*0850

*But you are not cross cancelling out.*0856

*You only cross cancel when you are multiplying the fractions.*0858

*But here you are solving proportions where it is an equal, not a multiplication.*0863

*You are going to multiply them together and you are going to make it this side.*0867

*From here, I have to find out what X is; 2 times something equals 50.*0874

*2 times 25 equals 50; think of 50 cents.*0880

*2 times 2 quarters... that is 25 cents... equals 50 cents.*0888

*Or you can also just divide this 2; X is going to equal 25.*0895

*If you want, you can just use division like this.*0906

*2 goes into 5 two times; 4; subtract it; you bring down the 1.*0910

*Bring down the 0; 2 times 10 is 6; X is 25.*0917

*Same thing here; let's cross multiply; cross products.*0927

*3 times M would be 3M; just write them together like that.*0939

*Make sure you don't do it with numbers.*0945

*If it was 3 times 4, then you can't put 34 because it looks like the number 34.*0946

*This means 3 times M; then 4 times 21; 4 times 21 is 84.*0952

*21 times 4; 1 times 4 is 4; 4 times 2 is 8.*0968

*From here, 3 times something equals 84.*0976

*That means I have to divide this 3.*0979

*I need to divide the 3 to get my answer; 84 divided by 3.*0984

*How many times does 3 go into 8?--two times; that becomes 6.*0993

*I am going to subtract and get 2; bring down the 4.*0998

*3 times 8 equals 24; 24, 0; my M is 28.*1002

*Make sure you write what the variable is; the variable equals 28.*1018

*Don't forget, when you are solving proportions, if you can do it mentally, then go ahead and do that using mental math.*1023

*Otherwise you are going to just multiply this.*1030

*These two across equal these two across; then solve for your variable.*1032

*That is it for this lesson; thank you for watching Educator.com.*1038

*Welcome back to Educator.com; for the next lesson, we are going to continue proportions.*0001

*We are going to actually write proportions and then solve them.*0005

*When we write proportions, it is easier if you first create a ratio that you can base your proportion on.*0011

*I like to call it a word ratio because you are going to look at what you have*0022

*and then create a word ratio meaning a part to a part.*0033

*You are going to find out what you are going to leave on the top*0039

*and what you are going to put on the bottom of your ratio.*0043

*Here I have my example, 2 miles in 20 minutes.*0048

*I want to find out how many miles it will be in 30 minutes.*0053

*I want to create my word ratio; for example, I could put miles over minutes.*0059

*That means all the numbers that have to do with miles is going to go on the top.*0071

*All the numbers that have to do with minutes is going to go on the bottom*0077

*because when we write a proportion, remember a proportion has to be two ratios that equal each other.*0081

*The first ratio I am going to write is going to have to do with this part right here.*0091

*2 miles in 20 minutes; remember ratio, I am comparing two things.*0096

*I am comparing the miles and I am comparing the number of minutes.*0103

*All the miles is going to go on the top.*0106

*That means I am going to write 2 miles... mi for miles*0109

*Over 20 minutes because that is on the bottom; 2 miles in 20 minutes.*0116

*Then I have to create my next ratio.*0126

*Remember I am making a proportion; I am making this ratio equal to this ratio.*0131

*That way I have a proportion, I can solve for whatever is missing, my X, my variable.*0138

*Again the miles is going to go on the top because that is what I set.*0147

*That is my word ratio; it is going to be X miles over... 30 minutes.*0154

*That is minutes; that is going to go on the bottom.*0162

*As long as I keep all the miles on the top and all the minutes on the bottom, I can create my proportion.*0169

*Let's say I created my word ratio so that it was minutes over miles.*0175

*That is OK.*0180

*As long as you keep all the minutes on the top and all the miles on the bottom,*0181

*you are still going to get the same answer.*0186

*You are still going to get the correct answer.*0189

*Again this ratio is equal to this ratio; that is how I get my proportion.*0192

*That is how I am going to write my proportion.*0198

*From here, I need to solve this out; I can cross multiply.*0200

*If I can solve it in my head, then I want to do that instead so I don't have to do all the work.*0208

*2/20 is going to equal X/30.*0214

*I just rewrote the proportion without all of the units so you can see it a little bit easier.*0223

*Here I can create an equivalent ratio; remember equivalent ratios from the previous lesson.*0232

*2/20 is the same thing as 1 over... because here I divided this by 2.*0241

*2 divided by 2 is 1.*0250

*If I want to do 20 divided by 2, then it is going to equal 10.*0253

*I can also do the same thing here.*0260

*I want to make this ratio the same as 1/10.*0263

*I can multiply this by 3 to get 30.*0269

*Then I have to multiply this top by 3 to get 3.*0273

*My X is going to be 3.*0279

*Here I just used mental math to solve for X.*0285

*I just made this equivalent ratio, 1/10, and then turned this into the same thing, 1/10.*0289

*If you want, you can use cross products instead; that is another method.*0297

*You are going to multiply all this together; make it equal to this.*0302

*2 times 30... actually let's go this way first.*0309

*It doesn't matter which way you go first.*0311

*20 times X is 20X; equal to... 2 times 30 is 60.*0313

*If I double 30, it is 60; 20 times what equals 60?*0325

*20 times 3 equals 60.*0331

*20, 40, 60; that is 3; X will become 3.*0335

*That is the same thing; it doesn't matter which way you solve.*0340

*As long as you make it so that this ratio is equal to this ratio.*0347

*Let's do a few examples.*0354

*We are going to write a proportion and then solve them out.*0358

*5 pounds for $15; find the cost for 4 pounds.*0362

*I want to first create a word ratio; word ratio, what am I comparing?*0369

*Or what am I using?--I am using pounds and I am using money.*0380

*I can say money or dollars on the top.*0385

*Then I am going to keep the pounds on the bottom.*0391

*It doesn't matter if you do pounds over money; that is fine too.*0395

*Here is my word ratio; that means when I create my proportion,*0398

*I am going to keep all the dollars on the top and then all the number of pounds on the bottom.*0402

*5 pounds for $15, here is my first ratio, comparing these two things.*0410

*$15, that is the dollars; that is going to go on the top; 15.*0415

*Over 5 pounds; that is going to go on the bottom because that is what I made my word ratio.*0420

*Find the cost for 4 pounds.*0429

*4 pounds, does the 4 go on the top or the bottom?*0432

*It is pounds; it is going to go on the bottom.*0435

*I want to find the cost; that is what I am looking for.*0440

*I am going to make that my variable; I can say X.*0442

*That is the money part; find the cost; cost is money.*0446

*That is going to go on the top.*0451

*Here I am going to solve for X; again you can solve this two ways.*0454

*You can find the equivalent ratio; I am going to simplify this.*0462

*This is going to become... divide this by 5.*0468

*15 divided by 5 is going to be 3.*0474

*5 divided by 5 is going to be 1; there was my equivalent fraction.*0478

*Same thing here; I want to make this the same as 3/1.*0487

*How did I go from 1 to 4?--this was multiplied by 4.*0498

*Or I can just do 4 divided by 4 is 1.*0502

*Same thing here; 3 times... whatever I do to the bottom, I have to do to the top.*0505

*X becomes 12; or again you can just do cross multiplying.*0511

*You can do 15 times 4 equal to 5 times X.*0520

*Then you can see what you have to multiply by 5 to get this number.*0524

*My X is going to be 12 because 12/4 is going to be 3/1.*0531

*That is the same thing as 15/5.*0538

*I have to look back and see what am I looking for?*0543

*I know that X is 12; but it is asking for the cost.*0546

*We know cost is money.*0551

*How much is it going to cost for 4 pounds? $12.*0555

*The next one, 15 feet for every 4 minutes; find how many feet in 10 minutes.*0562

*My word ratio, I am going to make it 50 over minutes.*0570

*My 16 feet is going to go on the top.*0581

*My 4 minutes is going to go on the bottom.*0584

*Equal it to how many feet?--find how many feet.*0588

*That is what we are looking for; feet, that is the top number.*0591

*That is X; over the number of minutes is 10.*0595

*Again you can look for equivalent fraction.*0606

*This is going to be the same... 16/4 is going to be the same as...*0610

*If you divide this by 4, divide this by 4, you are going to get 4/1.*0615

*I am going to use that fraction to help me solve for X.*0626

*1 times 10 equals 10; it is 4 times 10 is 40.*0635

*That means X has to be 40.*0640

*Again these two have to be equal; this is the same as 4/1.*0647

*That means this has to be the same as 4/1.*0653

*1 times 10 is 10; 4 times 10 has to be 40.*0659

*How many feet?--X is going to be 40 feet.*0667

*Example two, write a proportion and solve.*0680

*5 chocolate bars costs 7.50; find the cost of 2 chocolate bars.*0684

*My word ratio, chocolate bars; you can do money on the bottom.*0689

*Or you can just do money on the top and then the number of chocolate bars on the bottom.*0698

*It doesn't matter; there is my word ratio.*0702

*Chocolate bars; 5 chocolate bars; 5 on top; over money; 7.50 on the bottom.*0708

*Equal to chocolate bars... that is 2 on the top; over the amount of money on the bottom.*0717

*For this one, I can solve this proportionally.*0730

*You can also use this as a ratio.*0739

*Remember 7.50 for 5 chocolate bars; you can make that as a ratio.*0746

*Then find the unit rate; find how much it costs per chocolate bar.*0751

*If you remember from a couple of lessons ago, you can use unit rate also for the same problem.*0758

*Let's just go ahead and solve this using cross products.*0766

*I am going to multiply this and this; that is going to be 5X.*0770

*Again if you are multiplying number times variable, then you can just put it together like that.*0776

*Equals 7.50 times 2; 7.50 times 2.*0782

*If you want, you can just multiply it out like that.*0790

*0; 5 times 2 is 10; 2 times 7 is 14; add the 1; 15.*0796

*You know that this is 7.50; that is money; 7 times 2 is 14.*0808

*If you have 50 cents and you double it, that is a dollar.*0815

*You can think of it that way too; 5X equals $15.*0818

*I am not going to put the 0.00 because that is just change.*0825

*This is my whole number, $15; I can now find X.*0829

*5 times... I know 3 equals 15; X is going to be 3.*0836

*That means if for 5 chocolate bars, it costs 7.50,*0845

*for 2 chocolate bars, it is going to cost me $3.*0850

*I need to write my dollar sign here to give me the answer.*0855

*The next one, Sharon types 60 words per minute.*0864

*Find how long it will take for her to type 80 words.*0869

*My word ratio could be words over minute; 60 words per minute.*0874

*That is over 1 because the number of minutes is 1.*0886

*How long will it take... they are asking how long it will take.*0894

*They are asking for words or minutes?--they are asking for minutes; how long.*0899

*This will be X down here; then they are asking for 80 words.*0903

*Again you can use proportions; you can use cross products.*0911

*60 times X is 60X; equal to 1 times 80 is 80.*0919

*Remember if you want to find what 60 times X is and what X is, then you can divide the 60.*0929

*Anytime you have a number times 60, you have a number times a variable,*0939

*you can just divide that number to find X.*0942

*X is going to equal... I am going to cross out these 0s.*0947

*I am going to have 8/6; but then here I can simplify that.*0951

*Divide this by 2; divide this by 2; it is going to be 4/3; 4/3.*0957

*If I want to change this to a mixed number, this will be... 3 goes into 4 one whole times.*0971

*How many do I have left over?--1; my denominator is 3.*0982

*It will be 1 and 1/3 of a minute.*0988

*If you have a problem like this on your homework or at school,*1001

*it depends on how your teacher wants it, but you can change this to a decimal.*1008

*Or since it is minutes, you can take this fraction.*1012

*It is 1 whole minute and then some seconds; 1/3 is part of a minute.*1016

*You can just figure out how many seconds that would be by doing 60 divided by 3.*1022

*60 divided by 3; that is going to give you 20.*1030

*That means this is going to be 1 minute and 20 seconds.*1036

*Or you can just leave it like this if your teacher doesn't mind.*1040

*Then it is 1 and 1/3 of a minute.*1045

*The third example, Susanna estimates that it will take 4 hours to drive 600 kilometers.*1051

*After 3 hours, she has driven 500 kilometers.*1058

*Write a proportion to see if she is on schedule.*1064

*Basically they are asking if you make a ratio of this and you make a ratio of the next part,*1068

*are they the same?--that is all it is asking.*1076

*My word ratio, hours over kilometers.*1083

*It is going to be 4 hours over 600 kilometers.*1091

*We are going to see if this equals the same as 3 hours over 500 kilometers.*1102

*Let's see here; let's simplify these.*1117

*Here I can say that if I simplify this, 4 goes into 600 how many times?*1124

*Here is 1, 4, 2; I am just dividing it.*1141

*That is 5; 20; bring down this 0; 150.*1146

*If I divide this by 4 and I divide this by 4, I am going to get 1/150.*1155

*That means every hour, Susanna should drive 150 kilometers.*1167

*If 4 hours, she estimates she is going to be driving 600 kilometers,*1178

*that means every 1 hour, she is going to be driving 150 kilometers.*1184

*Is that the same thing as this?*1191

*If 1 hour, 150 kilometers, does she get this in 3 hours?*1193

*This is 1 times 3 equals 3 hours; 1 hour times 3 is 3 hours.*1203

*Does that mean 150 times 3 is 500?--let's see.*1210

*This times 3; 0; 5 times 3 is 15; add that; that will be 450.*1219

*No, if she drives 150 kilometers for every hour,*1234

*then in 3 hours, she should be driving 450 kilometers.*1240

*But she drove 500 kilometers; that means she is not really on schedule.*1246

*I mean, she is a little bit faster.*1252

*But according to what she has estimated, it is not the same.*1256

*So this one is no; she is not on schedule.*1260

*She is actually a little bit early because she drove more than what she thought she would be at.*1264

*This is not the same ratio.*1273

*If it is 4 hours for 600 kilometers, then in 3 hours, she should be driving 450 kilometers.*1276

*Because 1 hour is 150 kilometers; this needs to have the same ratio also.*1292

*1 hour is 150; this has to equal this too.*1304

*This one is no; she is early.*1311

*That is it for this lesson; thank you for watching Educator.com.*1318

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over similar polygons.*0002

*Polygons we know is some kind of shape.*0008

*If we have a triangle, triangles are polygons; squares, rectangles; those are all considered polygons.*0013

*Similar polygons means you have two polygons with the same shape.*0022

*They have to look exactly the same; but they are just different sizes.*0029

*One is going to be smaller or bigger than the other one.*0036

*But then they have to have the same exact shape.*0039

*When they are similar, it is a little symbol like this.*0046

*This means that this triangle here is similar to this triangle here.*0049

*It means that they have the exact same shape.*0054

*It means that one is not going to be any fatter and less taller and all that.*0057

*It is going to have the exact same shape.*0063

*But it is just going to be different sizes.*0066

*An example of similarities, if you are baby.*0072

*You are a baby; you have small hands; you have small feet; you are small.*0079

*As you get older, you grow; but everything has to grow proportionally.*0085

*Your hands grow and your feet grow the same amount.*0090

*If you are a baby and everything is small, as you grow older,*0094

*it is not like only your feet are going to grow but your hands stay the same size.*0100

*Everything has to grow according to how big and small or different let's say size is.*0105

*But then you are still going to have the same shape.*0113

*That is kind of an example of what it means to be similar.*0115

*Everything is proportional when things are similar.*0119

*Again if this is going to grow, if it is going to grow taller, then it also has to grow wider.*0125

*It has to grow in all areas just like a baby grows in all areas.*0130

*Again same shape but different size; then the corresponding sides are proportional.*0137

*Corresponding just means that the side that is basically related to each other.*0144

*This side and this side are called corresponding sides; corresponding sides.*0152

*It means that this side and this side are like the same.*0160

*They are being compared to each other.*0164

*Same thing here; this side with this side and this long side with this long side.*0166

*They are all corresponding.*0171

*That means I can create a ratio for each of these corresponding sides.*0175

*That means I can compare this one with this one.*0182

*4 to 6, remember that is a ratio; then it is all proportional.*0185

*Proportional means that this ratio is going to equal...*0192

*if I make a ratio for this, that is going to be the same.*0195

*For the third side too, this ratio to this is also going to be the same.*0200

*Just saying that all the sides, if you compare this side to this side,*0207

*that ratio is going to be the same as this side to this side.*0211

*It is also going to be the same as this side to this side.*0214

*We have three ratios; we only need two to make a proportion.*0218

*If you have a triangle, you are going to have three different ratios.*0227

*But you only need two.*0229

*You are only going to use the sides that they give you measures for.*0232

*Then you can create a proportion to solve for the missing side.*0238

*See how this all equals each other?--4/6 is equal to 4/6.*0244

*It is also equal to 6/9 because they all equal the same ratio of 2/3.*0251

*All of these ratios equal 2/3; that means these are all the same.*0260

*The first example is these two similar triangles.*0268

*You can draw a little similar symbol like that.*0276

*That means this triangle and this triangle have the same shape but just different size.*0278

*That means I can write a proportion and then find the value of X.*0287

*Here this side is corresponding with this side.*0292

*I can create a ratio comparing this to this.*0300

*The ratio will be 5 to X.*0305

*Again I want to write my ratio as a fraction because that is how I am going to solve my proportion.*0311

*This side to this side is 5 to X.*0316

*That means I can also create a ratio from this side to this side.*0318

*That will be 2 to 4.*0323

*Be careful, if you are going to make a ratio this to this,*0327

*then for the next ratio, the top number has to be from the same triangle.*0333

*If it is going to be this to this, then you have to make the next ratio this to that.*0338

*If you switch it around, then it is not going to be the same.*0343

*It is like saying boys to girls equals girls to boys.*0346

*You are flipping them; you are changing them; you can't do that.*0353

*If it is this triangle to that triangle, then your next ratio has to also be from this triangle to that triangle.*0356

*To solve this, you can use cross products.*0364

*Remember cross products is when you multiply across.*0369

*Or you can just simplify it and then use just mental math.*0372

*Here 2/4, this is the same as 1/2; how do I know?*0378

*2 divided 2 is 1; 4 divided by 2 is 2.*0385

*I can just make this also equal to 1/2.*0390

*1/2, that means the bottom number has to be double the top number.*0397

*5 over what?--what is X going to be?*0401

*If you multiply this by 5, you are going to get 5.*0405

*You have to multiply this by 5; you are going to get 10.*0407

*X has to equal 10; that means this side has a measure of 10.*0412

*Same thing here, we are going to write a proportion to find the value of X.*0425

*Here I can say this to this equal to this side to this side.*0432

*Or if I want, I can start off with this rectangle first as long as I stick to it for my second ratio.*0442

*5, corresponding side is X; 5/X equals... stick with the same one first... 7/14.*0451

*You can write it like that; or you can start with this one first.*0466

*It doesn't matter as long as you stick to that order.*0469

*7/14 is 1/2 because 7 divided by 7 is 1.*0475

*14 divided by 7 is 2.*0484

*That means I need to turn this also into 1/2.*0488

*1 times 5 is 5; 2 times 5 is 10.*0493

*X is going to equal 10.*0503

*If you want to practice cross products, again you are going to just do*0510

*5 times 14 which is going to be equal to X times 7.*0514

*I can write 7 times X.*0524

*You are going to just solve that out and then divide the 7.*0527

*You are going to solve for X that way.*0532

*You are still going to get 10.*0533

*70, 7 times 10 is going to equal 70.*0536

*For the third example, this is called a parallelogram.*0546

*It is not a rectangle because it is not perfectly going straight up and straight across.*0554

*It is not perpendicular; it is kind of tilting off to the side.*0559

*This is a parallelogram; but these are similar polygons.*0564

*Here this is corresponding with this side; this is corresponding with this side.*0572

*But they give you the other sides.*0581

*For a parallelogram, this side and this side are the same.*0584

*I can just write this as 12.*0589

*This side and this side are the same; this is going to be X.*0592

*When I write my proportion, I am just going to do the same thing.*0597

*Ratio of this to this side is 6 to X which is equal to 9 to 12.*0602

*Again I can figure out an equivalent ratio.*0613

*9/12 is the same as... let's divide this by 3; divide this by 3.*0619

*9 divided by 3 is 3/4.*0626

*That means this also has to be the same as 3/4.*0630

*3 times 2 equals 6; that means I have to multiply the 4 times 2.*0638

*X is going to give you 8; that means this side right here is 8.*0645

*Again you can just do cross product; 6 times 12 equals 9 times X.*0652

*Solve it that way.*0660

*For the fourth example, they give us a word problem.*0664

*We have to draw our own similar polygons.*0670

*A tree casts a shadow that is 10 feet long.*0676

*Let's see, I want to draw a tree; there is a tree.*0681

*I know my drawing is kind of bad; there is the ground; tree.*0688

*The shadow... let's say this is a shadow... is 10 feet long; this is 10 feet.*0696

*A person 5 feet tall is standing next to the tree.*0708

*Let's say the person is right here; draw a stick man.*0713

*This is still the same ground.*0720

*Person 5 feet tall is standing next to the tree and is casting a shadow.*0722

*Or let's say this person is 5 feet tall.*0727

*From here down to the ground is 5 feet.*0731

*Where this person is standing, his shadow is 3 feet.*0737

*The triangle formed by the person's height in the shadow...*0747

*That means height and shadow; this is a triangle; you can see that.*0751

*This triangle is similar to the tree and its shadow.*0761

*Then the triangle formed by this tree, here all the way down to this shadow.*0767

*These two triangles, this triangle here and this triangle here, are similar.*0778

*They want us to find... what is it?... the height of the tree.*0785

*How tall is the tree?--I am going to make this X, from here to here.*0796

*Because they said it is similar, I can make a proportion now.*0803

*I can say the 10 feet, the shadow, over the 3 because this side is corresponding to this side.*0808

*It is going to be equal to the tree's height.*0820

*Remember if you started off with this tree triangle, then you have to start it with the next one.*0823

*The tree height X over the person's height, 5.*0830

*From here, now it is a proportion; now I can just solve it out.*0838

*In this case, I can't simplify this.*0845

*I can't do the equivalent fraction method because this is already simplified.*0847

*There is no number that goes into both 10 and 3.*0851

*In this case, I just have to use cross products.*0855

*Here I want to do 3 times X.*0862

*3 times X equals 10 times 5 which is 50.*0865

*Again if I am going to solve for X, I need to divide this 3 because 3 times X is 50.*0874

*It is 50 divided by 3 to find the X.*0880

*If I want to find this, I have to do that.*0887

*Make sure this top number goes inside.*0892

*3 goes into 5 one time; 3, if I subtract it, I get 2; 0.*0895

*3 goes into 20 six times which is 18; I get 2.*0903

*Now that I have a remainder, I have to put my decimal point.*0912

*Bring down another 0; 3 goes into 20 again eight times.*0918

*18 again; 2; another 0; 8.*0926

*It depends on how many numbers after the decimal point your teacher wants.*0934

*But otherwise you can just probably leave it as 16.89.*0940

*Or maybe 16.9 if we are going to round this; round this from that number.*0947

*16.9; that will be in feet.*0954

*The X or the tree is 16.9, almost 17 feet tall.*0961

*Again create your proportion.*0969

*Make sure when you do your ratio, you are going to stick with the same side first.*0970

*It is this side to this side is your ratio.*0976

*Equals this side to that side ratio.*0979

*Then you just solve your proportion using cross products.*0983

*That is it for this lesson; thank you for watching Educator.com.*0988

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over some scale drawings.*0002

*A scale drawing is an enlarged or reduced drawing that is similar to the actual object or place.*0007

*You are basically going to compare two things--the actual object or place and the drawing.*0016

*It could be something that is enlarged or can be something that is reduced.*0026

*We just went over similar figures.*0031

*It is the same concept where you are going to compare something huge and something small.*0035

*Or vice versa, something small with something big.*0044

*Again these two things are going to be similar,*0049

*meaning they are going to have the same shape but just different size.*0053

*The scale is the ratio of the two.*0057

*A lot of times for a scale drawing, we use maps.*0062

*A map is one of the main examples of a scale drawing.*0067

*If you have a map of the city that you live in, then that would be drawn to scale,*0071

*meaning every inch or so on the map is going to represent however many miles in real life, the actual place.*0077

*That is one example of a scale drawing.*0089

*If you have let's say a person and you draw a picture of that person,*0093

*but you draw it to scale meaning you are going to draw that person the same size but just on paper,*0101

*then that would also be a scale drawing because that drawing is going to represent the actual person or object.*0108

*For example, if I have a map of... go back to the map example... two cities.*0118

*Let's say this is city A; that is one city; here is another city B.*0125

*This map is going to represent the actual place, city A and city B.*0136

*If we say that from here to here on the map, let's say this is 2 inches apart.*0146

*We know that in actuality city A is not 2 inches away from city B.*0156

*But on the map, if this represents 1 inch... this is also 1 inch.*0161

*If I say that 1 inch on the map represents 10 miles in real life,*0168

*then the ratio, the scale, is going to be 1 inch on the map to 10 miles in real life.*0176

*It is going to be the ratio between the drawing and the actual place.*0187

*You are going to use that to find, let's say I ask how away is city A from city B?*0194

*On the map, since it is 2 inches, how will I know how far away it actually is in real life?*0205

*If this is a ratio, I can turn this into a fraction.*0217

*1/10, 1 to 10, that is the ratio.*0220

*Then I am going to create a proportion.*0225

*2 inches to X, that is what we are looking for.*0228

*If 1/10 equals 2/X, what does X equal?*0235

*You can either make this using this equivalent fraction,*0241

*meaning turn this into the same fraction as 1/10 to find X.*0249

*Or remember we can use cross products.*0255

*We can multiply this way, 1 times X equal to 2 times 20.*0258

*If you multiply across this way and use cross products, then 1 times X is 1X.*0263

*Equals... 2 times 10 is 20.*0271

*1X is the same thing as X; so we know that X is 20; 20 miles.*0275

*That means city A and city B, they are actually 20 miles apart from each other.*0283

*Let's go through some examples; the first example is the map.*0289

*On the map, it is 1 inch; 1 inch represents 50 miles in real life.*0296

*The ratio is 1 inch to 50 miles.*0303

*If I want to turn this into a word ratio, remember I can say that this is the map.*0309

*Then I can say that this is the place or actual.*0316

*This would be like the word ratio.*0326

*Your word ratio is in words the ratio of what is going to go on top and what is going to go on the bottom.*0328

*Again the ratio is 1 to 50.*0334

*This next part, if it is 100 miles between two cities, how many inches is it apart on the map?--the 100 miles.*0343

*Is that going to go on the top or the bottom of this next ratio?*0353

*Remember you have to keep the ratio according to the word ratio.*0356

*It is actually 100 miles.*0362

*That would be on the bottom because that is the actual place.*0363

*100 goes on the bottom.*0367

*Then the map, how many inches is it apart on the map?*0369

*The map number is going to go on the top.*0374

*That is what we are looking for; you can call that X.*0376

*Now we can solve this.*0380

*If we are going to use cross products, 50 times X is 50X.*0384

*Remember number times letter, you just put them together like that.*0390

*Equals... 1 times 100 is 100.*0393

*Here remember to solve for X.*0399

*50 times X equals 100; 50 times what equals 100?*0402

*I know that X is 2 because 50 times 2 is 100.*0406

*That is inches; X is 2 inches; on the map, it is 2 inches apart.*0412

*The next example, the scale of a drawing of king kong is 1 inch to 3 feet.*0423

*If king kong is 54 feet tall, how tall is he in the drawing?*0430

*Again we have this ratio.*0437

*We are going to say drawing of king kong over the actual height of king kong.*0440

*This is our word ratio; this is what we are going to base it on.*0449

*The drawing is 1 inch over 3 feet.*0452

*You don't have to put these here because we are not going to use that to solve.*0459

*If you want, you can just put 1/3; that is fine.*0464

*Equals... king kong is actually 54 feet tall.*0467

*That is going to go on the bottom because that is the actual; 54 feet.*0474

*How tall is he in the drawing?*0480

*That is the top number; that is X; that is what we are looking for.*0483

*Again I can use this, cross products; 1 times 54 is 54.*0488

*Equals... 3 times X, 3 times X.*0501

*You can just write that as 3X as long as you know that that represents 3 times X.*0507

*To solve for X, remember if I want to get rid of this, I select the variable, get rid of the 3 by dividing.*0513

*I can divide this by 3; then I can divide this by 3.*0522

*Here to do 54 divided by 3, put 54 the top number inside.*0529

*3 goes into 5 one time; this is 3; subtract it; you get 2.*0538

*Bring down the 4; 3 goes into 24 eight times; X is 18.*0544

*That means 3 times 18 is 54.*0554

*Since that is the top number, that is the drawing number, I know that that is in inches.*0558

*King kong is 18 inches tall in the drawing.*0563

*The third example, a toy car is made to scale with the actual car.*0576

*If the ratio of the car to the toy is 15 inches to 0.5 millimeters,*0582

*and the toy is 6 millimeters long, what would be the length of the actual car?*0591

*The ratio of the car to the toy; that means my word ratio is going to be car over the toy.*0600

*The car to the toy is 15 inches to 0.5 millimeters.*0610

*I am going to create a proportion; the toy is 6 millimeters.*0620

*That is the toy number; I am going to put that on the bottom.*0627

*What would be the length of the actual car?*0632

*That is going to go on the top, X.*0634

*Again I am going to cross multiply.*0640

*Here 0.5 times X is 0.5 times X or just 0.5X.*0647

*Equals 15 times 6; we are going to have to solve that out.*0655

*15 times 6; this is 0; 6 times 1 is 6; plus 3 is 9; this becomes 90.*0661

*Again I want to know what I have to multiply to 0.5 or 0.5 to give me 90.*0674

*Then I would have to divide 0.5.*0683

*If I make this into a fraction, this is the same thing as divide; 0.5.*0686

*Again this is 90 divided by 0.5.*0694

*I am going to do that right here; 90 divided by 0.5.*0698

*Make sure that this top number is inside the house or inside this.*0705

*To divide this, if you have a decimal on the outside, remember you have to move it to the end.*0712

*I moved it one space.*0718

*That means from here, this end of the number, since I don't see a decimal, it is always at the end.*0721

*I have to move this number one space.*0727

*Then I have to fill this space with something; that will be 0.*0732

*This is my new decimal point right here; I am going to bring that up.*0739

*Then I have 3 spaces on top right here; now I can divide.*0744

*05 is the same thing as 5; 5 goes into 9 one time.*0750

*This is 5; subtract it; I get 4; bring down the 0.*0756

*5 goes into 40 eight times; that becomes 40.*0761

*If I subtract it, I get 0; then I have to bring down this 0.*0768

*5 goes into 0 zero times; that is just 0 and 0.*0773

*My answer becomes 180; X equals 180.*0779

*0.5 times X equals 90; that means 0.5 times 180 equals 90.*0788

*Again my car then because that is the top number, my X.*0796

*That represents the car length; that is going to be in inches.*0801

*My car is 180 inches long.*0813

*That is it for this lesson; thank you for watching Educator.com.*0819

*Welcome back to Educator.com; for the next lesson, we are going to go over probability.*0000

*Probability is the measure of an outcome compared to the total possible number of outcomes.*0007

*It is in the form of a fraction.*0015

*That means we are going to have a top number and we are going to have a bottom number.*0018

*We are going to have a numerator and denominator.*0021

*On the top, you are going to write the number of the desired outcome.*0026

*Basically what are you looking for?--what is it asking for?*0038

*That number is going to go on the top.*0041

*Then on the bottom, you are going to write the total number of possible outcomes.*0045

*Probability is the measure between the part, the part that you want*0060

*or the part that it is asking for over the whole thing or the total.*0066

*That is the ratio; the ratio is between the part to the whole.*0073

*For example, if I have a bag full of marbles, let's say I have one black marble in this bag.*0085

*I have 3 blue marbles; and I have 2 red marbles.*0096

*I want to know what is the probability, what are the chances of picking a blue marble?*0114

*If I close my eyes and reach into the bag and pick one marble,*0122

*what is the probability, what are the chances of picking a blue marble?*0126

*That would be the number of blue marbles is going to go on the top.*0133

*Blue marbles over... remember the bottom is always the total number, the total number of marbles.*0138

*This is going to be my probability.*0150

*How many blue marbles do I have?*0153

*I have 3; that is my desired outcome.*0155

*I am asking you what are the chances of me picking a blue?*0157

*There are 3 blue marbles; it is going to be 3 on the top.*0162

*Over... how many marbles do I have in all?*0167

*1, 2, 3, 4, 5, 6; 6; my probability is 3/6.*0170

*Since this is a fraction, I need to simplify it.*0179

*If 3/6, then I have to divide this by 3, divide this by 3.*0183

*This becomes 1; 6 divided by 3 is 2.*0190

*The probability of picking a blue marble from this bag of marbles is 1 to 2 or 1/2.*0194

*Let's go over some examples.*0204

*The first example, what is the probability of landing on orange?*0208

*Here I have a pie chart that is divided up into 1, 2, 3, 4, 5 different colors.*0214

*Orange, purple, yellow, green, red; what is the probability...*0223

*If I spin this spinner, what is the probability of landing on orange?*0227

*That means orange is my desired outcome; that is what I am asking for.*0234

*The orange is going to go on the top.*0239

*The total or the whole thing is going to go on the bottom.*0245

*How many orange sections do I have?--I only have 1.*0251

*Then I am going to put 1 on the top.*0256

*How many different sections do I have?--1, 2, 3, 4, 5.*0260

*Out of 5 sections, 1 of them is orange.*0266

*That is why the top number is 1.*0269

*On the bottom, I am going to put 5 because that is the total.*0272

*The probability of landing on orange is going to be 1/5.*0275

*Again it is part to whole; so 1 to 5 is the probability.*0281

*If I ask you what is the probability of landing on purple?*0289

*There is 1 purple out of total of 5 different colors.*0292

*It will be the same thing, 1/5.*0297

*The next example, each side of a dice has a number from 1 to 6.*0304

*What is the probability of rolling a 5?*0310

*We know what a dice is; it is a cube that looks like this.*0313

*Each side has a number representing dots usually.*0322

*One side will have 1; the other side will have maybe 2.*0330

*One side will have 3; another side might have 4 like that.*0334

*There are 6 sides; each side has a different number of dots.*0341

*I want to know what the probability of rolling a 5 is.*0350

*The desired outcome, the top number is going to be rolling a 5.*0354

*Then the bottom number is going to just be the total or the total number of numbers.*0363

*Even though this is a number, that is not the actual number I am going to write on the top for my probability*0371

*because there is not 5 of something; there is not five 5s.*0378

*How many 5s do I have on this?--how many sides have 5 dots?*0382

*Only 1 because each side has a different number.*0389

*There is only one side that will have 5 dots.*0393

*That means I have to put 1 here because it is the number of this desired outcome.*0397

*It is not what this number is.*0403

*It is not that this is a 5 and I have to write the 5 up here.*0406

*It is how many sides do I have with 5 dots?--that is just 1.*0409

*How many total number of sides do I have?--6.*0417

*because again when you are rolling a dice, you are looking to see what the top side is going to be.*0422

*How many dots is the top side going to have?*0429

*That is 1 out of 6.*0433

*Same thing if I asked you what is the probability of rolling a 2?*0437

*Isn't this roll a 2 right now?*0441

*How many sides have a 2?--only one side does.*0445

*It will be the same thing, 1 out of a 6 possible sides.*0449

*1/6 is the probability.*0455

*More marbles; there are 2 red marbles, 5 blue marbles, and 3 green marbles in the bag.*0464

*Let me draw my bag; I have 2 red; red, red.*0475

*I have 5 blue; 1, 2, 3, 4, 5; blue, blue, blue, blue, blue.*0484

*I have 3 green; 1, 2, 3; green, green, green.*0494

*What is the probability that the marble will be red?*0503

*That is my desired outcome; that is the part that is asked for.*0506

*That is going to go on the top; red over total; that is my probability.*0513

*How many reds do I have?--I have... it says here 2 red; red and red.*0522

*2 out of... how many do I have in all?--1, 2, 3, 4, 5, 6, 7, 8, 9, 10.*0534

*1, 2, 3, 4, 5, 6, 7, 8, 9, 10 total marbles.*0544

*My probability would be 2 out of 10.*0550

*Again this is still a fraction; I have to simplify it.*0554

*I know that because the top number and the bottom number are both even numbers,*0560

*I can divide both of them by a 2.*0563

*2/10 becomes... 2 divided by 2 is 1; over... 10 divided by 2 is 5.*0568

*The probability of picking a red is 1 out of 5.*0577

*The fourth example, there are 6 boys and 5 girls in a room.*0585

*If one student is chosen at random, what is the probability that the student will be a girl?*0592

*Again my desired outcome, girl over total.*0601

*I now that there are 5 girls; number of girls is going to be 5.*0611

*On the bottom, I am not writing the number of boys.*0620

*Be careful with this type of example because when it comes to ratio which we learned a few lessons ago,*0623

*if I ask for the ratio of girls to boys, then I would put 5 on the top and 6 on the bottom*0630

*because that would represent girls to boys.*0637

*But this is probability; this is girls to total number of students.*0640

*The bottom number for probability is always the total.*0647

*How many students do I have in all?--5 out of how many in all?*0652

*I would have to add the number of boys and the number of girls*0658

*to figure out how many total number of students there are.*0662

*If I have 6 boys and 5 girls, then I have 11 students in all.*0666

*The total I am going to write is 11.*0674

*Girls, 5; there is 5 girls; out of 11 students.*0677

*The probability is 5/11; this is the probability.*0685

*If they ask for the ratio of girls to boys, then it would be 5 to 6.*0692

*But again probability is different; total number has to go on the bottom.*0702

*That is it for this lesson; thank you for watching Educator.com.*0707

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over percents, fractions, and decimals.*0002

*We are going to learn how to change percents into fractions and percents into decimals.*0009

*First, a percent is a ratio that compares a number to 100.*0020

*We know that this symbol right here represents percent.*0026

*To change from a percent to a fraction, all you are going to do,*0031

*since we know that a percent is a ratio of the number to 100,*0036

*to change a percent to a fraction, all you have to do is drop this little symbol and put the number over 100.*0042

*If I have 17 percent, all I have to do is take this number and put it over 100 and then simplify.*0051

*That becomes my fraction.*0060

*If I have let's say 50 percent, to change this to a fraction, I am just going to take the number.*0064

*I am going to drop this percent sign.*0073

*Since I am changing it to a fraction, I no longer need the percent.*0075

*Going to put it 50/100; 50/100, I know I can simplify that.*0079

*I know that 50 goes into both numbers.*0087

*I can divide the top and the bottom by 50.*0090

*50 divided by 50 is 1.*0095

*100 divided by 50 is 2 because 50 fits into 100 two times.*0099

*50 percent into a fraction becomes 1/2.*0105

*That is how you change from a percent to a fraction.*0111

*To change from a percent to a decimal, we are going to divide it by 100.*0115

*We are actually going to put it over 100, but we are actually going to divide it.*0124

*To divide it, whenever you divide a number by 100, you just move the decimal point to the left two spaces.*0131

*If you get confused which way to move it, just know that percent is always a bigger number than a decimal.*0142

*Decimals are small; percents are always bigger.*0149

*In order to make this number, because it is a percent...*0155

*If I change it to a decimal, then I have to make it smaller.*0157

*Think that you are going to make it smaller.*0160

*When you make it smaller, you have to move the decimal over to the left so that the number will get smaller.*0162

*Again the decimal point here, if you don't see one, it is always at the end.*0170

*There is always an invisible decimal there.*0175

*I am going to write in my decimal point.*0179

*Then I am going to move it two spaces to the left.*0182

*It is going to go one; that is one number; that is one space.*0185

*Two, that is another space; 17 percent becomes 0.17.*0189

*Again just two spaces to the left because percent is bigger than the decimal.*0199

*You have to make it smaller by moving it to the left.*0204

*Another number, if I have let's say 5 percent, remember drop the percent sign.*0209

*Then you are going to move the decimal point; let me write it out here.*0216

*5, the decimal point is at the end of it, end of the number.*0221

*You are going to go one and then two; then put that point there.*0225

*But since there is an empty space here, I have to fill it in with a 0.*0230

*5 percent becomes 0.05; that is the decimal.*0235

*Again percent to a fraction, all you have to do is put the number over 100.*0243

*To change it from a percent to a decimal, you just move it to the left two spaces.*0250

*For fractions, if I want to change a fraction to a decimal,*0259

*I am going to think of this little bar, that fraction line, as divide.*0266

*It is just going to be A divided by B.*0271

*You are actually going to divide it.*0274

*It is going to be A divided by B.*0276

*Again fraction to a decimal, you are going to just divide the top number into the bottom number.*0280

*Here is a fraction here; 3/4, I am going to divide 3 and 4.*0286

*It is going to be 3 divided by 4.*0294

*Make sure, make sure this top number, even though it is smaller, this top number has to always go inside.*0296

*When you divide, 3 inside; 4 on the outside.*0303

*The top number gets to go inside.*0312

*This number is bigger than the number inside; but that is OK.*0318

*Remember here the decimal point is always at the end of a number.*0324

*If you don't see it, it is always at the end.*0327

*When you have a decimal point, you can always put 0s behind it as many as you want.*0330

*I can put a hundred 0s if I want to.*0338

*I am not going to; but I could if I want to.*0341

*It is OK for you to put 0s at the end of it*0345

*as long as it is behind the decimal point and it is at the end.*0349

*Here remember if I want to divide the decimal, I have to bring this decimal point up.*0353

*I know 4 goes into 3 zero times; it doesn't fit into 3.*0360

*But 4 goes into 30 how many times? *0365

*4 times 6 is 24; 4 times 7 is 28; 4 times 8 is 32.*0370

*I know that this is 7; 4 times 7 is 28.*0378

*If I subtract it, I get 2; bring down this 0.*0385

*4 goes into 20 five times; that becomes 20; you subtract it; it becomes 0.*0390

*3/4 then becomes this number here; becomes 0.75; that becomes my decimal.*0400

*Again fraction to decimal, you are going to do the top number divided by the bottom number.*0413

*Solve it out; you are going to get your decimal.*0419

*Usually it keeps going; let's say it doesn't give you a remainder of 0.*0423

*Then you end up having to keep going.*0430

*Usually you are going to only write it maybe two or three numbers after the decimal place.*0433

*It depends on what your teacher wants.*0439

*If your teacher says two numbers behind the decimal point, then two numbers.*0441

*Three numbers, then three numbers; make sure you round it though.*0447

*The last number, you are going to have to round it based on the number behind it.*0451

*You are going to either keep it as a 5 or you are going to round it up.*0455

*This right here, I can only write 2 because it stopped.*0460

*The remainder became a 0; I don't have to keep going.*0466

*That would just be my answer; I am done.*0469

*If I want to change a fraction to a percent, I have to first change my fraction to a decimal.*0473

*I am going to do exactly what I did up here.*0483

*Then I am going to change my decimal to a percent by moving it to the right two spaces.*0486

*I am going to move the decimal point over to the right two spaces.*0497

*Remember to change from a percent to a decimal, I moved it to the left two spaces.*0500

*Decimal point, left two spaces.*0514

*But if I am doing the opposite, if I am going to from decimal to a percent,*0523

*it is the same thing, but then I have to get bigger.*0532

*Remember decimal is smaller than the percent.*0534

*I have to go to the right to make it bigger; right two spaces.*0537

*Decimal place is going to go to the right.*0543

*Again change your fraction to a decimal.*0545

*3/5, if you divide it, it becomes 0.6; 0.6.*0551

*Then I am going to move it to the right two spaces.*0557

*It is going to go one, two; there is my new decimal point.*0560

*Again I have empty space; I have to fill it with a 0.*0565

*This 0 in the front, I can just drop that because 060 is the same thing as 60.*0568

*If a 0 is in the front, then that doesn't really matter.*0576

*You can just let it go; just drop it; erase it; this becomes 60 percent.*0579

*If I have another decimal, let's say I have 0.50.*0588

*Again you are going to take this; you are going to go one, two.*0598

*Leave it there; it is going to become 50 percent.*0601

*Now decimals.*0611

*To change a decimal to a fraction, you are going to count how many numbers you have behind the decimal point.*0614

*If I have 0s at the end here, remember you can add 0s at the end of a number, behind the decimal point.*0625

*You don't have to count those; just count the numbers here.*0631

*I have one, two; I have two numbers behind the decimal point.*0638

*That means I am going to put this number 15 without the decimal point over two 0s.*0642

*I have to put 1 in front of it; it is going to be 100.*0652

*If I have 0.155 and I want to change this to a fraction, I have one, two, three numbers behind the decimal point.*0656

*Then I am going to take that number; put it on top.*0670

*On the bottom, I am going to put three 0s.*0673

*That is going to be 1000.*0677

*One, two, three 0s with the 1 in front of it, that is 1000.*0679

*This would be your fraction; this is 0.15; 15/100.*0684

*Again since it is a fraction, you have to simplify it.*0691

*You are going to see a common factor.*0697

*What number goes into both 15 and 100?*0700

*I know that 5 goes into 15 and 5 goes into 100.*0703

*I can divide both top and bottom by 5.*0708

*This becomes... 15 divided by 5 is 3; over... 100 divided by 5 is 20.*0713

*That becomes your fraction.*0724

*Then to change a decimal to a percent, again we already went over this.*0730

*Decimal to a percent, you are going to remember take the decimal point and move it to the right two spaces.*0734

*Again if you get confused which one do I move to the right and which one do I move to the left?*0742

*It is always going to be two spaces.*0746

*But it depends on what you are changing it to.*0749

*Always think that to percent is bigger than the decimal; decimals are small.*0753

*Think of change like 15 cents; that is small; decimal points are small.*0758

*You want to make the number bigger.*0764

*The way to make the number bigger is to move the decimal point over to the right to make it whole numbers.*0767

*You go one, two; then decimal point at the end of 15.*0775

*It becomes 15 percent.*0780

*If I have a decimal point 0.155, to change it to a percent, you are going to go one, two again.*0783

*That is going to become 15.5 percent.*0791

*Even though you still see a decimal, this is still percent because you moved it two spaces.*0798

*Anytime you move the decimal point two spaces to the right, it becomes a percent.*0802

*Let's do some examples; write each percent as a fraction in simplest form.*0809

*Remember percents, you always just put it over 100.*0815

*No matter what, you are always going to just put it over 100.*0820

*Take this number 42; we are going to get rid of that percent sign.*0824

*We are changing it to a fraction; it is going to be 42/100.*0830

*Always just put it over 100.*0835

*These are both even numbers; I know that I can simplify.*0838

*Since they are both even... this one ends in a 2; this one ends in a 0.*0843

*They are both even numbers; I can divide each of them by 2.*0848

*42 divided by 2; 2 goes into 4 two times; becomes 4.*0853

*I am going to bring down the 2; 2 goes into 2 one time.*0864

*42 divided by 2 is 21; over... 100 divided by 2.*0868

*You are cutting it in half; 100, cut it in half is 50.*0875

*My answer is 21/50; that is simplest form.*0881

*Same thing here; 10 percent, we are going to write it over 100.*0892

*Percent to a fraction, you always just put it over 100.*0896

*If I have a 0 on top and a 0 on the bottom, I can just cross out the 0s.*0901

*This is going to be 1/10; 10 percent to a fraction is 1/10.*0907

*Same thing here; I take this whole number, 220; put it over 100.*0918

*0 on the top; 0 on the bottom; I can't do it like this.*0925

*If I have 0 here, 0 here, I can't cross out those two.*0932

*One has to be on the top and one has to be on the bottom for you to be able to cross it out.*0936

*This is 22/10; but then look.*0941

*This is called an improper fraction because the top number is bigger than the bottom number.*0946

*I need to change it to a mixed number.*0952

*In order for me to change it, I want to see how many times does 10 fit into 22?*0955

*There is going to be leftovers; but how many times can it fit into 22?*0963

*10 times 2 is 20; that is going to be two whole numbers.*0969

*How many are remaining?--2; there is 2 leftovers.*0975

*Over... keep the same denominator, 10.*0982

*2/10, I can simplify this because they are both even.*0989

*2 goes into both the top and the bottom; this becomes 2.*0992

*2 divided by 2 is 1; over 5; this is my answer.*0997

*Again all I did was put this number over 100, just like I always do when I change percent to fraction.*1002

*I crossed out the 0 at the top and the bottom 0; it is 22/10.*1009

*Then I just changed it to a mixed number; it became 2 and 2/10.*1016

*Then simplified this fraction; 2/10 became 1/5.*1023

*Write each as a decimal; percent to a decimal.*1031

*Again percent to a decimal, that is when you...*1037

*Decimal to percent or percent to decimal, that is just when you move the decimal point two spaces.*1040

*But percents remember are bigger than the decimal.*1046

*I have to turn this number, to change it to a decimal, I have to make this number smaller.*1049

*Remember smaller means move the decimal point to the left; that makes it smaller.*1054

*Take the 68 percent; I am going to put my decimal point here because I don't see it.*1061

*It is always at the end; then I go one, two, point.*1067

*To write it again, it just becomes 0.68; that is my decimal number.*1073

*Now I have a fraction.*1085

*To change it to a decimal, I just need to divide these two numbers, 5 divided by 8.*1086

*Let's do it right here so I have some space to work with.*1096

*Here 5 inside; the top number always goes inside; 8 on the outside.*1100

*Again I need to make this longer.*1108

*Decimal point is at the end of the number.*1110

*I can put as many 0s as I want as long as it is at the end of a number and behind the decimal point.*1114

*Bring the decimal up; 8 goes into 50 how many times?*1123

*I know that 8 times 5 is 40; 8 times 6 is 48.*1128

*8 times 7 is 56; so I know it is 6.*1137

*This is 48; if I subtract, then I get 2; I bring down the 0.*1141

*How many times does 8 fit into 20?*1150

*8 times 2 is 16; 8 times 3 is 24; it has to be 2.*1156

*That becomes 16; subtract it; I get 4.*1162

*Remember I can add another 0 if I would like; put a 0 there.*1167

*8 goes into 40 five times; that goes in evenly; I have 0 left.*1171

*My answer here, 5/8, becomes 0.625.*1180

*I could put a 0 here if I want to.*1188

*I could put a 0 here if I want to in the front because that just means zero whole numbers; zero 1s.*1191

*Percent, again percent to decimal, you are just moving the decimal point over.*1203

*Make it smaller; so two spaces to the left.*1207

*340, the decimal point is right here at the end; go one, two.*1212

*To write it again, it becomes 3.40 or 3.4.*1219

*I don't have to write the 0.*1224

*Remember it is behind the decimal point and at the end of a number.*1225

*For this one, again 3 divided by 50; let's do it here.*1233

*Put the top number inside; 50 goes on the outside.*1243

*Point, decimal at the end; I can add a few 0s if I want.*1253

*50 goes into 30... 30 is too small; it is smaller than 50.*1259

*50 goes into 30 zero times.*1266

*I put a 0 on top of this to represent that I am talking about 30.*1268

*That is 0 times 50 is 0; subtract it; I get 30.*1274

*Bring down this 0; how many times does 50 go into 300?*1282

*I know that 5 times 6 is 30; let's try that.*1292

*50 times 6; 0; 30; yes, 50 times 6 is 30.*1299

*50 times 6 is... I'm sorry... 300; then that is going to become 0.*1307

*I don't need to bring anything else down because I have no remainders.*1318

*3 divided by 50 is 0.06; this 0, you cannot drop.*1323

*You have to have this 0 because that is between a decimal and another number.*1331

*If the 0 was right here, then you can drop this.*1337

*You don't have to put that there because it is at the end of a number and behind the decimal point.*1341

*This 0 is not at the end of a number because there is another number here.*1346

*But this one you can put because that just means zero whole numbers, zero 1s.*1351

*That you can write there if you would like.*1357

*That is a decimal for that fraction.*1362

*Write each fraction as a percent.*1367

*Here anytime you want to go to percent, you always need a decimal.*1372

*I have a fraction; I need to change it first to a decimal*1381

*so that I can just move the decimal point two spaces and make that into a percent.*1385

*Change this to a decimal.*1393

*99 divided by 100, this is going to be... decimal point at the end; put a 0 here.*1396

*100 goes into 99 zero times; 100 goes into 990 nine times.*1415

*Bring the decimal point up because 9 times 100 is 900.*1426

*Subtract it; 090; bring down another 0; 100 goes into 900 nine times.*1433

*Here this is 0.99 in decimal; but remember we are changing it to percent.*1448

*I need to move the decimal over which way?--left?*1457

*No, right, because again percent is bigger than decimal.*1460

*You have to make the number bigger by moving it to the right.*1463

*This becomes 99 percent.*1466

*That would be the same thing from a percent into a fraction.*1475

*Remember how we just always put it over 100.*1478

*See how that is 99 over 100.*1480

*The next one, 6 divided by 5 first.*1484

* 6 inside; the top number goes inside; 5 on the outside.*1490

*Decimal point at the end; bring it up; 5 goes into 6 one time.*1496

*That becomes 5; subtract it; write the 1.*1505

*I can add a 0 here because it is behind the decimal point; bring it down.*1510

*5 goes into 10 two times; that becomes 10; subtract it; I get no remainders.*1516

*My decimal or this fraction is 1.2.*1525

*Again to change it to percent, I am going to move the decimal point right here.*1532

*Two spaces to the right to make it bigger; it goes one, two.*1536

*Again I have empty space here; I have to put a 0 there.*1542

*This is 120 percent.*1548

*This next one, this one is going to be a little bit harder.*1558

*14 divided by 15; remember that the top number has to go inside.*1563

*Decimal at the end; add 0s; 15 goes into 14 zero times.*1571

*Bring the decimal point up; how about to 140?-*1580

*Let's see; I know that if I multiply this by 10, I get 150.*1587

*Let me just try something a little bit smaller than 10 because this 140 is smaller than 150.*1597

*I am going to try 9; 15 times 9 is 45.*1604

*9 times 1 is 9; add 4 is 13; 135.*1611

*Isn't that only five away from 140?*1619

*I know that 9 has to be the correct number; that is the closest one.*1622

*That is 135; subtract it; I get 5; bring down the 0.*1629

*15 goes into 50 how many times?--15 times... let's see... 3 is 5.*1638

*How about 3?--it is 45; that is only 5 away from 50.*1654

*Then 3 has to be the correct answer; 3, you get 45.*1662

*Subtract it; I get another 5; you can add a 0; bring it down.*1670

*I know that again it is 3; 45; 5.*1677

*It is going to keep going; 0; bring down the 0; 3.*1686

*I can just stop here because I have enough numbers to give me my decimal.*1696

*Again if I just want to make it 0.93, two or three decimal places,*1701

*if I want to stop here, then I can just base this number on that, the one behind it.*1711

*It is smaller than 5; it will just stay a 3.*1716

*If this number where the arrow is pointing to, if that number was 5 or bigger, I can round it up to 0.94.*1720

*I can do 0.93 or I can do 0.933; decimal.*1727

*But again I am changing it to percent; this is going to go one, two.*1734

*There is the new decimal point right there; 93.3 percent.*1740

*I can drop the 0 because it is in the front; that doesn't mean anything.*1747

*There is my answer.*1753

*Here we have a table; the first problem, 50 percent.*1760

*I want to change it to a fraction and decimal.*1769

*Same thing here; this is a decimal.*1772

*I want to write this as a fraction and as a percent.*1774

*I have to fill in all these.*1780

*Percent to a fraction.*1783

*Remember anytime I want to change a percent to a fraction, I just put it over 100.*1786

*This will be 50/100; that is it; but I just have to simplify.*1791

*50/100, again 50 goes into both the top and the bottom; I can just divide.*1799

*This is just 1/2.*1807

*To change it to a decimal, you can do two things; you can move this.*1812

*Remember percent to decimal, you move it to the left two spaces.*1819

*Or from a fraction to a decimal, you just divide, top number divided by bottom number.*1823

*This is easier because all I have to do is move this decimal point.*1828

*It starts right here at the end; it goes one and two.*1833

*The decimal point is going to be right in front of the 5.*1838

*0.50; you can leave it like that.*1841

*Or you can drop this 0 because the 0 is at the end of a number.*1847

*It is behind the decimal point.*1852

*This will be 0.50 or 0.5; that is part two.*1854

*The next one, 0.07, that is the decimal.*1864

*I want to change it to a percent; again I am making it bigger.*1868

*That means I have to move the decimal point over to the right to make it bigger.*1872

*Two spaces; it is going to go one, two.*1877

*It is going to go right there, right behind the 7.*1881

*07 percent or 07 is the same thing as... let me just erase the 0.*1885

*7 percent is the same thing as 7 percent.*1896

*I can just leave it like that.*1899

*To change this to a fraction, remember percent to a fraction, you just put it over 100.*1903

*Or from a decimal to a fraction, remember you count the number of numbers behind the decimal point.*1912

*That is two; you are going to put two 0s in the denominator.*1917

*Either way it is the same thing; it is 7/100.*1921

*See if you can simplify; no, because there is no common factors.*1926

*No numbers that go into both 7 and 100; so that is it.*1931

*The third one, same thing here; percent to a fraction; put this number over 100.*1938

*8... let me write that over; 8/100.*1945

*Here we have both even numbers; divide this top and the bottom by 2.*1955

*8 divided by 2 is 4; over... 100 divided by 2 is 50.*1962

*Look, I can simplify this again because top and the bottom number are both even again.*1971

*Divide this by 2; I know that 4 divided by 2 is 2.*1977

*50 divided by 2 is 25 because that is half of 50.*1983

*This is my answer.*1988

*Then to decimal; here I make it smaller.*1992

*Move it to the left two spaces; it is going to go one and two.*1996

*See how from right here, it went one; then it went two with an empty space.*2002

*It has to be 0.08 because you have to fill in the empty space with a 0.*2008

*8 percent; at the end, one, two, decimal point; fill in this space with 0.*2016

*The last one, from a fraction to percent and decimal.*2027

*In order for me to go from a fraction to a percent, I have to give the decimal point first.*2035

*Divide; 4 inside; 5 outside; point; bring it up; put a 0.*2041

*5 goes into 4 zero times; 5 goes into 40... 5 times 8 is 40.*2052

*If we subtract it, you get a remainder of 0; that means I am done.*2064

*My decimal is going to be 0.8 or 0.8; it is the same thing.*2069

*Then from decimal to percent, make it bigger.*2077

*0.8; that means I have to move it to the right two spaces.*2082

*I go one, two; empty space; put a 0; becomes 80 percent.*2086

*That is it for this lesson; thank you for watching Educator.com.*2101

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to be finding percents of a number.*0002

*When we are finding a percent of a number, that means we are finding a portion of a number, some part of a number.*0008

*Whenever we have a percent of a number, let's translate the sentence into an equation.*0016

*Of means times; whenever you see the word of in the sentence, it means times.*0024

*Of like this is times.*0034

*When you see the word what, what means the unknown.*0036

*What is what we are looking for; that is going to be the variable.*0042

*What is the variable.*0045

*Whenever you see the word is, is means equal.*0048

*Whenever you see the word is, you are going to write that symbol.*0059

*Whenever you are finding a percent of a number, you are always going to be*0067

*multiplying the percent with the number because of is times.*0071

*Here what is a variable X; you can use whatever variable you would like.*0080

*You can use A; you can use B; you can use Z; it doesn't matter.*0085

*X is; is means equals; 30 percent.*0089

*30 percent, whenever we solve with percents, we have to change it to a decimal.*0097

*Remember we can't solve with a percent number.*0104

*Remember to change percents into a decimal, you are going to start from the end of the number.*0108

*If you don't see the decimal point, then it always belongs at the end of a number.*0122

*You are going to move two spaces to the left.*0128

*You know it is going to be to the left when you change it to a decimal because decimals are small.*0132

*Think of decimals as smaller than percents.*0136

*To change from a percent to a decimal, we are going to move the decimal point*0140

*over to the left because that is going to make the number smaller.*0144

*Two spaces; it is going to go one, two.*0147

*That is where the decimal point is going to go.*0152

*Here it is going to be 0.30; this is going to change to 0.30.*0156

*Of always means times.*0164

*Here I want to show that I am going to multiply this number to this number.*0171

*Be careful because we don't want to use X for times anymore because here we have X as a variable.*0176

*Instead of using X, use parentheses.*0184

*You also probably know about the dot as times; sometimes that is OK.*0190

*But between two numbers, you don't want to use dot because that looks like a decimal also.*0195

*Maybe if you write it a little bit too low, it might look like a decimal.*0201

*Whenever you are going to multiply two numbers together, it is always best to just use parentheses.*0207

*Writing the number in parentheses like that between two numbers,*0215

*it shows that you are going to be multiplying those two.*0219

*What is 30 percent of 100?*0224

*You know that you are going to be multiplying 0.30 with 100 to find the missing value X.*0225

*100 times 0.30 is going to be 0; then 0; 0; I'm sorry.*0235

*If you want to just multiply this 0 out, you can do that.*0250

*Then multiply this through under; put a 0 here; 3 times 0; 0; 3.*0254

*3; 0; 0; 0; be careful when you have so many 0s.*0264

*Within the problem, because we multiplied, how many numbers are behind the decimal point?*0269

*There is two numbers; one, two; we have two numbers behind the decimal point.*0274

*You are going to go to the answer starting at the end.*0279

*You are going to go one, two.*0284

*That is where the decimal point is going to go; 30.*0286

*X equals... all this was 30; there is a shortcut.*0291

*Whenever you multiply a decimal number, to a multiple of 10... that is 10, 100, 1000, 10000.*0296

*Whenever you have a number multiplied to a number with 1*0306

*and then 0s like 100, there is a shortcut way of doing this.*0311

*You can count how many 0s there are in that number; 100 has two zeros.*0316

*Then you are going to move this decimal point two spaces.*0324

*You know, since you are going to be multiplying, when you multiply, the number tends to get bigger.*0331

*You are going to go two spaces to the right*0337

*because moving the decimal point over to the right makes the number bigger.*0340

*You are going to one, two; that is going to give you 30.*0344

*Remember if you don't see a decimal point, it is always at the end right there.*0348

*If you were multiplying this number by 10, let's say you are multiplying it by 10.*0356

*10 has only one 0.*0363

*You would only move the decimal point over once to the right.*0366

*That would be 3.0 which is the same thing as 3.*0371

*Remember if there is a 0 at the end of a number behind the decimal point, it is as if it is not there.*0374

*3.0 is the same thing as 3.*0379

*If you are multiplying by 1000, 1000 has three 0s.*0388

*You would have to move the decimal point over three spaces.*0394

*Remember if you have an empty space, you have to fill it in with an extra 0.*0398

*That is the shortcut when multiplying by a number that is a multiple of 10; 10, 100, 1000, and so on.*0404

*You just have to count the number of 0s and then move the decimal point over that many spaces.*0413

*What is 30 percent of 100?--that is 30; 30 is 30 percent of 100.*0420

*We are just going to do a few more examples.*0433

*Here 50 percent of 18 is... here 50 percent.*0435

*Again remember whenever we have to solve using percents,*0443

*we have to change it to a decimal because you can't solve with a percent.*0448

*If you have a percent, then you need to change it to a decimal.*0453

*Remember percent to decimal.*0458

*If we have a percent, 50 percent, you want to move the decimal point*0466

*over two spaces between percent to decimal or decimal to percent.*0471

*From percent to decimal, you start at the end right here.*0479

*You have to move two spaces to the left because remember decimals are small numbers.*0484

*You want to turn this number into a smaller number.*0488

*To do that, you have to move the decimal point over to the left.*0492

*If you move it to the right, then the number gets bigger.*0496

*So move it to the side that is going to make it smaller.*0499

*Two spaces, it is going to be right here.*0504

*One, two; drop the percent sign; becomes 0.50.*0506

*Or because this 0 is at the end of a number behind the decimal point, it is as if it is not there.*0512

*You don't even have to write it.*0519

*0.50 is the same thing as 0.5; it is the same thing.*0520

*This is 0.50 or 0.5; of means times.*0528

*Remember if I am going to multiply two numbers together, I want to write it in parentheses.*0535

*Times 18 is; is is equals; then the number X which is what we are solving for.*0538

*As long as you know that you have to multiply the percent with the number,*0554

*because of is between the two numbers, again 50 percent you change to a decimal.*0560

*As long as you know that you have to multiply these two, you don't have to write it into this equation*0565

*because all we are doing is finding percents of a number; of meaning times.*0573

*You just have to be able to multiply these two numbers together.*0579

*The reason why I am saying write an equation is because for the next lesson,*0582

*we are going to have to solve for maybe this number or solve for the percent.*0589

*This number will be given to you.*0598

*In that case, when you are solving for another number,*0600

*this is the easiest way for you to be able to write an equation*0603

*and know what your variable is, what it is you are solving for,*0608

*because a variable is always what you are solving for.*0611

*If you don't want to do it this way, then just make sure you remember to multiply.*0616

*Of means times; you are going to be multiplying the number with the other number.*0622

*0.5 times 18; I am going to write that here; 18 times 0.5.*0631

*8 times 5 is 40; 5 times 1 is 5; plus 4 is 9.*0640

*Within here, how many numbers are behind the decimal point?*0648

*I only have one; starting right here, you are going to go inwards one.*0650

*My answer is going to be 9.0 or 9.*0657

*9.0 and 9 is the same thing.*0663

*Again 0 is at the end of a number behind the decimal point so you can just drop it; 9.*0666

*50 percent of 18 is 9; 50 percent means half.*0673

*50 percent means half; what is half of 18?*0679

*Half of 18 is 9 meaning if you had let's say 18 pieces of candy.*0684

*You had to split it in half; you can only take half of them.*0690

*You take half; your brother or sister has to take the other half.*0695

*Then you would take 9; your brother or sister would take the other 9.*0699

*50 percent means half.*0704

*All you have to do is 50 percent of 18, you just figure out what half of 18 is.*0705

*The next one, here 8 percent; from percent to a decimal.*0712

*Again you are going to start right here.*0723

*You are going to move two spaces to the left.*0724

*You are going to go one and two.*0727

*Point... there is an empty space right here.*0731

*You have to fill it with a 0; 0; 8.*0735

*Then again you are going to drop the percent because it is no longer a percent.*0740

*0.08; 8 percent is 0.08; of means times; 6.*0743

*Again write it in parentheses when you are multiplying two numbers.*0752

*Is equals the number X or blank.*0759

*0.08 times 6; 0.08 times 6; 8 times 6 is 48; this is just 0.*0764

*How many numbers do I have behind the decimal point?*0785

*I only have two; there is none right here.*0789

*Decimal point is at the end right here; I only have two.*0791

*From the end, I am going to go in two spaces; one, two; right there.*0795

*8 percent of 6 is 0.48.*0803

*The next one, 99 percent of 100 is.*0815

*99 percent changes to a decimal because you are solving with it; 99 percent.*0819

*You are going to go one, two; it is going to be 0.99.*0826

*Of means times; 100 in parentheses; equals something; 0.99 times 100.*0836

*Let's do our shortcut; remember if you want, you can multiply it out like this.*0846

*Remember whenever you multiply a number that is a multiple of 10, meaning 10, 100.*0852

*Or I'm sorry; not a multiple of 100; but if you have a power of 100; 10, 100, 1000, and so on.*0859

*What that means is any number with a 1 with a bunch of 0s; 10, 100, 1000, 10000, and so on.*0866

*Since I have 100 and 100 has two 0s, I am going to take the decimal point.*0878

*I am going to move it two spaces.*0887

*But remember I have to move it to the right two spaces*0889

*because when you multiply by 100, that means your number has to get bigger.*0892

*To make this number bigger, I am going to move it to the right.*0897

*One, two; that is where my decimal place is going to go.*0900

*That is going to be 99.0.*0904

*Or remember if it is at the end of a number, you can just drop it.*0909

*You can make it invisible; 99; 99 percent of 100 is 99.*0914

*Whenever you find a percent of 100, it is always just that number.*0921

*If I have 1 percent of 100, that is going to be 1.*0925

*If I have 2 percent of 100, that is going to be 2.*0931

*If I have 100 percent of 100, that is going to be 100.*0935

*Whenever you take a percent of 100, it is just going to be this number without the percent sign.*0938

*68 percent of 100 is 68; that is only if this number is 100.*0945

*Let's just do a few more; what is 25 percent of 60?*0958

*This is what we are solving for; I can make that into my variable.*0966

*Use a question mark; you can do a little blank; is equals.*0971

*25 percent; percent to decimal; I am going to change 25 percent.*0980

*Start right here at the end; you are going to go in one, two.*0993

*It is going to be 0.25; of means times.*0997

*Again since you are multiplying two numbers together,*1007

*you are going to write it in parentheses like that.*1009

*To solve this, I have to... to find the what, to find this, I have to multiply 0.25 with 60.*1011

*60 times 0.25; this is 0; 6 times 5 is 30; put a 0 here.*1023

*2 times 0 is 0; 2 times 6 is 12.*1035

*Add them together; 0; 0; 5; 1.*1043

*How many numbers are behind decimal points?--I only have one, two.*1049

*You are going to start here; you are going to go one, two.*1056

*Remember 0s are at the end of a number behind the decimal point.*1060

*This is just going to be 15; X, this unknown, is 15.*1063

*That means 15 is 25 percent of 60.*1069

*What number is 10 percent of 10?*1078

*Again what number is the same thing as just what or blank or question mark.*1081

*You are looking for the number.*1087

*Is 10 percent; one, two; 0.10; of 10.*1089

*I know that 10 times 10 is 100.*1107

*You can just move the decimal point over that many times.*1111

*Or we can just use our shortcut.*1113

*Since I am multiplying a decimal by 10, how many 0s do I have in 10?*1116

*I only have one 0.*1121

*I am going to move this decimal point over one time.*1124

*It is going to be 1.0; but that is the same thing as 1.*1128

*You can just drop this 0.*1136

*If the decimal place is at the end of a number, then you can just make that invisible.*1139

*You can just write that as 1; that means 1 is 10 percent of 10.*1145

*Find 5 percent of 40; we want to find 5 percent of 40.*1156

*5 percent; this right here is the same thing as this.*1162

*We still have to change this; be careful.*1168

*Just because you don't see the percent sign doesn't mean that you can just drop this number down.*1171

*It is still a percent; you have to change this to a decimal.*1176

*5 percent; again start at the end; you are going to go one, two.*1189

*Going to be point, space right here, 0, 5; of is times; 40.*1196

*You are going to multiply these two numbers together; 0.05 times 40.*1212

*0 times 5 is 0; 0 times 0 is 0; put a 0 down there.*1220

*4 times 5 is 20; 4 times 0 is 0; add the 2.*1229

*Be 0, 0; remember we are adding them down; 200.*1239

*How many numbers total are behind decimal points?--I only have two; these two.*1245

*I am going to go from here; I am going to go one, two; in two spaces.*1251

*My answer is 2.00 which is remember the same thing as just 2.*1258

*My answer is going to be 2; 5 percent of 40 is 2.*1265

*100 percent of 18; 100 percent of 18.*1276

*When I say 100 percent, I am trying to say all of it.*1282

*This is like saying all of 18; 100 percent of 18 is all of 18.*1289

*What is all of 18?*1295

*All of 18, if you have 18 pieces of candy, what is all of it?*1297

*How many would be considered all of it?--18.*1301

*100 percent of 18 is just 18.*1304

*If you want to just solve it out, 100 percent into a decimal is going to be one, two, 1.0.*1308

*Or remember this is the same thing as 1.*1318

*Even though you don't see a decimal here, numbers always have a decimal point.*1323

*It is just if you don't see it, if it is invisible, it is always at the end.*1329

*100 percent is the same thing as 1 in decimal.*1334

*It is like saying 1, or 1.0 if you want, times 18.*1338

*What is 1 times 18?--isn't that 18?*1346

*Again shortcut, if you have 100 percent, you are saying what is all of it?*1352

*All of 18 is 18.*1357

*1 percent of 2000; into a decimal.*1361

*Right here, you are going to go one, two; the decimal point.*1372

*Empty space; you are going to fill it in with a 0; and 1.*1378

*1 percent in decimal is 0.01; be careful that you don't just make it 0.1.*1383

*Of, times; 2000; multiply it out; times 0.01; this is 0; 0; 0; 2.*1392

*Then this is just all 0s so I don't have to write that in.*1414

*It is like adding 0s.*1418

*If I add 0s to this number, it is just going to be that same number.*1419

*From here, how many numbers are behind decimal points?--two.*1426

*Start from here; you are going to go one, two, decimal point.*1430

*My answer here is going to be 20.00.*1435

*Again if the 0s are behind the decimal point at the end of a number, you can just drop them.*1446

*I can drop this, drop this; it will be 20 point.*1450

*But then I can just drop that, make that decimal point invisible if it is at the end.*1454

*It will just be 20.*1458

*The final example, a bag of candy contains 40 pieces.*1467

*If Susanna ate 20 percent of the candy or everything in the bag of candy, how many pieces did she eat?*1475

*A whole bag contains 40 pieces; she ate 20 percent of it.*1487

*How many pieces did she eat?--look at this; 20 percent of the candy.*1491

*What is the candy?--how many pieces does the bag of candy contain?--40 pieces.*1504

*It is like saying she ate 20 percent of the 40 pieces.*1510

*20 percent, again if I want to solve with this number, I have to change it to a decimal.*1516

*20 percent to decimal; I am going to start here; go one, two; is 0.20.*1523

*Of is times; you are going to multiply what?*1537

*The 40 because she ate 20 percent of all the pieces; times 40.*1541

*Here thing is going to be...*1554

*Remember if the 0 is at the end of a number and it is behind the decimal point, I can just drop it.*1557

*This is the same thing as 0.2 times 40.*1562

*I can't drop this 0; be careful here.*1566

*Don't drop the 0 because this 0 is not behind the decimal point.*1569

*Decimal point is right here; right there.*1574

*Since it is not behind the decimal point, I can't drop the 0.*1579

*But this one, I can; that will just make it easier to multiply.*1583

*40 times 0.2 is going to be 80.*1587

*I only have one number behind the decimal point.*1595

*Start here; you are going to go in one space.*1598

*0.2 times 40 is going to be 8.*1601

*Good thing with word problems is that you can estimate if your answer sounds correct.*1607

*Let's say I forget to count how many numbers I have behind decimal points and I just leave it as 80.*1613

*80 can't be my answer because if the bag of candy contains only 40 pieces, Susanna ate 20 percent of it.*1620

*Would it make sense if my answer was 80, that she ate 80 pieces?*1631

*There is only 40 pieces; I know this answer sounds correct.*1637

*It seems correct; it seems reasonable; remember 50 percent is half.*1646

*If Susanna ate 100 percent of the candy, that means she would have eaten all of the candy.*1653

*Then my answer would just be 40.*1658

*If Susanna ate 50 percent of the candy, remember 50 percent is half of the number.*1661

*If this said 50 percent, then you would have to just find half of 40.*1671

*She would have eaten 20 pieces of candy.*1676

*She only ate 20 percent; that means she has to have eaten less than half.*1680

*If 20 is half, we know 8 is reasonable then because it has to be less than half.*1685

*That is it for this lesson; thank you for watching Educator.com.*1695

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over solving percent problems.*0002

*To solve a problem that involves percents, we want to first translate whatever the sentence is into an equation.*0008

*Whenever you have a number, you are going to write that down in your equation.*0018

*If you have a percent, you need to change it to a decimal.*0022

*You see the word of; that means times; you are going to be multiplying.*0027

*When you see the word what or what number, that means you are going to have a variable.*0032

*That is what you are going to be looking for.*0037

*That is what you are going to be solving for.*0039

*This is almost the same as what we just did the last lesson.*0041

*But now we are going to be looking at variables and solving equations.*0047

*We are going to have to do a little bit deeper into these problems.*0053

*The first set of examples; for example one, 15.*0060

*Remember if we have a number, we are just going to write it straight down into our equation.*0068

*Is; the word is remember means equals; 25 percent; this is write down.*0074

*This we are also going to write straight down.*0083

*But because it is a percent, in our equation, we need to make it into a decimal.*0085

*Percent to decimal, remember we have to move two decimal spaces to the left*0091

*because think of percent as a bigger version of a decimal; decimals are small numbers.*0105

*Whenever you are converting from a percent to a decimal, you have to get smaller.*0112

*The way for you to get smaller is to move the decimal point over to the left.*0118

*The decimal point is over on this side.*0127

*If you don't see a decimal point, it is always at the end of the number.*0130

*We are going to move it one, two spaces.*0134

*The decimal point of 25 percent is going to be 0.25.*0139

*Of course obviously we have to drop the percent sign.*0145

*25 percent is 0.25; that is what I am going to write in my equation.*0148

*0.25; of means times; times what number; this is my variable X.*0154

*Be careful here because whenever you write a dot for times, that looks like the decimal point.*0168

*The best way to represent multiplying is to either write it in parentheses.*0174

*If you are going to show that you are going to be multiplying two numbers, write them in parentheses.*0179

*Or if it is a number with a variable, a letter, then you can just put them together.*0184

*0.25X, that would mean 0.25 times X.*0191

*Here is our equation now; this is what I am solving for.*0198

*15 is 25 percent of what number?--this is the number that I am looking for.*0202

*When I solve for X, remember that since this is 0.25 times X, I need to do the inverse operation.*0207

*0.25 times X, the inverse operation is divide.*0219

*In order to solve for X, I have to divide 0.25.*0224

*0.25 over itself is going to be 1.*0232

*Whatever I do to one side, I have to do to the other side of the equal sign.*0236

*I have to divide this side also by 0.25; this, it looks like a fraction.*0241

*But fractions are division problems; this is the same thing as 15 divided by 0.25.*0247

*Another way to explain why we have to divide, let's say I have 8 equals 4X.*0255

*If I have 4 times a number equals 8, what do I know about X?*0268

*Isn't 4 times 2, 8?--so I know X has to be 2.*0275

*In the same way, I have to solve for X and I can just divide this number by this number.*0285

*Divide this by 4; divide this by 4; 8 divided by 4 is 2.*0292

*That gives me 2 equals X.*0298

*Here 15 divided by 0.25, I need to actually solve that out in order to find X.*0305

*Let's review over how to divide numbers with decimals.*0313

*If I am going to divide these two numbers, remember that the top number is what goes inside when you divide.*0321

*15 on the inside; 0.25 on the outside.*0330

*The decimal point for this number is at the end because we don't see one.*0337

*It is always at the end.*0342

*If I need to add 0s to this number, I can because*0349

*I can always add 0s to the end of a number behind a decimal.*0352

*If it was before the decimal, I can't because then that will just become 150 instead of 15.*0358

*As long as it is behind the decimal and it is at the end of a number,*0364

*you can add as many 0s as you want.*0370

*I can add two 0s; I can add three; I can add ten; however many I need.*0372

*This number, we want to change to a whole number.*0379

*I need to move the decimal point over two spaces to the right to make the decimal point at the end.*0384

*If I move two decimal places for this number, then I have to move this two decimal places over here.*0392

*Then I am going to take that decimal point up.*0399

*25 goes into 15 zero times; 25 goes into 150 how many times?*0403

*Think about quarters; 25 cents or 25 is like a quarter.*0412

*How many of those fit into 100 or a dollar?--four.*0421

*Four quarters is a dollar; think of 150 cents.*0425

*25 cents goes into 150 cents or 1 dollar 50, how many of them?*0428

*That would be six.*0434

*If you want to just check that, this is 12, 13, 14, 15.*0437

*That becomes 0 when you subtract it; I can bring down this 0.*0449

*25 goes into 0 zero times; I have to fill in this space right here.*0454

*That is 0; subtract it; that is nothing.*0461

*I don't have to bring down another 0 because my remainder is 0.*0464

*My answer then, if I do 15 divided by 0.25, is this number up here, 60.*0471

*This number, if there is a 0 in front of the number like that, then that doesn't mean anything either.*0479

*I can just drop the 0; that would just be 60.*0485

*This 0 I cannot drop.*0489

*This 0 has to stay there because if I drop it, my number is going to change to 6.*0491

*We know that 6 is not the same as 60.*0498

*This 0, it is not after the decimal point so we can't drop that 0.*0501

*My answer just becomes 60.*0507

*If you need to review over this, you can either go back to that lesson, dividing decimals.*0513

*Or we are going to do a few more problems that involve dividing decimals.*0520

*The next one, again we are going to change this into an equation.*0527

*1; is is equals; 4 percent.*0535

*4 percent, again we have to change it to a decimal.*0540

*Be careful; 4 percent is not 0.4.*0543

*Again 4 percent to decimal; the decimal point is at the end right here.*0549

*I go one, two; then put the decimal point there.*0556

*I have an empty space that I have to fill.*0561

*I have to fill that with a 0; it is going to be 0.04.*0563

*Again at the end here, one, two, decimal point; 0.04.*0568

*Of means times; blank, that is what we are looking for; that is my variable.*0580

*I am going to put X there.*0589

*Remember if I put number with variable, that means times.*0591

*This represents... I don't have to put a dot here.*0595

*I don't have to use parentheses when it is number with variable.*0597

*Again how do I solve for X?--look at this example again.*0603

*If we are going to do 8 equals 4 times a number, I can take this number, divide it by this number.*0607

*8/4; that is going to give me X.*0614

*Then I have to do this number divided by this number.*0617

*Remember that this over this becomes 1.*0626

*This whole side, my right side, just becomes 1X.*0630

*1X is the same thing as X.*0635

*That is probably a little bit hard to understand, 1X being the same as X.*0639

*But it is like me saying I have an apple.*0645

*If I say I have an apple, you know I only have 1 apple.*0650

*Even though I didn't say I have 1 apple, you just know because how many A's do you see?*0655

*You see one; an A is the same thing as 1A.*0661

*An apple is the same thing as 1 apple.*0667

*Just think of that as having 1X; again we have to divide that; 1.*0672

*Be careful, the top number is going to go inside.*0683

*0.04, the bottom number, is going to go on the outside.*0689

*Again I have to move this decimal point over one, two spaces to the right.*0693

*That means I have to take this decimal point; it is at the end.*0697

*Go one, two spaces; I have to fill these in with 0s.*0702

*There is my new spot for my decimal point; I bring it up.*0709

*4 goes into 10 how many times?--4 times 2 is 8.*0717

*4 times 3 is 12; 12 is too big; it only fits into 10 twice.*0724

*2 times 4 is 8; subtract it; I get 2.*0732

*I am going to bring down this 0.*0738

*4 goes into 20 how many times?--five times.*0741

*4 times 5 is 20; subtract it; I get a remainder of 0.*0746

*I can stop there; my answer becomes 25.*0753

*I don't have to put that decimal because it is a whole number and it is at the end.*0758

*My answer X is 25; right here, this is 25.*0764

*Again 1X is the same thing as X; what did this become?*0773

*This became 25; if 25 is X, then I can just say that X is 25.*0778

*It is the same exact thing.*0786

*The next one, 20 equals 100 percent; to decimal.*0792

*Again start at the end; you are going to go one, two; right there.*0806

*It is going to be after the 1; 1.0.*0810

*Remember if the 0s are at the end of a number behind the decimal point, then I can just drop it.*0814

*Isn't this the same thing as 1?--I can just write 1.*0819

*100 percent as a decimal is 1; times; of means times.*0824

*What number, that is my variable; 1 times X; 20 equals 1X.*0832

*Remember 1X is the same thing as X because if I have 1 apple,*0841

*that is the same thing as just saying I have an apple.*0846

*If you want, you can go ahead and divide the 1 just like we did the other problems.*0850

*20 divided by 1 is 20.*0856

*Whenever you have a number divided by 1, it is always itself.*0860

*20 equals X; or I can flip this around.*0864

*If 20 equals X, then isn't that the same thing as X being 20?*0870

*Either way, that is correct; we just want to know what the number is.*0877

*The number is 20; or you can say 20 is the number.*0881

*It is the same exact thing.*0886

*Let's do a few more examples; these are a little bit different.*0889

*What percent of 50 is 10?*0895

*Now the variable, what we are solving for, is a percent.*0900

*Be careful here; what percent, I am going to make that X.*0907

*Times, times; 50, 50; is, equals; 10, 10; X times 50 equals 10.*0913

*Remember you can change this if you want to 50X just like we did the other problems*0929

*because a number times a variable, you just put it together with the number in front.*0935

*50X equals 10; it is the same thing.*0939

*How do we solve for X then?--how do we get what X is?*0944

*Remember my example?--let's say I have 3 times X equals 6.*0949

*You can do this in your head and know that 3 times 2 equals 6.*0959

*Another way for you to solve it is to do 6 divided by this number; this divided by this.*0964

*Same thing; we can just do 10 divided by 50.*0972

*It is not 50 divided by 10.*0975

*It is this number divided by this number, the one that is multiplied to the variable.*0978

*I can show you this way; 50/50, that is 1.*0985

*10/50, that is what you have to do; 10 divided by 50.*0993

*Again fractions are the same thing as division; 10 divided by 50.*0997

*A shortcut way of doing this is if you are dividing two numbers*1009

*with 0s at the end of it, you can just cross out the 0.*1014

*If there is one 0 up here and one 0 down here, you can just cross out*1020

*one 0 from each of the numbers as long as there is 0s in both numbers.*1022

*But we are just going to go ahead and just divide it this way.*1027

*50 divided by 10; it is not going to go into this number.*1032

*This number is too big to go into this number.*1037

*I am going to have to use my decimal point.*1040

*Do I move it at all?--no, because there is no decimal point here.*1042

*I can just bring it up, bring it straight up.*1047

*I can add 0s at the end behind the decimal at the end of a number.*1050

*Now I can just look at this, 1-0-0, 100.*1057

*50 goes into 100... 50 plus 50 is 100; or 50 times 2 is 100.*1061

*Think of 50 cents; 50 cents goes into 100 cents how many times?*1072

*100 cents is the same thing as a dollar.*1080

*50 cents goes into a dollar twice; this becomes that.*1082

*Subtract it; you get 0; no remainder; that is my answer.*1087

*I don't have to bring down anymore 0s because my remainder became 0.*1092

*When I divided this, my answer became 0.2; X equals 0.2.*1099

*Here is the thing though; they are asking for percent.*1114

*Even though this is my answer, this is my answer as a decimal.*1119

*They want it in percent; they are asking what percent.*1124

*They are not asking what decimal; what percent?*1127

*I have to change this number to a percent; decimal to percent.*1129

*Remember decimal is a small number; percents are larger.*1141

*I have to go from a small to a larger.*1146

*That means I have to move the decimal point over two spaces; but which way?*1148

*If I go to the left, I am going to get smaller.*1154

*But if I go to the right, then I start getting whole numbers.*1158

*I make the number bigger.*1162

*0.2, to make it into a percent, I need to make it bigger.*1165

*I need to go to the right; one, two.*1169

*I have to fill this space with something.*1173

*0.2 as a percent will be 2-0 and then percent.*1177

*The decimal point is right here; it is at the end.*1185

*If it is at the end, remember you can just... it doesn't have to be there.*1187

*You can make it invisible.*1190

*Then we have to write the percent sign because we changed it to a percent.*1193

*My answer is then 20 percent; that is my answer.*1196

*20 percent of 50 is 10.*1204

*Another one; again what percent, make that X, your variable.*1208

*Times 75; is is equals; 7.5.*1214

*Again we have to do this number divided by this number; 7.5 divided by 75.*1224

*Again if you want to see it, I can show you this way.*1235

*because this you have to get rid of by dividing it.*1239

*This 75/75 is 1; X times 1 is just X.*1244

*X equals; then I have to actually divide that to find the answer.*1251

*Here I don't have to move the decimal point anywhere because it is at the end.*1258

*This decimal point will just come straight up.*1263

*75 goes into 75 how many times?--once.*1267

*1 times 75 is 75; subtract it; I get 0.*1274

*I don't have to go any further; 0.1 is my answer; X equals 0.1.*1280

*But again remember it is asking for percent.*1285

*Be careful that you don't forget to change it to a percent.*1288

*I am going to put that here to represent decimal.*1294

*To change it percent, I am going to go one, two, point.*1301

*1; fill this space with a 0; put the percent sign.*1306

*0.1 in decimal becomes 10 percent; X equals 10 percent; there is my answer.*1314

*The last one for this; again what percent X times... of is times... 4 equals 4?*1325

*This one we can just do in our head; what times 4 equals 4?*1340

*Isn't this 1?--1 times 4 equals 4; 4 times 1, it equals itself.*1345

*I don't even have to solve this; I can just make X equal to 1.*1352

*If the problem is fairly easy, you can just do it in your head, then go ahead.*1356

*There is no need for you to do all the work unless your teacher wants you to show the work.*1360

*Then X equals 1; since X equals 1... I didn't mean to box this.*1367

*That is not my final answer so I don't want to box it.*1381

*Since X equals 1, I need to change it to percent.*1385

*How do you change a 1, a whole number, into a percent?*1391

*Again where is the decimal point?--I don't see it.*1395

*If I don't see it, it is invisible, it is at the end like that.*1399

*Go one, two, point here; I have two spaces to fill.*1404

*This becomes 1-0-0 percent; this X equals 100 percent.*1411

*The next example, we are just going to do a few more, just different types though.*1428

*The other examples, they were the same kind, all the problems on that page.*1433

*15 equals what percent, X, times 150; let me rewrite this equation out.*1439

*Since this is 150 times X, let me just write it here.*1452

*15 equals... remember whenever I do a variable times a number,*1458

*I want to write it together but with the number in front; 150X, like that.*1463

*Then to solve for X, remember I have to do this number divided by this number.*1471

*I am going to divide this side by 150.*1478

*Whatever I do to one side, I have to do to the other side.*1480

*That way this becomes 1X or 1 times X; 1X.*1486

*That is the same thing as X.*1492

*15 divided by 150; no decimal point here; I don't have to move it.*1495

*Instead I need to draw that in; bring it up; add 0s.*1509

*150, we know it doesn't go into 15.*1518

*If you want, you can put a 0 up here; if not, then that is fine.*1523

*Just remember that it is now the next three or just these three.*1525

*150 goes into 150 one time; that becomes 150.*1530

*If you subtract it, it becomes 0; I drew an extra 0 there.*1536

*But you don't even have to bring it down because it is just going to be 0s.*1542

*Remember 0s at the end of a number behind the decimal point means nothing.*1545

*0.1 is my answer; X is going to equal 0.1.*1551

*You can also say 0.1 is going to equal 1X or X; same thing.*1558

*But I can switch it like this; it is asking for percent.*1564

*I need to take this decimal point and go one, two; fill in this space.*1572

*It is going to be 1-0 percent; 0.1 is the same thing as 10 percent.*1579

*The next one, 30 percent, this is written out as percent.*1592

*But it still means the same thing as percent like that.*1599

*30 percent, I have to change that to a decimal.*1602

*30 percent becomes... at the end, you go one, two, right there.*1606

*0.30 or 0.3 because remember it is 0s at the end behind the decimal.*1613

*It doesn't mean anything.*1619

*0.3 or 0.30; of, times; 12; it is number times number.*1621

*I can't write it next to each other like how I do numbers with letters.*1629

*I have to put it in parentheses.*1633

*Is is equals; what number, this is my variable.*1637

*All I have to do to figure out what X is is multiply those two numbers.*1645

*0.30, you know what?--we know that the 0 means nothing.*1654

*Let's just make it easier and just not even have that 0.*1660

*One digit number is easier to multiply.*1665

*I am going to put the 12 on the top; 0.3 on the bottom.*1669

*Multiply it; 2 times 3 is 6; 3 times 1 is 3.*1673

*From here, I only have one number behind the decimal point.*1680

*Start at the end; go place the decimal in there.*1685

*What I just did, when you multiply decimals, you have to count...*1692

*This decimal point is at the end, right here.*1695

*You count to see how many numbers are behind the decimal point.*1698

*Here I only have one; then I start at the end.*1702

*I only go one inwards; that is where the decimal place goes.*1705

*X is 3.6; to finish this equation, 3.6 equals X.*1711

*3.6 is the number; or I can say the number is 3.6.*1723

*The next one, 5 percent, change that to a decimal.*1734

*Start here; go one, two; it is point... fill in that space.*1741

*It is 0.05; it is not 0.5; 0.05.*1745

*0.05 times my unknown which is X; X equals 4.*1752

*Again I have to solve for X which means I have to divide.*1764

*4 divided by 0.05; 4 divided by 0.05.*1768

*I have a decimal point here; I have to move it one, two.*1779

*There is my decimal point; I am going to move it one, two.*1784

*Bring it up; fill in these spaces with 0s; 5 goes into 4 zero times.*1788

*5 goes into 40... 5 times we know 8 is 40.*1797

*Write 40 down here; subtract it; 0.*1804

*I am not finished with the number yet because I still have space up here.*1808

*Bring down the 0; 5 goes into 0 zero times.*1814

*That is why for this, I have to keep going.*1818

*Even though this became a 0, I have to bring down the 0 and solve it again*1820

*because there was an empty space before my decimal point.*1826

*In that case, you have to continue.*1831

*If it is after the 0 like in this problem right here... I'm sorry.*1833

*If it is after the decimal point, then I can stop once I get 0 as my remainder.*1838

*But for this, if there is a space here before the decimal point,*1844

*then I have to go again until I fill in those spaces.*1849

*Here this is 80; X is 80; they are not asking for percent.*1852

*5 percent of 80 is 4.*1861

*The last one, 100 percent of 3448 is what number?*1866

*They are asking for 100 percent of this number.*1874

*100 percent is all of it, is the whole thing.*1877

*100 percent of this number is just this number.*1883

*If you want to solve it out how we solved out the rest of them,*1888

*100 percent as a decimal, again move the decimal point over one, two spaces.*1891

*That becomes 1 or 1.0 which is the same thing as 1.*1898

*Times; times 3448; I am going to change this to parentheses.*1905

*3448 equals what number?--X; 1 times this number is just that number.*1912

*I can say 3448 is the number or the number is 3448.*1926

*That is it for this lesson; thank you for watching Educator.com.*1948

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over simple interest.*0002

*Simple interest has to do with savings account.*0010

*When you take money and you want to save it,*0015

*you take it to the bank and you put it into a savings account.*0018

*That money that you deposit, that you put into the bank, is the principal.*0023

*The initial amount, the money that you have, that you give to the bank, is called the principal.*0026

*Deposit just means you putting in.*0037

*You are going to put it into the savings account.*0039

*Because you are giving it to the bank, the bank uses that money.*0042

*They try to make more money.*0049

*Since they are using your money, it is like they are borrowing your money.*0052

*They end up paying you what is called interest.*0055

*Interest is the money the bank pays you based on the interest rate.*0061

*Again you take your money.*0066

*You put it in the bank, in a savings account to try to save the money.*0067

*The bank pays you money just for putting your money into their bank.*0072

*That is interest; interest is also money.*0078

*It is the money earned; the money the bank paid you; the amount the bank paid you.*0082

*Interest rate is the percent.*0090

*They are going to give you a percent of that money.*0093

*This is in percent; that is the interest rate.*0097

*Simple interest is a type of interest; it is only based on the principal.*0104

*Only based on the initial deposit, the amount that you first deposit.*0113

*When they calculate the interest only based on that money, that is called simple interest.*0119

*There is going to be different types of interest.*0126

*We are only going to go over simple interest.*0129

*Simple again is when the bank pays you interest based on just the principal amount.*0132

*The formula for the simple interest is the principal, how much you deposit,*0144

*times the rate, the percent the bank is going to pay you,*0154

*and T for time, number of years that they are going to pay you.*0162

*PRT means P times R times T; the principal times the rate times the time.*0167

*You are going to multiply those three together.*0176

*That is going to give you the amount that you earned in interest, how much you made from the bank.*0178

*Interest again is in dollars because if you are making that money, then it is in dollars.*0188

*The principal; again it is money; dollars.*0193

*The rate; the rate is the percent; and T for time in years.*0197

*If it is 2 years, then T is going to be 2.*0208

*5 years, T is 5.*0211

*Since the bank is paying you, the bank is paying you a percent, you would want a high percent.*0215

*The higher the percent, the greater your interest, how much you are going to make.*0222

*Just keep all that in mind; I equals PRT; P times R times T.*0228

*Let's go through our examples; find the simple interest.*0235

*P, the principal, is 200 dollars.*0240

*The formula again is I, the interest, equals P times R times T.*0242

*I, amount that you make, is in dollars; P is in dollars.*0252

*Rate is in percent; T is in years.*0257

*Principal, the amount that you deposit, is 200 dollars.*0264

*I want to find the interest; I is what I am looking for.*0267

*Equals; P which is 200; R... remember since R is in percent,*0270

*anytime you use a percent to solve something out, you have to change it to decimal.*0282

*We can't solve out any numbers that is in percent form.*0287

*We have to change it to decimal form; 10 percent; remember percent to decimal.*0291

*Think of it as the number has to get smaller because decimals are small.*0303

*10 percent, I have to move the decimal point two spaces over to the left*0307

*because that is going to make the number smaller.*0313

*From here, I am going to go one, two; remember the decimal point.*0315

*If you don't see one, it is always at the end of the number.*0318

*You are going to go one space, two spaces.*0322

*It is going to be 0.10 or 0.1.*0325

*Remember when the 0 is at the end of a number behind the decimal point, then it is nothing.*0330

*You can just drop; it will be 0.1 or 0.10; it is the same thing.*0335

*I can write 0.10 times T; how many years?--12 years.*0340

*I wrote these numbers in parentheses; that means multiply.*0352

*If you have a bunch of parentheses together like that, that means multiply.*0358

*The reason why I don't use the X symbol for times, since we are using variable,*0362

*you don't want to use that little X to represent times because X can be a variable.*0370

*Now you want to just it either in parentheses...*0378

*Actually that is the only way you should do it if you are multiplying numbers together.*0382

*How do I find the interest?--I have to just multiply these numbers together.*0389

*200 times 0.10; 200 times 0.1; remember 0.1.*0393

*I just want to make the number smaller.*0401

*Again since 0 is at the end of a number behind the decimal point, I can just drop it.*0404

*0.1; this is 0; 0; 2.*0409

*How many numbers do I have behind the decimal points?*0416

*One; I am going to start at the end here.*0419

*I am going to go in one space; it will be 20.0.*0421

*Again this 0 is at the end of a number behind the decimal point.*0427

*I can just drop it; this is actually the same thing as 20.*0431

*200 times 0.10 was 20; I have to now multiply the 12; 20 times 12.*0435

*0 times 2 is 0; 2 times 2 is 4; put a 0 right here.*0444

*1 times 0 is 0; 1 times 2 is 2; need to add.*0451

*0 plus 2 is 2; 4 plus 0 is 4; 0 plus 0 is 0.*0458

*The interest is going to be 240 dollars.*0467

*Over 12 years, you are going to be making this much money.*0481

*Find the simple interest earned over 5 years*0491

*when the principal is 500 dollars and the interest rate is 5 percent.*0494

*The time, we know that this is time because it is saying over 5 years.*0502

*T equals 5 years.*0506

*It makes it easier if you are just going to write down what each variable is.*0510

*Principal is 500; P is 500 dollars.*0514

*The interest rate is 5 percent; it is not I.*0524

*I is the amount that you earned or amount that you have.*0530

*Rate, the interest rate, is the percent; look for this number, rate.*0535

*R equals 5 percent.*0542

*Simple interest equals the principal, PRT, 500.*0550

*Times R which is again 5 percent; change it to a decimal.*0562

*It is going to go from here one, two; 0.05.*0578

*Then time; how many years?--5 years.*0587

*500 dollars times the rate 0.05; for 5 years; you just multiply those three out.*0593

*500 times 0.05; 0 times 5, 0; 0; 25.*0601

*Here 0 times all these, it just all becomes 0s.*0617

*If you want, you can just write them in; it is not going to change.*0621

*It is going to be 2; 5; 0; 0.*0626

*From these two numbers, how many numbers do I have behind the decimal point?*0631

*I have two; I am going to go to this side.*0635

*I am going to go one, two; this is 25.*0638

*0s are at the end of a number behind the decimal point.*0643

*500 times 0.05 is 25; that is actually 25 dollars.*0646

*This is actually saying the principal, how much you deposit, how much you put in,*0653

*times the interest rate, this is how much they are going to pay you, 25 dollars per year*0659

*because when you multiply that, that just becomes 1 year.*0665

*But then since you have 5 years, you are going to take the 25 dollars.*0669

*You are going to multiply it by 5 because they are going to pay you for 5 years.*0675

*5 times 5 is 25; 5 times 2 is 10; plus 2 is 12.*0680

*No decimal points or numbers behind decimal points; 125 is my interest earned.*0690

*I, the amount that I make, is 125 dollars.*0698

*This is how much the bank is paying me.*0707

*For putting 500 dollars into the account for 5 years with 5 percent interest.*0709

*This is how much I make in those 5 years.*0715

*The next example, Samantha deposited 100 dollars into a savings account with an interest rate of 2 percent.*0722

*Find how much simple interest she earned over 8 years.*0732

*She took 100 dollars into the bank; that is 100.*0739

*The interest rate is 2 percent; the rate R.... it is not I even though it is interest rate.*0747

*It starts with an I, but it is the rate; it is 2 percent.*0757

*How many years?--the time is 8 years.*0764

*That means she put in 100 dollars into the savings account for 8 whole years.*0771

*She left it in there for 8 years.*0775

*That bank had to pay her for all 8 years.*0776

*Again I am solving for I; equals the principal, 100, times the rate, 2 percent.*0782

*Change it to decimal; start at the end; you are going to go one, two.*0792

*Be careful, 2 percent is not 0.2.*0800

*It is 0.02 because you have to fill in that space; 0.02 times 8 years.*0802

*If I multiply just the principal times the rate,*0815

*that is going to give me how much I am going to make in 1 year.*0818

*That is why I have to multiply it by 8 because they have to pay Samantha interest for all 8 years.*0822

*It is times 8.*0831

*100 times 0.02; 0; 0; 2; again 0 multiplying by all that is nothing.*0833

*It just becomes that; if you want, you can draw in all your 0s.*0845

*You add it; 200; see it is the same thing.*0851

*Whenever you have a 0 that you have to multiply to all the numbers, it is just nothing.*0854

*It is just 0s; it doesn't change anything.*0858

*How many numbers do I have behind decimal points?*0863

*I have two; I am going to start here and go one, two.*0866

*It is 2 dollars; 2.00.*0871

*These 0s you just drop because it is at the end of number and it is behind the decimal point.*0875

*There is a shortcut you can do.*0881

*Whenever you are multiplying by 100 or 10 or 1000, any multiple of 10,*0882

*10, 100, 1000, 10000, 100000, 1 with a bunch of 0s,*0891

*you can move the decimal point however many 0s there are.*0897

*Since there is two 0s here for 100, you can move this to make it bigger*0904

*because remember when you multiply, you tend to make the numbers bigger.*0910

*Then you just move this over one, two spaces.*0913

*Let's say you are multiplying this number by 10.*0918

*10 only has one 0; you would move the decimal point over once.*0921

*If you are multiplying it by 1000, you have three 0s.*0926

*You would have to move the decimal point over three times.*0929

*0; then you fill in that extra space with a 0; that is a shortcut.*0932

*This is going to be 2; this was 2; times the 8.*0938

*2 times 8 we know is 16; I equals 16.*0949

*Put the dollar sign in there; interest is always in money, how much you made.*0957

*That means the bank, by Samantha putting 100 dollars into the savings account at the bank,*0963

*and then paying her 2 percent of that 100 for 8 years,*0970

*she is going to end up making a total of 16 dollars.*0978

*Let's say we want to find out how much she has overall, she has total.*0987

*She left the money into the bank; the bank has her 100 dollars.*0993

*The bank also paid her 16 dollars.*0999

*How much is she going to have in all?*1001

*She is going to have that 100 of her money that she put in the bank*1004

*and that 16 dollars that the bank paid her.*1009

*In all, to find the total amount that she has, you can just do*1013

*principal, how much she deposited, plus the amount that she earned.*1021

*That is 100 dollars plus the 16 dollars.*1029

*How much is she going to have in all?--116.*1034

*That is just to see how much she has in all, how much total.*1042

*Amount that she deposited plus the amount that she made from the bank.*1046

*That will be her total.*1051

*It has to be greater than the principal amount if it is the savings account*1052

*because she made money so then her remaining balance has to be greater than how much she put in.*1057

*Let's go over one more example.*1067

*If the simple interest earned in 4 years is 10 dollars*1070

*and the interest rate is 3 percent, how much is the principal?*1076

*Look at what they are asking for.*1084

*They are asking how much is the principal?--they are asking for the P.*1085

*The simple interest earned in 4 years is 10 dollars.*1092

*Simple interest... that is I... is how much?--10 dollars.*1096

*The rate is 3 percent; how many years?--4 years.*1105

*This seems really difficult because you are used to solving for I.*1121

*The formula, it has you solving for I; I equals PRT.*1126

*They are asking for P.*1134

*They give you the I; they are asking you for the P.*1135

*Whenever they do this, it is OK.*1139

*Just all you have to do is follow the formula.*1140

*Just plug in the numbers according to where it is in the formula.*1145

*I, we have I; we know what I is; I is 10.*1152

*Write 10 instead of I; equals; P is what we are looking for.*1156

*Leave in P because we don't know what P is.*1164

*R; R is 3 percent; we can replace R; 3 percent becomes... one, two, 0.03.*1168

*T, time, is 4; this looks pretty difficult, right?*1185

*I know that I have this and this that I have to multiply because this is times.*1197

*Parentheses means times; P times this times this.*1201

*I can't multiply P times this number because it is a variable.*1205

*But I can multiply this and this together.*1209

*4 times... you know what, let me just do it the other way.*1214

*0.03 times the 4; 3 times 4 is 12; 0; plus 1 is 1.*1222

*How many numbers are there behind decimal points?--two.*1231

*You start here; you go one, two; there is my number, 0.12.*1235

*It is as if this whole thing right here, when I multiply these two numbers together, it gave me 0.12.*1241

*Let me just write it again and write 0.12 instead of that number.*1248

*It is still P times this number times this number.*1255

*But then because I can solve these two out, it is just multiplied together.*1259

*I can solve it out; that is 0.12; then how do I solve for P?*1263

*This is also 0.12 times P; how do I solve for P?--remember my example?*1269

*If I have 6 equal to P times 3, I know that P is 2 because 2 times 3 is 6.*1277

*I can also say that 6 divided by 3 is P.*1291

*I can take this number and divide it by this number.*1296

*I can take 10 and divide it by 0.12.*1299

*If I take 10 divided by 0.12, I can solve for P.*1304

*I can figure out what my P is.*1309

*To divide decimals, if you have a decimal on the outside,*1312

*you have numbers behind it besides 0, then you need to go one, two.*1317

*You moved it two spaces because we have to get rid of the decimal point.*1322

*Decimal point here is at the end; I have to go one, two.*1327

*Decimal point is right there; fill these in with 0s; bring it straight up.*1332

*12 goes into 10 zero times; 12 goes into 100... let's see.*1340

*I am going to try to say 8; let's do it over here.*1350

*Let's see; 12 times 8 is 16; 8 times 1 is 8; plus 1 is 9.*1358

*You can just guess and check; you can try guessing 5.*1366

*You can try guessing 10; then see what the best number would be.*1369

*12 times 8, it is over this 0, is 96; subtract it.*1377

*100 minus 96 is going to be 4; bring down the 0.*1385

*12 goes into 40 how many times?--12 times 3 is 6...36.*1392

*12 times 4 is 48; this one is too big.*1403

*Then I know it has to be the 3; plug in the 3 in here.*1410

*That is 36; subtract it; that is 4 if I subtract it.*1416

*I can bring down a 0; I can divide it again; that is also 3.*1425

*36; 4; look it is a repeating number; 0; 3; here I can stop.*1439

*I know I am going to probably keep getting the same number 3.*1455

*It is a repeating number.*1459

*But I can stop because I am dealing with money; I am looking for principal.*1461

*If I am looking principal, then it has to be in money.*1468

*Money, we know that there is only two numbers behind the decimal point*1472

*because that is how much cents there are or pennies there are.*1476

*83.33 would be the same thing as 83 dollars and 33 cents.*1482

*P, when I divide this number by this number, I get 83 dollars and 33 cents.*1491

*Let's go over what I just did.*1507

*This problem gave me time, gave me the simple interest; they gave me I.*1510

*They gave me the interest rate; they are asking for the principal.*1520

*I just list it out, what I am looking for and what was given to me.*1526

*Then I plug everything into the formula.*1531

*I substitute in these numbers for these variables.*1534

*I, the interest rate is 10 dollars.*1541

*I am going to put in 10 instead of I; equals.*1542

*P, I don't know; I am going to leave the P.*1545

*R, I know is 3 percent; I change it to a decimal; put it in as R.*1548

*T, I know as 4; I am going to put in 4 instead of the T.*1555

*There is my equation; again I am solving for P.*1559

*Here I can multiply because everything is multiplied together; P times R times T.*1564

*Since I can't multiply P times a number, I can do number times the number.*1570

*These two I can multiply together; that is what I did; multiply them; get 0.12.*1574

*Then to find what P is, remember if you have this example, I can do this number divided by the 3.*1581

*I am going to do 10 divided by 0.12; you just divide it.*1589

*When you divide it, make sure you move the decimal point over twice.*1597

*You have to move this decimal point over twice; bring it up; divide it.*1600

*You end up getting 83 dollars and 33 cents; that is the principal.*1605

*That is how much was deposited to make 10 dollars with a 3 percent interest over 4 years.*1611

*That is how much was put into the bank.*1620

*That is it for this lesson; thank you for watching Educator.com.*1626

*Welcome back to Eduator.com.*0000

*For the next lesson, we are going to go over discount and sales tax*0002

*and how to calculate the amount of discount and how much sales tax we have to pay.*0005

*We all have bought something that was on sale.*0011

*We all have had to pay sales tax so this should be a little bit familiar.*0014

*First let's go over discount.*0021

*The discount we know is how much we have to subtract from our total amount that we have to pay.*0023

*It is the amount of decrease; decrease is getting less.*0031

*We are going to have to subtract from the regular price.*0035

*Before we subtract however much we are going to be saving,*0038

*we have to be able to figure out how to find how much we are going to save.*0043

*Meaning if you are going to buy something,*0048

*let's say you want to buy a soda and that soda is 10 percent off.*0051

*How are you going to know how much you are going to subtract?*0057

*How much less you are going to pay for that soda?*0060

*Discount is the percent of the discount multiplied to the regular price.*0065

*How much you are supposed to be paying for that*0074

*multiplied to the percent will be how much you are going to save.*0077

*To find how much you are going to be paying, your new price, your sale price,*0084

*it is going to be the regular price, how much you were supposed to pay,*0091

*minus how much you are going to save, the discount amount.*0095

*Again discount is money; percent of discount, we know it is percent.*0099

*Times the regular price; this is also money.*0110

*The sale price, we know this is all going to be in money.*0115

*The important thing to remember is that to figure out how much your discount is going to be,*0123

*you have to multiply the percent times the regular price.*0129

*All of this is going to equal this, the discount.*0134

*Sales tax; sales tax is the amount that you have to pay based on the total cost.*0146

*There is a percent; you have to pay a percent of that total cost.*0157

*When you multiply the rate, that percent, with the total cost,*0163

*that is going to give you how much you have to pay in sales tax.*0173

*For sales tax, once you figure out you have to pay however much in sales in the tax,*0182

*then you have to add it to your total cost.*0189

*How much your total balance came out to, how much everything came out to, plus the sales tax.*0195

*You have to pay that together; that will be how much you now owe.*0200

*Again the rate, the percent, times the total cost is going to give you*0207

*how much you have to pay, how much additional amount that you have to pay.*0212

*That is all going to equal that right there.*0218

*Let's do a few examples.*0224

*A pair of shoes that regularly sell for 50 dollars are on sale for 10 percent off.*0226

*Find the discount; regular price is 50 dollars.*0232

*If it is not on sale, then you would be paying the 50 dollars.*0239

*10 percent off; find the discount.*0243

*We want to know how much we are going to be saving.*0245

*To find the discount, you are going to multiply the regular price, the 50 dollars,*0252

*times it by the percent of the discount; that is 10 percent.*0264

*Remember whenever you use percents in some kind of equation,*0272

*when you are solving with percent, you have to change it to decimal.*0281

*10 percent in decimal, you put the decimal point at the end*0284

*because we don't see one so it is always at the end.*0292

*You are going to move it two spaces to the left because remember decimal is small number.*0294

*Think of decimal as small.*0303

*You have to make the number smaller by moving it to the left.*0305

*This is going to be 50 times 0.10 or 0.1; remember 0.*0309

*If it is at the end of a number and it is behind the decimal point, then you can drop it.*0318

*That is going to equal the discount.*0325

*50 times 0.10 or 0.1; I will just put 0.10.*0328

*That is 0; 0; 1 times 0 is 0; 1 times 5 is 5.*0337

*You can put a 0 there; you can put a 0 there.*0345

*We add; it is going to be 5; 0; 0.*0348

*How many numbers do you have behind decimal points?--we have two.*0352

*Start at the end here; you are going to go one, two.*0355

*My discount is going to be 5.00 which is the same thing in money.*0362

*It is going to be 5 dollars.*0371

*That is how much the discount is going to be.*0373

*That is how much you are saving because again 10 percent of the regular price is 5 dollars.*0374

*That is the discount amount.*0381

*That is all they are asking for; find the discount; 5 dollars.*0383

*The next example, a math textbook is 5 percent off.*0389

*If the original price is 100 dollars, what is the sale price?*0393

*The textbook originally cost 100 dollars; 100 dollars.*0398

*To find the discount... they are asking for the new price.*0406

*The sale price is the new price.*0410

*After you take away how much you are saving, that is going to be the new price.*0412

*Before we do that, we have to know what we are going to subtract.*0417

*What is the discount amount?*0421

*Discount is going to be the original price, the 100 dollars*0424

*multiplied to the discount rate, percent of discount; 5 percent.*0431

*Again we have to change this to a decimal.*0440

*You can't solve anything out with percents.*0442

*5 percent to decimal is going to be... start here.*0445

*You are going to go one, two; 0.05.*0449

*That is 100 times 0.05; then just multiply it; 100 times 0.05.*0457

*That is 0 times 5 is 0; that is 0; 5 times 1 is 5.*0470

*Here it is just 0, 0, 0; it is not going to change anything.*0477

*We can write it in if you want; fill in the empty spaces with 0s.*0482

*Add it; it is going to be 5, 0, 0.*0486

*I don't have to add this; 0 in the front is nothing.*0490

*How many numbers do I have behind decimal points?--I have two only.*0494

*Here nothing; here two; in all, I have two.*0500

*Start here; you are going to go one, two; place it there.*0506

*This is going to be 5 dollars.*0511

*Whenever you multiply a decimal by 100, remember you can just take this decimal point.*0523

*Whenever you multiply a decimal number or any number with a number that is a multiple of 10,*0532

*meaning 10, 100, 1000, 10000, 100000, then however many number of 0s you have*0538

*is how many spaces you are going to move to the right.*0548

*Because if you are multiplying, then you are getting bigger so you have to move to the right.*0551

*That will be two 0s; it is going to go one, two.*0556

*It is going to be 5.05 which is that right there.*0560

*Let's say we are going to multiply this number 0.05 times 10.*0565

*Let's say you are going to multiply it by 10.*0571

*10 only has one 0; then you would move this decimal place over one time.*0573

*It would be 0.5 or 0.5 if you multiply it by 10.*0579

*If you multiply it by 1000, you have three 0s.*0584

*You would move the decimal point over one, two, three times.*0587

*That will be 50; that is a shortcut.*0591

*Again that is only when you have 1 with 0s, a number like 10 or 100 or 1000, so on.*0595

*My discount amount is 5 dollars; that is how much I am saving.*0602

*I have to take the original price, how much I am supposed to be paying.*0607

*The sale price then is my 100 dollars or the original price of 100 dollars*0613

*minus however much I am going to be saving, the discount.*0624

*My new amount, the new price that I have to pay, 100 minus 5 is 95 dollars.*0630

*If you want to solve it out, you can change this to a 10.*0645

*This becomes 9; 10 minus 5 is 5; bring down the 9.*0653

*95 dollars; that is my new price.*0663

*Let's go over sales tax now.*0677

*A shirt cost 10 dollars; find the sales tax if the rate is 10 percent.*0681

*Now we have to actually pay more because sales tax we have to*0688

*add to our balance or add to how much we have to pay.*0691

*The original cost is 10 dollars.*0697

*To find how much the sales tax is going to be, I am going to take that 10 dollars.*0700

*Then multiply it to the 10 percent, the rate, the sales tax rate.*0713

*Again I want to change this percent to a decimal; 10 percent to decimal.*0721

*Start here; you are going to go one, two; 0.10 or 0.1.*0727

*Again the 0 is at the end of a number behind the decimal point.*0733

*You can just drop it; 10 times 0.10 or 0.1.*0736

*Look we can use our shortcut rule because we have a decimal*0749

*or we have a number that is being multiplied to 10, 1 with a 0.*0752

*How many 0s do I see here?--just one.*0758

*I can take this decimal point; I can just move it over one space.*0761

*If I were to multiply this number by 100, I have two 0s.*0769

*I can move this over two spaces to the right.*0775

*Be careful you don't move it to the left.*0778

*If you move it to the left, you are going to make your number smaller.*0780

*You have to move it to the right so that you want a bigger whole number.*0784

*Again 100, you are going to move it two spaces over.*0791

*If it is 1000, you have three 0s in 1000.*0792

*You are going to move it over three spaces.*0796

*You have to fill in your empty spaces with 0s.*0798

*Let's get rid of that 100; 10 times 0.10 or 0.1 is 1.0.*0803

*Remember I move the decimal place over once because of that number.*0816

*It is 1.0 which is the same thing as 1.*0820

*My sales tax is going to be 1; let me move this over.*0826

*Give my dollar sign some room; my sales tax is going to be 1 dollar.*0834

*That is how much I have to pay in sales tax*0838

*for my shirt that costs 10 dollars if the tax rate is 10 percent.*0840

*The next example, we are going to buy a CD that costs 14 dollars.*0850

*It is not on sale even though that is better on sale.*0855

*It has 10 percent sales tax; let's see, sales tax.*0860

*What are we looking for?--total amount that we are going to end up paying.*0874

*Before we figure out the total amount, we need to know how much we are going to pay for sales tax.*0878

*The total due or the cost is going to be 14 dollars.*0887

*Times it by the sales tax rate which is 10 percent.*0893

*Again change this to a decimal; this is 14; this is one, two.*0900

*That is 0.10 or 0.1; 0.10, you can just drop the 0 if you want.*0907

*0; 0; 1 times 4 is 4; 1 times 1 is 1.*0920

*Put 0s in those spaces; add them; 0 plus 1 is 1.*0928

*0 plus 4 is 4; 0 plus 0 is 0; 140.*0933

*How many numbers do I have behind decimal points? I have two.*0938

*From here, I am going to go one, two.*0943

*It is going to be 1.40; that is money.*0947

*A dollar forty is how much I have to pay in addition to my 14 dollars I have to pay for the CD.*0959

*Total due, total is going to be the 14 dollars plus the dollar forty.*0969

*14 is the same thing as 14.00; plus 1.40.*0982

*When you add numbers with decimal points, you have to make sure*0991

*the decimal points are lined up, the two are lined up like this.*0997

*All the rest of the numbers are aligned also.*1002

*This is 0; this is 4; bring down the decimal point.*1006

*4 plus 1 is 5; 1; bring it down.*1012

*How much am I paying?--15 dollars and 40 cents.*1016

*That is how much I have to pay for a 14 dollar CD if I have to pay for sales tax.*1025

*That is it for this lesson; thank you for watching Educator.com.*1032

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over intersecting lines and angle measures.*0002

*Remember a line is always straight and it is never ending.*0009

*Meaning it goes on forever; that is what these arrows are for.*0015

*It shows that it is going on forever this way and that way.*0017

*To name a line, we can use the points that are on the line.*0023

*To name a line using the points, we need at least two points.*0031

*Here point A and point B; can write it as A, B.*0035

*Then you are going to draw a little line above it like that.*0045

*That shows line AB.*0049

*Because it doesn't matter which way it is going,*0052

*whether I name it AB or BA, I am still talking about the same line.*0055

*It goes on forever in both directions.*0063

*I can also say BA with a line over it to show that it is a line.*0067

*This is how you represent, how you name this line.*0074

*This whole thing is also called L; I can also name this as line L.*0081

*When you usually name a line, it is usually in cursive.*0092

*That is why it is a cursive L; line L.*0095

*Three ways; AB using the points, two points on the line.*0098

*AB with the line above, AB; or BA, same thing.*0104

*Or if the whole thing has a name L, then you can just call it line L.*0110

*When you have two lines that are intersecting or two lines*0119

*that cross each other like this, they are intersecting lines.*0121

*They are two lines that intersect; they intersect at point P.*0126

*This is a point; that is the point where they touch; that is point P.*0129

*This is line L; line N.*0136

*For this, you can also name this as line CD; line DC just like we did here.*0140

*But P is also on that line; I can also name this as line PD; PD with a line above it.*0148

*That can also be used to name this line; just any two points on that line.*0157

*If I say line CD or line PD, I am talking about the same line.*0164

*It doesn't matter which one.*0169

*Again that is intersecting lines, when they cross each other.*0172

*For angles, this right here is an angle; B is a point on one side.*0178

*C is a point on the other side of the angle; there is two sides.*0189

*This point right here where those two sides meet, that is the vertex.*0194

*That is called the vertex; point A is the vertex of that angle.*0197

*When I name this angle, I can say angle; that just shows an angle.*0204

*I use the points; I need three points on this angle.*0211

*If I just say BC, then that doesn't tell me what angle I am talking about.*0216

*Or that doesn't even give me an angle; I have to say BAC; angle BAC.*0220

*Again if you are going to use points to represent the name of an angle, then you have to use three points.*0230

*I can also say angle CAB.*0239

*Make sure your angle is going like that; it is not going like this.*0249

*If you noticed for these two names, both of these, A the vertex is the middle point.*0255

*It is BAC; angle CAB; I can't say angle BCA.*0263

*Angle BCA is not the correct name for it.*0269

*Angle BCA, that is not a name for this angle.*0272

*The vertex has to be the middle point when you name it.*0278

*Another name, just like the previous slide where we had line L,*0286

*the name of the line was L so we can also name it line L.*0292

*For this one, if it says 1, usually the angles, if there is a name for it, it is a number.*0297

*That number 1 right there, that is talking about this angle.*0306

*So I can also say angle 1.*0309

*The degree of an angle is the angle measure.*0318

*Measure is talking about how narrow or how wide open the angle is.*0322

*This right here, if I say this is a 90 degree angle, it is a perfect right angle.*0333

*Meaning this is vertical and this is horizontal.*0339

*This little box right here says that it is a 90 degree angle.*0344

*This is 90; to represent degree is a little dot right there.*0348

*That is 90 degrees.*0354

*If I have a straight line, a straight line measures 180 degrees*0356

*because it is like 90 and then it is another 90.*0367

*If I were to draw a 90 degree angle from here, it will be half way.*0370

*This is 90; this is 90; together it makes 180.*0373

*If I start from here and I go all the way around a full circle, that is 360.*0383

*You can also use this to represent a 360.*0392

*This right here was 180; that is 180.*0400

*This again is 180; together it is 360.*0404

*All of it together, the whole full circle going all the way around,*0408

*from starting point and then going all the way back to that same point, it is 360 degrees.*0411

*Again right angle is 90.*0418

*Two right angles make a straight line; that is 180 degrees.*0420

*Two straight lines, going this way and then going another this way, is 360.*0426

*That is a full circle; a full circle is always 360 degrees.*0430

*There is three types of angles when it comes to classifying.*0440

*The three types would be acute angle... this is when...*0444

*Remember this is a 90 degree angle; that is a 90 degree angle.*0450

*Acute angle has to be smaller than a right angle.*0458

*It has to be smaller than 90; this is less than 90 degrees.*0463

*That makes up an acute angle.*0472

*Right angle we know is perfectly 90 degrees.*0475

*An angle that is greater than 90, greater than 90 degrees, is called an obtuse angle.*0484

*The right angle would be like that right there; this is 90.*0496

*It is going more than 90; it has to be bigger than 90.*0502

*These are the three types of angles.*0506

*So that you don't confuse the acute angle with the obtuse angle, we know a right angle is perfectly 90.*0510

*Acute angle and an obtuse angle; notice how the acute angle is a lot smaller than the obtuse angle.*0515

*Acute angles are small; they are smaller than 90.*0521

*Think of it as a cute angle because it is small.*0524

*Acute angles are small; obtuse angles are big; three types of angles.*0529

*When we compare two different angles to each other, some angles have a relationship.*0541

*The first angle relationship is a vertical angle.*0551

*If we have intersecting lines, two lines that are intersecting, there is four angles that are formed.*0557

*We have this angle, this angle, this angle, and this angle.*0564

*There is four angles that are formed by intersecting lines.*0567

*When you look at the opposite angles, the top one and the bottom one, those are called vertical angles.*0570

*Remember when we talked about how to name angles.*0578

*This is angle 1; this is angle 2.*0582

*We can name angles by using the points on the angle.*0587

*Or if it is labelled as 1, 2, then we can say that that is angle 1 and this is angle 2.*0589

*This is different; don't get it confused with angle measure.*0597

*Because angle measure, that is how many degrees that angle is and it has that little degree sign.*0599

*This is not degrees; it is not 1 degree.*0609

*It is angle 1; this is angle 2.*0611

*Angle 1 and angle 2 are vertical angles; that is the relationship between the two.*0615

*Again if they are intersecting lines and then they are opposite, this one and this one are vertical angles.*0620

*This one and this one are also vertical angles.*0626

*If this is angle 3, this is angle 4, then angles 3 and 4 would also be vertical angles.*0631

*The next type of relationship is called adjacent angles.*0638

*Adjacent, think of it as next to.*0643

*They are angles that are next to each other.*0647

*They have to have a common vertex and side; they share two things.*0650

*The vertex, we know that a vertex is this part right here.*0655

*That is the vertex; they have to have the same vertex and a side.*0658

*Angles 1 and 2 here are adjacent because they are next to each other.*0666

*This is the vertex of angle 1; this is the vertex of angle 2.*0671

*They have the same vertex; and they share a side.*0675

*This is the side that they share; these would be adjacent angles.*0679

*Adjacent angles don't always have to be from intersecting lines.*0686

*If I have let's say like this, angles 1 and 2, these would be adjacent angles*0690

*because they share the same vertex and the same side and they are next to each other.*0701

*Same vertex; same side; angles 1 and 2 here are also adjacent.*0708

*Complementary angles.*0715

*Complementary angles are two angles that when you add them together becomes 90.*0717

*It has to be 90 for it to be complementary.*0723

*Again two angles that add up to 90 degrees.*0727

*Here angle 1 and angle 2, if you add them together, it is going to become 90 degrees.*0730

*If I were to take this angle and place it so that it is like this, this would be angle 1.*0737

*See how it forms a right angle; 90 degrees is a right angle.*0746

*Any two angles that add up to 90; they don't all have to be adjacent.*0752

*It doesn't have to be like this for it to be complementary.*0755

*I can have one angle here; I can have another angle over here somewhere.*0758

*As long as they add up to 90 degrees, they would be complementary angles.*0767

*Supplementary angles are any two angles that add up to 180.*0775

*Here angle 1 and angle 2 would add up to 180.*0782

*If I were to put it together, notice how they would line up to be a straight line.*0786

*This is angle 1; this is angle 2.*0796

*If you add them together, see how this would be a straight line, 180 degrees.*0799

*Remember how we said if I have a straight line, it is as if I have two 90 degree angles.*0806

*This is 90; this is 90; together they add up to 180.*0817

*A straight line is 180.*0822

*If I have two angles that form a straight line, then they are supplementary angles.*0825

*They don't have to be together; they don't have to be adjacent.*0834

*They can be just like the complementary angles.*0838

*They can be two angles that are split; one angle here, one angle over there.*0841

*As long as they add up 180, they are supplementary angles.*0847

*Again two angles that are opposite to each other when they are*0851

*formed by intersecting lines are called vertical angles.*0855

*Angles 1 and 2, since they are opposite angles, they are vertical.*0859

*Adjacent angles are two angles that are next to each other.*0865

*They have to share a common vertex and a side.*0869

*An example of nonadjacent angles, meaning two angles that are not adjacent, would be like that.*0873

*This is angle 1; this is angle 2.*0883

*Even though they are next to each other, they are not adjacent because they don't share the same vertex.*0886

*This is the vertex of angle 1; this is the vertex of angle 2.*0892

*For this, this is not adjacent.*0897

*They have to be next to each other and share the same vertex.*0901

*Complementary angles are two angles that add up to 90 whether they are together, adjacent, or not.*0908

*Supplementary angles are two angles that add up to 180 whether or not they are adjacent.*0916

*To remember between complementary and supplementary, C comes before S in the alphabet.*0923

*C, A-B-C, and then S is way down there; C comes before the S.*0932

*90 comes before 180 if you were to count; 90 comes before 180.*0937

*C before S; 90 before 180.*0943

*C, complementary angles are 90 degree angles; supplementary are 180.*0946

*That is just one way for you to remember between complementary and supplementary.*0953

*Our examples, the first one, write two other names for AB, line AB.*0961

*Line AB is this line right here.*0969

*To find two other names... I didn't label them; this is L, N, and P.*0974

*Here I can say, since that is AB, I need to find two other names.*0987

*I can say BA; line BA; that is one other name.*0995

*Then I can say line P; line P.*1002

*Again the names for lines are usually in cursive; line BA and line P.*1009

*Name two intersecting lines; line AB and line AC are intersecting.*1020

*Line AB with line DE is intersecting.*1030

*I can also say line P with line L or line P with line N; any of those.*1036

*But just make sure that it is not line AC with line DE.*1043

*They could intersect eventually because remember these lines are never ending.*1049

*They go on forever; if they are not parallel, eventually they can meet sometime.*1053

*But in this diagram, it doesn't show them intersecting.*1062

*We can just say line AB with a line; this one with line maybe DE.*1067

*You can also say BE; it doesn't matter; DE; any two points on the line.*1079

*DE; those are two intersecting lines; I can also say line P with line L.*1085

*Classify each angle and name the relationship between the two.*1103

*This angle; classify, remember there is three types of angles.*1108

*The acute angle, a right angle, and obtuse angle; this is less than 90.*1111

*I know that because a 90 degree angle is a right angle; that is 90.*1118

*This would be an acute angle.*1125

*This one is greater than 90; it is 135 degrees.*1133

*That is definitely greater; this is an obtuse angle.*1139

*The relationship between these two, I know they are not vertical; they are not adjacent.*1147

*They are probably either going to be complementary or supplementary.*1155

*Let's add these up; this one, 45 degrees plus 135 degrees.*1158

*135 plus 45; 7, 8; they add up to 180 degrees.*1169

*Because they add up to 180, that would make them supplementary angles.*1182

*If they were to add up to 90, then that would be complementary angles.*1199

*The next one, determine the angle relationship between the pair of angles.*1206

*The first is angle 1 or angle 2.*1210

*Again be careful that these are not the angle measures.*1215

*There is no way that this can be 1 degree, 2 degrees.*1218

*These are the names of the angles.*1222

*This angle and this angle here, what is the relationship between them?*1226

*They are next to each other; they share the same vertex and a side.*1233

*These are adjacent; adjacent angles.*1237

*The next one, angle 3 and angle 4, see how they are opposite angles.*1247

*They are formed by intersecting lines; these are vertical; vertical angles.*1254

*The fourth example, name the measure of angle 1; here we have a right angle.*1274

*This angle along with this angle together form that right angle.*1284

*I want to find the angle of this measure right here.*1290

*I know this whole thing is 90.*1292

*If I take 90 and I subtract the 50, don't I get measure of angle 1?*1297

*I can say the measure of angle 1... a shortcut for me to say that is measure of angle 1.*1303

*You know angle 1 is like that.*1310

*But when I am talking about the angle measure, the degrees, then I could put M for measure.*1312

*This just says measure of angle 1; I am talking about the number of degrees.*1320

*Measure of angle 1 plus... this is 50 degrees.*1327

*Together, if I add them together, it becomes 90 degrees.*1335

*How do I solve for measure of angle 1?--I can subtract 50.*1341

*That way measure of angle 1 is 40 degrees.*1348

*This is 40; this is 50; together they add up to 90.*1354

*We know that these two angles are adjacent because they are next to each other.*1360

*They share the same vertex and a side.*1364

*They are also complementary because they add up to 90.*1367

*This angle with this angle together are complementary angles.*1372

*Here straight line.*1377

*That means together measure of angle 1 plus 83 degrees has to add up to 180 degrees.*1381

*Straight line is always 180.*1391

*Again I am going to put measure of angle 1 plus...*1394

*This angle plus this angle, 83 degrees, equals a total of 180 degrees.*1400

*I am going to subtract the 83 degrees.*1411

*Measure of angle 1 is... this is 97 degrees.*1417

*Here these two we know are supplementary because they add up to 180.*1434

*90 so they are complementary; 180 so they are supplementary.*1443

*These are also adjacent angles; they are next to each other; same vertex, side; adjacent.*1447

*That is it for this lesson; thank you for watching Educator.com.*1455

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over angles of a triangle.*0002

*Remember a triangle is a polygon with three sides; three straight sides.*0009

*Which means that there are three angles; those sides form three angles.*0015

*All triangles have three angles.*0021

*Here is one; here is another one; there is a third.*0027

*To name this angle here, we can say angle BAC.*0031

*That would be this angle right here; angle BAC.*0040

*But since the A is a vertex and there is only one angle*0045

*that this is a vertex for, we can just call this angle, angle A.*0052

*This one, I can just call angle B; this is angle C.*0059

*Again only if the point A is a vertex for just a single angle.*0065

*Let me give you an example of what it is not.*0071

*If I have an angle like that, I have two adjacent angles; this is A.*0075

*I can't call this angle, angle A, because there is three different angles formed here.*0084

*There is this small angle; there is this angle; there is this big angle.*0089

*This point, this vertex, is a vertex for three different angles.*0095

*In this case, you cannot call it angle A; you can't say angle A.*0099

*You would have to name the other three points like this one.*0106

*You would have to name, if this is B and this is C, then you have to say angle BAC or like that.*0110

*But again this one, because in a triangle, there is only three angles and three vertex.*0119

*You can just name this as angle A.*0129

*If I say angle A, I am talking about this angle here; angle B; angle C.*0131

*Within the three angles of a triangle, remember each angle has an angle measure, the number of degrees.*0139

*All three angle measures is going to add up to 180,*0148

*like the supplementary angles where we have two angles that form a straight line.*0152

*That adds up to 180.*0156

*Here the three angles of a triangle also add up to 180.*0159

*If this is 60, this is 60, then what I can do is add these two up and subtract it from 180.*0168

*Here if I want to write an equation, I can say measure of angle A.*0179

*Remember this M is for measure; it is to show the number of degrees.*0186

*Measure of angle A plus the measure, the number of degrees, of angle B*0190

*plus the measure of angle C is going to equal 180 degrees.*0199

*We know what the measure of angle A is; how many degrees is angle A?*0212

*We know it is 60; this whole thing is just 60 degrees.*0216

*Measure of angle A is just 60; I can just replace this with 60.*0221

*Do I know measure of angle B?--no; I can just leave that there.*0225

*Plus the measure of angle C is also 60.*0231

*That is all going to add up to 180.*0236

*Again I can just add these two together which is this and this.*0240

*That is going to be 120; plus this unknown adds to 180.*0244

*I can subtract this from 180; 180 minus these two; whatever is left over.*0256

*From the 180 total, if I add these two together*0263

*and then figure out how many degrees are left over from the 180,*0269

*then all of that, all of those left over degrees have to go to angle B.*0273

*I am going to subtract; measure of angle B is going to be 60 degrees.*0278

*The leftover degrees from the 180 is 60; then this also has to be 60.*0290

*That is how you are going to solve for the missing angle measure.*0300

*Remember if we are going to be solving for the missing angle measure,*0305

*then we have to know two of the three angle measures.*0309

*I can't only have the measure of angle A and then find both B and C*0317

*because they are going to be different angles; they could be different angle measures.*0323

*I don't know how many are going to go here and how many are going to go here.*0330

*To find the missing angle measure, you have to have two out of the three like this one.*0334

*I have measure of angle A, 70 degrees.*0343

*I have the measure of angle B; that is 60 degrees.*0348

*I want to find the measure of angle C, meaning I want to find how many degrees is in angle C.*0352

*Again I can just take these two, add them together; how many from the 180?*0359

*I know that this plus this plus this all have to add up to 180.*0364

*This and this are used up.*0371

*However many are left over all have to go to angle C.*0373

*I can say 70 degrees plus this 60 plus the measure of angle C.*0379

*This is the proper way to write it.*0387

*I can't just write C because you are talking about the measure, meaning how many degrees.*0389

*It is all going to add up to 180.*0394

*Again I am going to add these two together.*0398

*This will be 130 plus the measure of angle C.*0400

*130 being used up plus the leftovers is going to equal 180.*0410

*Remember I subtract 180 with this number.*0415

*That way measure of angle C is going to be 50 degrees.*0422

*That means this has to be 50.*0426

*60 plus 70 plus 50 is going to add up to 180.*0429

*That is the missing angle measure.*0434

*Determine the angle measures if the angle measures could be the angle measures of a triangle.*0441

*Three angle measures for the three angles of a triangle.*0448

*If they add up to 180, then they can be the correct angle measures of a triangle.*0455

*But if not, if they don't add up to 180,*0460

*that means they can't be the three angle measures of a triangle.*0462

*The first one, I am going to take 50 plus the 90 plus the 40.*0466

*Just add them all up; I know that 0 plus 0 plus 0 is 0.*0473

*5 plus 9 is 14; plus 4 is 18; yes, they add up to 180.*0479

*That means these three angle measures can be the angle measures of a triangle.*0489

*This one is yes.*0497

*The next one, 45 plus 48 plus the 95.*0504

*5 plus... you can add this 5.*0516

*5 plus 5 is 10; plus 8 is 18; put up the 1; 8.*0520

*Already I know that it is not going to add up to 180*0528

*because the last digit has to be 0 and it is not.*0533

*This is 1 plus 4 is 5; plus 4 is 9; that plus 9 is 18.*0537

*This is 188; this is too much.*0546

*That means it can't be the angles of a triangle; this one is no.*0550

*Remember the angles of a triangle have to add up to 180.*0557

*The third example, find X.*0565

*We want to find the measure of this angle right here.*0568

*I have this triangle.*0574

*Remember all three angles of a triangle have to add up to 180.*0578

*But this one is what I am looking for; this is the missing angle measure.*0583

*I don't have this angle measure either.*0586

*If I need to find the third angle measure, I need to have the other two.*0589

*I have this one; I need to have this one also.*0594

*If I don't have this, then I don't know how many goes here.*0598

*I need to find this one first.*0603

*I have to use another method to find this angle measure.*0606

*I know that this right here, this straight line...*0615

*This is from the last lesson, the previous lesson on angles and lines.*0621

*If this is the line here, this one doesn't have an arrow.*0629

*Just do that; here is where it goes up.*0635

*Remember this, two angles right here, they are adjacent angles.*0643

*But they are also supplementary because it is a straight line.*0652

*It is straight; a straight line has an angle measure of 180.*0656

*This whole thing together is 180; that means this one plus this one is 180.*0662

*This is given that it is 135 degrees.*0671

*If this one together with this small one is 180, then I can just subtract it.*0675

*180 minus the 135 to see what this angle measure is going to be.*0680

*180 minus 135; this is going to be 45 degrees.*0686

*That means this has to be 45 because again this angle with this angle together forms a straight line.*0698

*That has to be 180; they are supplementary angles.*0705

*Now that I found this angle and I have this angle, I need to find the measure of this angle.*0711

*I can just say that X... this is just angle measure so I can just leave it as X.*0719

*I don't have to say measure of angle X because that is not a name.*0728

*That is the number of degrees. *0732

*X degrees plus 53 degrees plus 45 degrees all add up to 180 degrees.*0734

*See how they are all in degrees.*0745

*Again I am going to add these two together to see how many of the 180 I am using up.*0749

*Then see how many are left over to be X.*0753

*This is 53 plus 45 is 98 degrees.*0761

*That means X degrees, this many degrees, plus 90 degrees together is 180 degrees.*0768

*Again I am going to subtract this from 98; I get 82 degrees.*0778

*Right here, X is 82 degrees; this has to be 82.*0796

*That way this plus this plus this, the three angles of a triangle, are going to add up to 180.*0804

*That is it for this lesson; thank you for watching Educator.com.*0812

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over classifying triangles.*0001

*Depending on the angles and the sides of the triangles, they have different names.*0008

*First by angles, to classify, we know that we have three angles and we have three sides.*0016

*Depending on the three angles of the triangle, we are going to have different names for them.*0025

*The first one is called an acute triangle.*0034

*Notice how all the angles, this angle here, this angle here,*0039

*and this angle here, are all less than 90 degrees.*0044

*It means that they are small; all angles are less than 90 degrees.*0050

*Remember a 90 degree angle is a right angle.*0064

*All three of them have to be less than 90.*0067

*It is called an acute triangle.*0072

*The next one is when you have one right angle.*0076

*The other two angles are going to be acute.*0083

*From within a triangle, if only one angle is a right angle, then it is called a right triangle.*0087

*Only one angle is going to be 90 degrees.*0097

*It is only one angle because you can only have one angle be a right angle.*0104

*There is no way you can have a triangle where there is two angles that are right angles.*0110

*Again a triangle can only have one right angle; you just can't do it.*0118

*If that is one of the right angles of a triangle,*0126

*if I draw this one as a right angle, that is not going to be a triangle.*0129

*It is not possible to have two right angles in a triangle.*0133

*Again one right angle; and that becomes a right triangle.*0138

*The third type of triangle by its angles is an obtuse triangle.*0143

*That means one of the angles, this one right here, is going to be greater than 90.*0149

*One angle is greater than 90 degrees; this means bigger.*0158

*One angle is bigger than 90 degrees; the other two are going to be acute.*0164

*Just like the right triangle, it is impossible to have more than one angle of a triangle be obtuse.*0172

*That is my obtuse angle.*0182

*If draw another obtuse angle let's say like that, there is no way I can have a triangle.*0186

*because remember a triangle only has three sides.*0191

*You can only have one obtuse angle within a triangle.*0196

*If you do have one obtuse angle, then it becomes an obtuse triangle.*0201

*An acute triangle, a right triangle, and an obtuse triangle are all types of triangles by angles.*0207

*A right triangle we know is when we have a right angle.*0220

*To remember between an acute triangle and an obtuse triangle,*0224

*think of what this spells: a-cute, a cute triangle.*0229

*If these are small angles, then we tend to think that they are cute.*0237

*You can think of it that way.*0242

*Acute triangle is when all three angles are small so then it is a cute triangle.*0244

*Obtuse is just a larger angle; that would be the obtuse triangle.*0249

*Next is classifying triangles by size.*0257

*Again depending on their sides, they are going to have different names.*0262

*If all three sides are the same, then it is an equilateral triangle.*0267

*These little marks right here, that shows that it is the same.*0275

*If this side, this side, and this side all have one mark each,*0280

*that means all three sides are the same, are congruent.*0285

*That means this is an equilateral triangle.*0291

*If this is 10 inches, then this has to be 10 inches and that has to be 10 inches.*0293

*It is equilateral; this means equal; lateral means side.*0297

*It is like equal sides; equilateral triangle.*0304

*Three sides are congruent; three sides are the same.*0310

*Isosceles triangle, the next one.*0322

*Isosceles is when you have two out of the three sides being the same.*0324

*This side and this side are congruent; congruent just means the same.*0332

*This side and this side are the same; that is an isosceles triangle.*0338

*This is all three sides; this is two out of the three sides.*0343

*Two sides are the same.*0347

*The third one is a scalene triangle.*0355

*A scalene triangle is when no sides are the same.*0358

*This one, this one, and this one, they are all different.*0363

*All sides are different; scalene; equilateral, isosceles, and scalene triangle.*0367

*This is classifying triangles by its sides, depending on the sides.*0386

*The first example is to classify the triangle by its angles and sides.*0396

*Look at the angles first.*0402

*If you look at the angles, this is an acute angle.*0405

*This is an acute angle; and this is an acute angle.*0408

*We can tell it because they are all smaller than 90 degrees.*0411

*They are all smaller than right angles.*0414

*That name for a triangle with all three acute angles is an acute triangle.*0419

*By sides, look at the sides.*0435

*This one is congruent to this one is congruent to that.*0438

*All three sides are the same; that is an equilateral triangle.*0442

*This is by angles; this is by sides; this type of triangle has two names.*0452

*This one, this is acute, acute, and acute; therefore this is an acute triangle.*0461

*By its sides, we have two that are the same so this is isosceles triangle.*0474

*Angles and sides; let's sketch each figure; an isosceles right triangle.*0487

*We have to make sure that it is... isosceles is when we have two sides being the same.*0499

*A right triangle is when we have one right angle.*0506

*I need to draw a triangle with one right angle and these sides being the same like that.*0512

*Just to mention, if this is a right isosceles triangle, we know that this is 90 degrees.*0527

*Then this angle and this angle will actually be exactly the same because this is isosceles.*0536

*See how the distance from here to here and from here to here are the same.*0542

*This angle and this angle will be exactly the same also.*0549

*This is an isosceles right triangle.*0555

*The next one is scalene and obtuse triangle; a scalene obtuse triangle.*0558

*Scalene is when no sides are the same.*0564

*Obtuse is when you have one angle that is larger than 90.*0570

*Obtuse angle; and then scalene means that no two sides are the same.*0577

*Draw one short; draw one longer; this one is going to be the longest.*0584

*That is a scalene obtuse triangle.*0589

*Classify the triangle by its angles and sides.*0598

*Here my angles first; let's do angles; let's see; this is acute.*0602

*This looks like it is obtuse because it looks like it is greater than 90.*0610

*This is acute.*0618

*Just because I have one obtuse angle, that is going to make this whole triangle an obtuse triangle.*0619

*By its sides, we have two sides that are the same.*0634

*This is going to be an isosceles triangle; obtuse triangle and isosceles triangle.*0638

*The next one, by its angle, I have one right angle; acute and acute angle.*0651

*When you have only one right angle, that makes the whole triangle a right triangle.*0660

*By its sides; it doesn't show that any two sides are the same.*0671

*Looks like this is the shortest one.*0677

*This is the next one; that one is the longest.*0679

*This is going to be scalene; scalene triangle.*0683

*For the fourth example, given the measures of the angles of a triangle,*0696

*classify the triangle by its sides and measures.*0701

*Here just based on the angle measures, we need to figure out*0708

*what type of triangle it is by its angles and by its sides.*0719

*Here look at that; that is a 90 degree angle.*0725

*That means I am going to have a right triangle.*0728

*This is 90; the other two angles are going to be 45 and 45.*0734

*If I were to draw this like that, like a house, this is 45 and this is 45.*0742

*They are the same.*0752

*That means these two sides are going to be the same*0754

*because the distance from here to here and the distance from here to here*0759

*have to be the same for these two angles to be the same.*0765

*This is going to be an isosceles right triangle.*0772

*This is by its sides; this is by its angles.*0785

*The next one, 30 degrees, 100 degrees, and 50 degrees.*0790

*If you look at that one, that is greater than 90; greater than 90.*0797

*Let me just draw this again so that way you can see this a little bit clearer.*0812

*This will be the 100 degree angle.*0826

*This is going to be the 100 degree angle.*0831

*Notice how I drew this side really long and this side really short.*0837

*If I draw this really long, then see how this angle gets skinnier so it gets less.*0843

*That means that one is going to be the 30 degree one and this one is going to be the 50 degree one.*0850

*In order for me to have different angle measures for this one and this one,*0855

*these two sides have to be different because I have to draw one long so it gets skinnier.*0861

*Then my angle would be less; it would be smaller.*0868

*This we know first of all is going to be an obtuse triangle.*0873

*But with the sides, because I had to draw one long, one longer than the other*0879

*so the angle would be smaller than the other, it is going to be a scalene triangle.*0883

*This is a scalene obtuse triangle.*0888

*Scalene by its sides; obtuse because of that 100 degree angle.*0900

*That is it for this lesson; thank you for watching Educator.com.*0908

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over quadrilaterals.*0002

*Remember a quadrilateral is a four-sided figure.*0006

*Any polygon with four sides is a quadrilateral.*0009

*That means the four sides has to be straight sides, and they have to be enclosed.*0016

*This is a type of quadrilateral.*0029

*Any shape that has four straight sides and no open areas is a quadrilateral.*0031

*Special types of quadrilaterals are listed out here.*0040

*The first one is a parallelogram; a parallelogram looks like this.*0045

*It has two pairs of opposite sides being parallel and congruent.*0053

*Again opposite sides are parallel and congruent.*0060

*Parallel means that they are slanted exactly the same way so that they will never touch.*0065

*If this line and this line were to keep going forever and ever, they are never going to touch.*0071

*That is what it means to be parallel; they are also congruent.*0079

*To show two sides are parallel, you can draw arrows like that.*0087

*One arrow with one arrow here shows that those two sides are parallel.*0092

*I can also say that these two lines are congruent.*0097

*I draw two marks like; remember that means they are congruent.*0100

*Then to show that these two sides are parallel and congruent, instead of drawing just one arrow*0105

*because one arrow is for these two, I have to draw now two arrows.*0111

*Now I am saying all the sides with two arrows are parallel to each other.*0117

*To also show that these two sides are congruent, I have to draw two marks instead of just one*0126

*because all the ones with one are congruent so all the ones that have two are congruent.*0131

*This is a parallelogram; again opposite sides are parallel and congruent.*0137

*These two sides are parallel and congruent; these two sides are parallel and congruent.*0142

*That is a parallelogram.*0147

*A type of parallelogram is a rectangle.*0152

*A rectangle is a type of parallelogram because parallelogram just has opposite sides parallel and congruent.*0157

*Rectangle has opposite sides parallel and congruent.*0164

*It is a parallelogram with four right angles.*0170

*It has all the properties of a parallelogram plus it has four right angles.*0174

*Opposite sides are parallel; these are parallel and congruent; parallel and congruent.*0181

*And it has four right angles; it is a special type of parallelogram.*0186

*The next type of parallelogram is a rhombus.*0199

*Rhombus, opposite sides are parallel and congruent; plus it has four congruent sides.*0203

*All sides are sides are congruent.*0215

*Again rhombus is a type of parallelogram.*0220

*It is not a type of rectangle; it is a type of parallelogram.*0222

*This is that; and then parallelogram with rhombus also.*0226

*The next one, square, we know what a square is.*0234

*But square is also a parallelogram.*0238

*But more specifically, it is a type of rectangle and it is a type of rhombus.*0241

*Square is like all of the above; why?*0247

*Not only does it have parallel and congruent sides,*0250

*it also has four right angles and it has four congruent sides.*0253

*The square, it has this one; it has this one; and it has this one.*0260

*That is a square.*0273

*The last one, the trapezoid; trapezoid is not a parallelogram.*0280

*Remember parallelogram has to have both pairs being parallel and congruent.*0288

*Trapezoid only has one pair; that means only this and this one are parallel.*0295

*One pair of parallel sides; that is the only requirement for a trapezoid.*0303

*Only one pair of parallel sides is trapezoid.*0309

*Two pairs of parallel sides and it is a parallelogram; these are obviously not parallel.*0312

*If these two sides were to keep going on forever, then they are going to eventually intersect.*0317

*Or they are going to eventually meet; so this cannot be a parallelogram.*0322

*Let's look at this flowchart; this right here is a parallelogram.*0336

*We have parallel and congruent; these two sides being parallel and congruent.*0341

*This is a parallelogram; there are two types of parallelograms.*0349

*This is a rectangle; this is a rhombus.*0361

*By the way, when you have more than one... rhombus is singular, when you only have one.*0370

*When you have more than one, it becomes rhombi; rhombi is the plural for rhombus.*0376

*Rectangle, rhombus; two types of parallelograms because a property of parallelogram...*0383

*As long as it has two pairs of opposite sides parallel and congruent, then it is a type of parallelogram.*0389

*This one also has that property; this one also has that property.*0398

*This one has to have four right angles; then it is a rectangle.*0403

*This one has all the parallelogram properties; plus it has four congruent sides.*0408

*Four right angles; four same sides.*0415

*Then when you combine all those properties together, it actually becomes this, a square.*0427

*Notice that a square has four right angles.*0438

*And it has four congruent sides, four same sides.*0443

*Square is always a rectangle; a square is always a rhombus.*0450

*So square is always a parallelogram.*0457

*Parallelograms are sometimes going to be rectangles and sometimes going to be rhombi.*0463

*Or it can just be a parallelogram.*0470

*Same thing here; rhombus can be a rhombus; or sometimes it can be a square.*0473

*When you look at this flowchart, if you are going downwards,*0479

*meaning you are comparing a parallelogram let's say to a rectangle, isn't it only sometimes?*0484

*Parallelogram is sometimes a rectangle because it can also be a rhombus.*0489

*When you are going down on the flowchart, it is going to be sometimes.*0495

*When you go up on the flowchart, isn't a rhombus always a parallelogram?*0502

*because the rhombus always has the properties of a parallelogram.*0507

*If you are going up on the flowchart, if you are comparing*0515

*like a rhombus to a parallelogram, a rhombus is always a parallelogram.*0517

*A square is always a rectangle because it always has four right angles.*0521

*A square is always going to be a rectangle; this is always.*0524

*Let's look at let's say a trapezoid.*0533

*A trapezoid doesn't fit anywhere on this flowchart.*0537

*Why?--because it starts off with parallelogram.*0540

*Parallelograms have to have two pairs of parallel sides and congruent sides.*0543

*Trapezoid only has one; so trapezoid goes over here to the side.*0551

*Parallel sides; that is a trapezoid.*0557

*Is a trapezoid ever going to be a parallelogram?--no, they are two different things.*0570

*One pair of parallel sides; two pairs of parallel sides.*0575

*Trapezoid, parallelogram?--never; a trapezoid to a rectangle?--never.*0579

*On the flowchart if you go left or right, how about rectangle with rhombus?*0586

*Are they ever going to be the same?--no, so this is never.*0592

*Again when you are going downwards, it is sometimes.*0599

*It is like classifying let's say animals; let's say quadrilaterals is like animals.*0605

*Parallelograms are types of quadrilaterals; let's say parallelograms are like dogs.*0613

*Parallelograms are like dogs; don't we have different types of dogs?*0622

*We can have Maltese; we can have Chihuahuas.*0629

*We can have whatever, any types of dogs.*0632

*The different types of dogs can go there.*0637

*From there, we can classify even further; that is kind of how it goes.*0639

*If I go as a dog, always, sometimes, never a Maltese; isn't it just sometimes?*0644

*Again when you go downwards, it is sometimes.*0651

*But then again, is a Maltese always, sometimes, never a dog?--isn't it always?*0655

*If I go side by side, it is going to be never.*0663

*Let's say over here where the trapezoid belongs, if I write birds.*0666

*Birds and dogs, they don't have anything to do with each other.*0676

*They are two different things.*0679

*If I ask you when is a bird a dog?--never.*0681

*That is how this flowchart works; this is just an example.*0687

*Let's do our examples; give the most exact name for the figure.*0698

*Here we have four congruent sides; what has four congruent sides?*0704

*We know that a square has four congruent sides.*0708

*But then again square also has to have four right angles.*0712

*This can be a rhombus; this can also be a parallelogram.*0716

*But a more exact name would be rhombus.*0721

*How about this one here?--this looks like a rhombus.*0729

*But I don't know that all four sides are congruent.*0733

*I know that these two are.*0736

*All I can say, because all I notice is that these two are congruent.*0741

*I do have two pairs of parallel sides; so then this is a parallelogram.*0746

*The last one, it looks like a rectangle; but I am not sure.*0757

*Here only one pair of parallel sides.*0766

*I don't see that any of the sides are congruent or the same.*0769

*No, it doesn't seem like any two sides are the same.*0775

*That is all I have; just that it is parallel; one pair.*0778

*This must be a trapezoid.*0784

*Just because it looks like a rectangle, it doesn't mean that it is.*0790

*Looks like maybe this side and this side, they are not parallel.*0793

*This side looks a little bit longer than this side; we can't really assume.*0800

*Just based on the facts, this being parallel to that and that is it, it would be a trapezoid.*0804

*The next one, if a parallelogram has four right angles, then it is a...*0813

*What do we know has four right angles?*0817

*We know a square has four right angles and a rectangle has four right angles.*0822

*But isn't a square a type of rectangle?--then this has to be a rectangle.*0827

*because if I say rectangle, then I am also including squares because a square is a type of rectangle.*0836

*A rhombus is a type of what?--yes, it is a quadrilateral.*0843

*But more specifically, it is a type of parallelogram.*0850

*If a quadrilateral has one pair of parallel sides, then it is a... *0860

*If it is two, then it is a parallelogram; one then it is a trapezoid.*0865

*The next example; always, sometimes, or never.*0878

*Let's see; a trapezoid is always, sometimes, or never a rectangle.*0883

*Remember that example where I said trapezoid is like a bird and a rectangle is a type of dog.*0889

*A bird is never going to be a dog.*0899

*It is going side by side on the flowchart.*0902

*It was trapezoid here; you know let me write it in red.*0905

*The flowchart starts off as quadrilaterals; we have trapezoids here; we have parallelograms here.*0916

*Parallelograms, the two types are rectangles and the rhombus.*0938

*These two have a square; here is your flowchart.*0954

*Again if you are going side by side, meaning if there is no arrows connecting them, then it is never.*0962

*Quadrilaterals is like saying animals.*0971

*Trapezoids is a type of animal; it is like a bird.*0974

*Parallelograms are like dogs; they branch out to the different types.*0978

*We said Chihuahuas and Maltese or whatever you want to say.*0983

*This can be, I don't know, maybe a type of Chihuahua or something.*0993

*That is kind of the idea of the flowchart.*1000

*Trapezoids, the birds, can never be type of a dog; so this is never.*1002

*A rhombus is always, sometimes, never, a parallelogram.*1012

*Maltese is always, sometimes, never, a dog; isn't it always?*1017

*If we are going to go up, then it is always.*1021

*A rectangle is always, sometimes, never a square.*1027

*Rectangles can just be rectangle; sometimes it could be a square.*1033

*Is a square always, sometimes, never a rectangle?*1044

*Because a square is a type of rectangle, it always has to be a rectangle.*1046

*That is it for this lesson; thank you for watching Educator.com.*1058

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over the area of a parallelogram.*0002

*First let's talk about area.*0008

*An area of a figure is the number of square units it encloses.*0010

*Another way to think of area is how much space it covers.*0017

*Let's say you have to cover your book.*0023

*That is all area because you are covering something.*0028

*It is how much space that you are covering.*0031

*If you have a hole in your jeans and you need to patch it up,*0033

*that is going to be area because it is the space that you are covering.*0037

*This, square units, it means how many 1 unit squares it covers.*0044

*This rectangle here, if I say that there are 8 square units,*0055

*that means each one of these squares, if it has a measure of 1 unit.*0066

*Units can be like centimeters, inches, whatever; this is 1; this is 1.*0076

*The area of this right here is 1 square unit.*0081

*How many square units is in this rectangle?--1, 2, 3, 4, 5, 6, 7, 8.*0086

*The area is 8 square units; it is how many square units it is covering.*0092

*If I say this is 1 inch, then this is 8 inches squared.*0106

*8 square units is 8 inches squared; that is area.*0115

*We know the area of a rectangle is base times height.*0121

*Area equals base times height.*0125

*A rectangle is a type of parallelogram; we learned that in the previous lesson.*0129

*A rectangle is a type of parallelogram; that formula applies to rectangles and to parallelograms.*0133

*Here this is a rectangle.*0143

*If this is the base, this is the height, we just multiply this side with side and we get the area.*0145

*We figure out how much space this is covering.*0151

*For parallelogram, if I maybe let's say I cut this whole part out.*0157

*This is the height; height always has to be perpendicular to the base.*0169

*This is the height; this is the base.*0173

*This whole thing is the base; height, base, perpendicular.*0176

*If I cut this piece out, say I am going to cut this out.*0182

*I take it over to this side; I glue it over here.*0190

*This is all going to be right here; then what do you get?*0201

*This part I cut out; then isn't this part a rectangle?*0206

*All this then becomes a rectangle; this is gone; this was moved over here.*0213

*A parallelogram covers the same amount of space as a rectangle.*0226

*So the formula is still the same.*0231

*Just make sure if you are going to find the area of a parallelogram,*0232

*you have to make sure that the height is perpendicular.*0236

*The height is from here to here; that is the height.*0240

*This right here cannot be the height.*0245

*It is like when you measure how tall you are,*0248

*if you measure your height, you have to be standing up straight.*0250

*You can't be slouching; you can't be leaning over to the side.*0254

*Same thing; the height of this parallelogram is not the side that is leaning over.*0258

*It has to be straight perpendicular; that is the height.*0265

*The first example, we are going to find the area of this rectangle.*0273

*We know it is a rectangle with four congruent sides, meaning four sides are the same.*0276

*That means this is actually a square; a square is a type of rectangle.*0283

*If this is 5, this is also 5.*0289

*The area is base times height which is 5*^{2} or 5 times 5.0294

*We know that is 25; then units, centimeters.*0307

*For area, because we are looking at how much space it covers,*0313

*it is centimeters squared because we are looking at base and height, two dimensions.*0317

*The area of this is 25 centimeters squared.*0324

*Find the area of the parallelogram.*0331

*The first one, this is 9 inches, 7 inches, and 6 inches.*0336

*The area is base times height.*0343

*Again remember the height and the base, they have to be perpendicular.*0347

*If I want to measure how tall the height of this perpendicular, I can't measure thi8s.*0352

*I can't measure it this way.*0357

*I have to make sure I measure it perpendicular, straight up and down.*0359

*The base will be 9; the height is going to be 6.*0366

*The area is 54 inches squared.*0374

*The next one, same thing; this is a parallelogram with four congruent sides.*0385

*We know that this a rhombus; the area is base times height.*0393

*Let's see; what is the base?*0405

*Even though we know that that is 2, that has nothing to do with our base.*0408

*The base is from here to here; that is 10.*0412

*Our height, even though the height is given to you on the outside of it,*0418

*it still measures from top straight down, perpendicular.*0422

*The height is 8; the area becomes 80 meters squared.*0428

*The next example, the base of a parallelogram is 10 inches.*0445

*The height is twice the base; find the area of the parallelogram.*0450

*If I draw a parallelogram, say there is my parallelogram.*0456

*The base is 10 inches; the height is twice the base.*0463

*Make sure you don't label this the height; the height has to be perpendicular.*0471

*You can draw a dotted line like that; that is twice the base.*0480

*Twice means 2 times the base; double the length of the base.*0486

*This is 2 times 10 which is 20.*0492

*The base is 10; it is twice; 2 times bigger, then it is 20 inches.*0498

*Area of this parallelogram is base times height; the base is 10.*0504

*The height is 20; 10 times 20 is 200.*0512

*It is in inches; it is inches squared.*0523

*The final example, find the area of the shaded region; we have two rectangles here.*0537

*This is the big one; here is the smaller one that is inside.*0546

*We are just trying to find the area of just the blue part, the shaded part.*0554

*That means I need to do two things.*0561

*I have to find the area of both rectangles; then I have to do what?*0566

*It is like saying... let's say I have a piece of paper.*0571

*Let's say this big rectangle is the piece of paper.*0574

*If I find the area of that this big rectangle,*0582

*that is going to be the area of that piece of paper, the whole thing.*0585

*But then I cut a rectangle out of that paper; it becomes white.*0589

*How would I figure this out?*0602

*I need to find the area of the big rectangle.*0605

*That is going to be everything.*0610

*If I find the area of the big rectangle, it is going to be this whole thing.*0611

*That is our piece of paper.*0616

*If I cut out another rectangle piece right there like that,*0618

*don't I subtract it?--because it is no longer there.*0625

*This base right here is empty; it is not being covered.*0629

*You have to subtract it; subtract the small rectangle; you are cutting it out.*0634

*That is going to be the area of the shaded.*0640

*Again the whole thing, the area of the big one is going to be 20 times 9.*0643

*The base times the height; 20 times 9.*0655

*That is... 20 times 9; 2 times 9 is 18.*0659

*Then I can just add a 0 at the end of that.*0665

*That is how you multiply numbers.*0667

*If I have a 0 at the end of a number that I am multiplying,*0668

*then I can just put that 0 at the end of my answer.*0672

*It is 20 times 9; you can just do that too; 0 and then 18.*0676

*That is where that 0 comes from; meters squared.*0681

*That is the area of this big one.*0688

*I can't say that is my answer because remember you cut out that little piece.*0691

*This part is not covering anything; it is an open spot.*0695

*To find the area of this rectangle, this is the area of just the first one.*0702

*Let's say that is the first one.*0710

*The area of the second one is 10; the base is 10.*0712

*Times, the height is 3; the area is 30 meters squared.*0721

*This is the part that we cut out.*0731

*I have to subtract it because it was originally covering this much space.*0734

*But then I cut out this much; I have to subtract it.*0740

*My area of the shaded becomes then 150 meters squared.*0745

*That is it for this lesson; thank you for watching Educator.com.*0760

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over the area of a triangle.*0002

*The formula for the area of a triangle is base times height divided by 2 or 1/2 base times height.*0008

*Here we have a parallelogram.*0018

*We know that area of a parallelogram is base times height.*0021

*Here is a rectangle; the area of this is base times height.*0029

*If I take this parallelogram and I cut it in half, let's say I cut it this way.*0035

*Then I have two equal halves; I then have a triangle.*0043

*One of these triangles would be the whole thing, the whole parallelogram, cut into half.*0049

*This triangle is base times height divided by 2 because I cut it in half.*0058

*Same thing here.*0066

*If I take this rectangle and I cut it in half, I am going to get the triangle.*0067

*That is why the formula for the area is the base times the height divided by 2.*0080

*Because it is cut in half; base times height, cut in half.*0085

*Here are a couple of triangles; we are going to find the area.*0095

*Again remember area is how much space it is covering.*0098

*We are going to see how much space this triangle is covering.*0101

*The area of this triangle has a formula of 1/2 base times height or base times height divided by 2.*0110

*The base we know is 8; remember base times height.*0122

*It is still the same as the previous lesson when we talked about parallelograms.*0126

*The base and the height have to still be perpendicular.*0131

*When we talk about height, we are talking about the perpendicular height from the highest point to the lowest point.*0134

*It has to be perpendicular; they have to be perpendicular to each other.*0141

*The base is 8; the height is not this side, this side right here.*0146

*It has to be this height; that is 6 inches.*0153

*It is all of that divided by 2.*0160

*8 times 6 is 48; divided by 2.*0164

*This looks like a fraction; but it is also divide.*0169

*48 divided by 2 is 24; our units is in inches; it is inches squared.*0172

*Because it is area, any time you are talking about area, it is always units squared.*0184

*That is the area of this triangle here.*0190

*The next one, area equals the base times the height divided by 2.*0194

*This looks like half of our rectangle we drew.*0203

*That rectangle; it is half of that.*0207

*Base times the height; the base is 5; the height is 10.*0212

*We know that is 10 because it is perpendicular.*0219

*But because we are only looking at half of it, the triangle part, we are going to divide that by 2.*0223

*Area equals 50 divided by 2; 50 in half is 25; centimeters squared.*0229

*Next, find the area of the figure.*0248

*There is no formula to figure out the area of this whole thing in one formula.*0255

*We have to break this up into two parts.*0262

*We know the area of a triangle.*0265

*We know the formula for the area of this rectangle.*0270

*If I put it together, I am going to add the area of this triangle to the area of this rectangle.*0274

*First the area of the triangle.*0290

*I am going to do a triangle plus a rectangle is going to equal...*0293

*All this plus all that is going to equal triangle with that.*0301

*Area of the triangle, triangle first, is 1/2 base times height or base times height divided by 2.*0313

*The base is 6 right here.*0329

*Even though the base is not the one on the bottom,*0332

*this has to be the base because the height and the base have to be perpendicular.*0337

*If you want, you can just redraw this triangle so that this becomes the base like that.*0344

*If this is 6, this side is this side right here.*0352

*Then that is the triangle; this can be 8.*0357

*But just because it is moved, it is rotated where this is right here, it doesn't change the area.*0362

*Base is 6; the height is 8 meters; divided by 2.*0373

*6 times 8 is 48; divided by 2; half of 48 is 24.*0382

*That is meters squared.*0391

*For the rectangle... because this is only the area of this.*0395

*To find the area of the rectangle, it is just base times the height and not divided by 2.*0400

*The base is 10; the height is 6; they are perpendicular; that is fine.*0408

*This is 60 meters squared; remember what you have to do.*0422

*Take the area of the triangle; add it to the area of the rectangle.*0429

*This is the rectangle.*0435

*It is going to be 24 meters squared plus 60 meters squared.*0442

*Together it is going to be 84 meters squared.*0452

*That is the area of this figure right here.*0458

*For this one, we are going to find the area of the shaded region.*0467

*This is different than the previous one because we had*0472

*two shapes that were put together to make up a figure.*0476

*This is different; this is overlapping.*0480

*Here we are just finding only the area of this right here, all this blue.*0484

*In this case, let's say we have a paper.*0492

*This blue, this whole rectangle here, let's say that is our piece of paper.*0499

*We have a piece of paper that is going to be blue like that.*0508

*We are going to take scissors and we are going to cut out a piece of it; that triangle piece.*0512

*Don't you remove some of the area?--you are uncovering some of the area.*0519

*You have to subtract the triangle there; the area of this minus the triangle.*0524

*That is going to give you that whole thing, cut out the triangle, all of this.*0536

*The previous one we had to add because they were put together.*0549

*But this one, we are going to subtract.*0552

*The area of this rectangle first; a rectangle.*0555

*We know that the formula for the area of a rectangle is base times height.*0564

*The base is 20; the height is 10.*0570

*That is going to be 200 inches squared.*0577

*Then we have to find the area of this triangle because how much space is the triangle using up?*0586

*Because that is how much we have to take away.*0595

*The triangle, area is a base times height divided by 2; the base is 5.*0598

*Again even though this is not the bottom, that is not the base,*0615

*we can still call that the base as long as the base and the height are perpendicular.*0620

*5; and then the height is 6; over 2.*0626

*5 times 6 is 30; divided by 2 is 15; inches squared.*0631

*We have the area of the rectangle and the area of the triangle.*0642

*Let's take the area of a rectangle and subtract, take away*0646

*the area of the triangle to see what is left in blue.*0651

*It is going to be 200 inches squared minus 15 inches squared.*0657

*If you do 200 minus 15, you are going to get 185 left.*0666

*185 inches squared, this will be the area of the shaded region.*0674

*That is it for this lesson; thank you for watching Educator.com.*0686

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over the circumference of a circle.*0002

*First let's go over some special segments within circles; the first is the radius.*0008

*Radius is a segment whose endpoints are on the center and on the circle.*0018

*One endpoint is on the center; the other endpoint is on the circle.*0029

*Here this segment CB or BC, doesn't matter which way, is a radius.*0034

*CA, that is also radius; I can say CA; I can say CB; I can say EC.*0043

*Those are all radius; each of those are radius; plural for radius is radii.*0056

*The next special segment is the diameter.*0071

*Diameter is a segment whose endpoints are on the circle.*0073

*It has to pass through the center.*0081

*It is like two radius put together like this back to back*0085

*to form a straight segment where each of the endpoints are on the circle.*0089

*That is a diameter; here EB, that is a diameter.*0095

*That is the only one for here.*0106

*DF, even though that segment has endpoints on the circle, it is not passing through the center.*0110

*So that is not considered a diameter.*0118

*That is actually called a chord; chord is like a diameter.*0120

*Diameter and chords, they both are similar in that they have their endpoints on the circle.*0128

*But diameter has to pass through the center; chords do not.*0134

*This is a chord; this is a diameter.*0139

*Again this is a radius; radius; this is a diameter; chord, this is a chord.*0145

*Endpoints on the circle without passing through the center, that is a chord.*0164

*The circumference is like perimeter.*0174

*We know perimeter is when you add up all the sides of some polygon.*0178

*Circumference acts as a perimeter.*0185

*But it is like the perimeter of a circle because circles, we don't have straight sides.*0187

*Instead of calling this perimeter, we call it circumference.*0196

*But it is pretty much the same thing; it is like you wrap around.*0200

*It is like if we need to build a fence around this garden.*0205

*We would call that perimeter because you are going around like this.*0214

*Let's say your garden is round like this.*0219

*Then it is not called perimeter anymore; it is called circumference.*0222

*But it is the same concept, same idea; distance around the circle.*0224

*You find that by multiplying the radius by 2 and multiplying that by π.*0230

*It is 2 times π times the radius.*0239

*When you multiply numbers together, see how we are just multiplying three numbers together.*0244

*2 and the π and the radius.*0248

*Whenever you multiply, it doesn't matter the order.*0250

*If we want, we can do 2 times π times the radius.*0253

*Or we can do radius times 2 times π or π times radius times 2.*0257

*The order doesn't matter when you multiply.*0263

*In short, this is circumference; it is 2πr; 2πr.*0270

*Since the order doesn't matter, since we are multiplying these three numbers together,*0278

*I can do 2 times r times π.*0284

*2r; if I take a radius and I multiply it by 2... that is one.*0290

*Here is another one; this is 2 times the radius.*0297

*R plus r is same thing as 2 times r; doesn't this become the diameter?*0302

*If we take 2 radius, this can also be diameter.*0308

*We can also say circumference is diameter times π; this actually has two formulas.*0315

*Circumference can be 2 times the π times the r.*0322

*Or it can be, since 2 times r equals the diameter, 2 times the radius is the diameter.*0327

*We can just say the diameter times π.*0333

*Doesn't matter which one we use; it depends on what they give us.*0341

*If we are given the radius, then we can just use 2πr.*0345

*They give us the diameter; you can just go ahead and multiply that by π.*0349

*Since you have to divide it, find the r, and then you have to multiply the 2 anyways.*0353

*If you are given radius, just use that.*0362

*If you are given diameter, just use that.*0363

*Again 2 times the π times the radius; π is 3.14; 3.14.*0366

*It is actually longer; but you only have to use 3.14.*0378

*The first example, we are going to name the given parts of the circle.*0388

*First is the chord.*0392

*Remember chord is a segment whose endpoints are on the circle.*0395

*But it doesn't pass through the center.*0400

*There is an endpoint on the circle; there is an endpoint on the circle.*0404

*A chord, you can say ED; it doesn't matter if I say DE.*0409

*Or I can say AB or BA.*0417

*The diameter, both endpoints on the circle; one, two; it passes through the center.*0422

*BE would be diameter; another one, AD is a diameter.*0434

*For radius, remember radius is endpoint on the center, endpoint on the circle.*0446

*That would be a radius; I can say CD; I can say BC.*0455

*I can say AC; I can say EC; I can say CE.*0465

*Find the circumference of the circle; this in the circle, this is the center.*0476

*This is the radius; this is the radius; it is 5.*0484

*The circumference of a circle is 2 times π times r, the radius.*0490

*It is 2 times, π is 3.14, times 5.*0500

*If you want, we can multiply this and this first.*0514

*Remember the order doesn't matter; it doesn't matter.*0517

*You can multiply this times this and then to that; it doesn't matter.*0520

*But I know that if I multiply 2 times the 5, then I get 10.*0523

*10 is an easy number to multiply with; C is 10 times 3.14.*0529

*I need to multiply these two numbers together; 3.14 times 10; 0.*0539

*1 times 4 is 4; 1 times 1 is 1; 1 times 3 is 3.*0549

*From here, since I am multiplying, how many numbers do I have behind decimal points?*0557

*I only have two.*0561

*I am going to in my answer put the decimal point in front of two numbers.*0563

*It is 31.4 or 31.40; it is the same thing.*0570

*I can just drop the 0 if I want to because it is behind the decimal point and at the end of a number.*0576

*Circumference is 31.4 or 31.40.*0584

*When I multiply by 10, there is a shortcut way of doing this.*0591

*When you multiply by 10, you see how many 0s there are.*0595

*10 has only one 0; you take the decimal point.*0600

*You are going to move it one space because there is one 0.*0605

*To determine if you are going to move it to the left or to the right,*0611

*if we are multiplying, don't we have to get a bigger number if we are multiplying by 10?*0615

*Our number has to get bigger.*0621

*If I move the decimal point to the left, my number is going to get smaller because 0.3 is not the same.*0623

*I want a bigger whole number.*0632

*I have to move it to the right to make my number bigger.*0635

*It is going to be 31.4.*0637

*Let's say I am going to multiply by 100; 100 has two 0s.*0642

*Then you would move it two spaces to the right to make it bigger.*0647

*It is going to be one, two; it is going to be 314.*0650

*That is going to be my answer; that is my circumference here.*0655

*Let's move on to the next problem.*0661

*Find the circumference of each circle with the given measure.*0666

*The first one, the radius is 9 inches.*0669

*Circumference equals 2πr, 2 times π times r.*0674

*2, π is 3.14, the radius is 9.*0684

*Again I like to multiply these two numbers first; you don't have to.*0697

*You can multiply this times this and then take that and multiply it to this again.*0701

*18; 2 times 9 is 18; times 3.14.*0707

*Now I have to multiply these two numbers; it is 3.14 times 18.*0713

*4 times 8 is 32; times 1 is 8; plus 3 is 11.*0721

*This is 24; 25; 1 times 4 is... I put a 0 up there.*0731

*1 times 4 is 4; 1 times 1 is 1; 1 times 3 is 3.*0738

*I can go ahead and add; 2 plus 0, 2.*0745

*This is 5; this is 6; this is 5.*0750

*Within my problem, how many numbers do I have behind decimal points?*0761

*I have two; from here, I am going to go one, two.*0765

*Place two numbers behind the decimal point for my answer.*0771

*My circumference becomes 56.52 inches.*0774

*It is not inches squared; only area is squared, units squared.*0789

*Circumference, you just leave it as 62.52 inches.*0794

*The next one, the diameter is 16 centimeters.*0800

*Remember if this is the formula, 2 times r becomes the diameter.*0804

*2 times radius is diameter.*0813

*I can just go ahead and say this formula is the same thing as diameter times π.*0816

*Diameter is 16; 16 times 3.14; here 3.14 times 16.*0826

*6 times 4 is 24; 6 times 1 is 6; plus 2 is 8.*0843

*6 times 3 is 18; place the 0 there; 1 times 4 is 4.*0851

*Times 1 is 1; 1 times 3 is 3; add; this is 4.*0859

*8 plus 4 is 12; 8, 9, 10; 3, 4, 5.*0865

*From here, I have two numbers total behind decimal points.*0874

*I am going to go one, two; for this one, my circumference is 50.24 centimeters.*0879

*My circumference here; and this is my circumference here.*0895

*That is it for this lesson; thank you for watching Educator.com.*0901

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over the area of a circle.*0002

*First to review over area, remember it is how much space it is covering up.*0007

*The area of a circle is when you have a circle and you see how much space it is using.*0013

*For example, let's say you have a hole in your jeans and you want to cover it up.*0024

*You cut out a circle from another pair of jeans let's say.*0033

*Then you stitch it on to your jeans to cover up your hole.*0040

*That, however much that circle, that patch is covering up, that is area.*0046

*It is just how much you are covering; how much space you are using.*0052

*Remember if you are measuring the distance around the circle, that is called circumference.*0056

*We have circumference which is the distance around the circle*0064

*and then area which is all of this, how much space it is using up.*0074

*The formula to find the area of a circle is π times the radius times the radius again.*0083

*In other words, the area is πr*^{2}; r^{2}.0090

*Be careful; this is not r times 2.*0097

*It is an exponent; that means it is r times itself that many times.*0100

*It is r*^{2}; radius times the radius.0105

*Circumference is 2 times π times r.*0110

*In this case, remember how we multiplied the 2 and the r together first.*0118

*In this case, this is 2 times r or r times 2.*0123

*This is not r times 2; this is r times r.*0126

*Remember keep in mind the difference between the formula for the circumference and the area.*0131

*First let's find the area of this circle.*0139

*The formula of area is π times r*^{2} or π times r times r.0143

*Remember π is 3.14; π, I am going to put in 3.14.*0152

*The radius is 4; 4*^{2}; again be careful; this is not 4 times 2.0162

*This is 4 times itself; 4 times 4.*0173

*Also for order of operations, because we have two different operations*0182

*meaning we have two different things we can do.*0190

*We can multiply; or we can do the exponent.*0192

*The order of operations, remember please excuse my dear aunt sally.*0196

*Parentheses, exponent, multiplication, division, addition and subtraction.*0203

*It is always parentheses first; exponents next; multiplication and division; addition and subtraction.*0211

*See how the exponent comes before multiplying.*0218

*Be careful; you do not multiply these two numbers first.*0224

*You always have to take care of the exponent first; then you can multiply.*0230

*3.14 times... 4*^{2} is 4 times 4 which is 16.0240

*Again remember do not multiply 3.14 times 4 and then square it.*0249

*If you do that, you are going to get the wrong answer.*0254

*Here I want to multiply 3.14 times 16.*0257

*4 times 6 is 24; 6 times 1 is 6; plus 2 is 8.*0266

*6 times 3 is 18; I put a 0 right here.*0272

*1 times 4, 4; 1 times 1, 1; 1 times 3, 3; then add.*0279

*4 plus 0 is 4; this is 12; 8, 9, 10; 3, 4, 5.*0290

*Since I am multiplying, I look at my problem.*0303

*I see how many numbers are behind the decimal point.*0307

*I only have two numbers behind decimal points.*0310

*In my answer, I am going to place two numbers behind the decimal point which is right there.*0314

*My answer becomes 50.24; I cannot forget my units; here it is inches.*0319

*Area is always squared; units squared; not numbers squared; units squared.*0333

*50.24 inches squared is my answer; that is the area of this circle.*0339

*Next example, here I am given that the diameter...*0348

*Remember diameter is a segment whose endpoints are on the circle; on the circle; on the circle.*0356

*And passes through the middle, the center of the circle.*0364

*This is a diameter; the diameter is 20 meters.*0368

*To find the area of a circle, area equals πr*^{2}, radius squared.0374

*I need to find the radius; I have the diameter; but I want the radius.*0382

*How do I find the radius if I am given the diameter?*0390

*The whole thing is 20; that is the diameter.*0393

*I know the radius is from the center to this point right there.*0395

*The radius is half the diameter.*0401

*If the whole thing is 20, then the radius has to be half of that which is 10.*0403

*Now I know my radius is 10.*0413

*I can go ahead and plug in my numbers and solve for my area.*0415

*π is 3.14; the radius is 10*^{2}.0420

*Again order of operations says we have to take care of the exponents before multiplying.*0428

*Area equals... I am going to leave this for the next step.*0437

*10*^{2} is not 10 times 2; it is not 20; be careful.0442

*It is 10 times 10 which is 100; remember the shortcut.*0446

*If we want to multiply by 10 or 100 or 1000 or 10000,*0457

*then you just count the number of 0s in that number.*0464

*Here I have two 0s; 100 has two 0s.*0467

*You are going to take this decimal point then.*0473

*Whenever you multiply a number to 100 or 10 or 1000, count how many 0s there are.*0477

*There is two; I am going to place this decimal point.*0483

*I am going to move it two spaces then.*0488

*Two 0s so I am going to move it two spaces.*0490

*Do I move it to the left or to the right two spaces?*0493

*Since I am multiplying by 100, this number has to get bigger.*0500

*The way to make the number bigger is to move the decimal point over to the right*0504

*because you want the whole number to be a bigger whole number.*0508

*I have to move it to the right two spaces; go one, two.*0512

*My answer then becomes... that is the new spot for my decimal point.*0516

*It is 314 is my answer; 314.*0522

*Again two 0s here; move it two spaces to the right.*0529

*It was here; it moved over to here, the end.*0534

*Since it is at the end, I don't have to write it.*0537

*It is just 314 point... same thing as if not being there.*0539

*314, you can leave it like that.*0545

*We are done solving; but I have to add my units now.*0550

*It is meters; area is always squared; units squared; 314 meters squared.*0552

*My third example, we are going to find the area of the shaded region.*0564

*I have this rectangle and a circle here that is cut out.*0572

*All this is missing; that is area.*0582

*If I cut it out, then don't I have to take it away?*0587

*I have to subtract it; it is as if I have this whole rectangle.*0591

*It was whole before the circle was cut out.*0599

*Find the area of the whole thing.*0602

*Then you are going to subtract the area of the circle.*0605

*That is going to become what you have left, the area that is shaded.*0609

*Imagine if this rectangle was like a piece of paper and you cut out a circle.*0617

*You have to figure out what is that area of the circle you cut out to see what you are taking away.*0625

*Find the area of rectangle; find the area of the circle; subtract it.*0632

*You will get the area of the shaded region.*0637

*The area of the rectangle; this is the rectangle.*0640

*Area is base times height or length times width; length times the width.*0645

*That is 8 times 7 which is 56 centimeters squared.*0659

*Centimeters squared is the area of this rectangle; that is that.*0674

*The area of the circle, πr*^{2}; π is 3.14; the radius is 2; 2^{2}.0683

*I am going to take care of this first.*0705

*Area equals 3.14... I am going to leave that; solve that out; that is 4.*0707

*3.14 times the 4; let's do that over here; 3.14 times 4.*0716

*4 times 4 is 16; 4 times 1 is 4; plus 1 is 5.*0723

*This is 12; I have two numbers behind the decimal point; one, two.*0730

*I need to place two numbers behind the decimal point in my answer.*0737

*Area equals 12.56 centimeters squared.*0742

*Now I have the area of the whole thing and then the area of the circle.*0752

*I need to take away the circle from the rectangle.*0755

*It is going to be 56 minus 12.56; I need to do that.*0760

*56 minus... remember when you subtract decimals, you have to line them up.*0774

*Where is the decimal in this number?*0782

*If you don't see it, it is always at the end right there.*0784

*Minus 12 point... make sure only when you add or subtract, the decimals have to line up... 56.*0788

*I am missing numbers here.*0800

*If I am missing numbers here, it is at the end of a number behind the decimal point, I can add 0s like that.*0802

*When I subtract, this is going to borrow; this becomes the 10; this becomes 9.*0811

*Borrow; 5; is that big enough?--yes.*0821

*10 minus this 6 is 4; 9 minus 5 is 4; point.*0827

*5 minus 2 is 3; 5 minus 1 is 4; it is 43.44.*0837

*This is 43.44 centimeters squared is my answer.*0849

*Again just find the area of the rectangle; then find the area of the circle.*0860

*I subtract it; I have to take the circle away; I have to subtract it.*0866

*Make sure your decimals line up when you subtract; you get this as your answer.*0872

*That is it for this lesson; thank you for watching Educator.com.*0881

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over prisms and cylinders.*0002

*A prism is the first type of polyhedron or solid that we are going to go over.*0009

*Polyhedron, that sounds like a big word; but poly means many.*0017

*We are used to hearing the word polygon.*0029

*Polygon remember is like a shape where we have many sides; polygon.*0031

*Polyhedron, it is a way of saying many faces.*0038

*We are going to go over face in a bit.*0045

*But polyhedron is just when you have a three-dimensional figure, three-dimensional object.*0046

*Each side has to be straight; it has to be a segment.*0054

*That is called a polyhedron.*0062

*Another name for these three-dimensional objects are called solids.*0069

*Solids, we can have round or circular sides.*0077

*But polyhedrons, that is the only difference between them.*0084

*They are pretty much the same thing.*0086

*Solids and polyhedron, they are both talking about three-dimensional objects.*0087

*Solids, they could be circles; the sides could be circles.*0091

*For polyhedrons, each side has to be straight.*0096

*They have to be segments, line segments; many faces.*0099

*Prism is the first type.*0107

*Prism is when you have a three-dimensional object with two opposite faces that are parallel and congruent.*0110

*It is talking about faces.*0122

*Each one of these sides like this right here, this right here, and this right here, those are all faces.*0124

*All the sides of the prism, the three-dimensional solid, is a face.*0138

*When two of those faces are parallel and congruent, the two opposite faces are parallel and congruent,*0148

*then you have what is called a prism.*0158

*Those two faces that are parallel and congruent, those two are called bases.*0162

*This base, we can label as the top face and the bottom face.*0174

*You can't really see a bottom face.*0183

*Those can be labeled as bases because see how these two sides are congruent and they are parallel.*0194

*So those could be labeled as bases.*0203

*Anytime you have just two opposite faces being parallel and congruent, you have a prism.*0211

*This is called a rectangular prism because the bases are rectangles.*0217

*You are probably thinking that these two sides, the left and the right side, are also parallel and congruent.*0224

*The front and the back sides, this front and the back, are also parallel and congruent.*0231

*Rectangular prisms, you can actually label any two opposite faces as your bases.*0237

*Let me give you an example of one that is not rectangular prism.*0246

*Say I have two, three, four, five; and then I have...*0250

*When I have something like this, this is also a prism.*0266

*The bases would be this top face and this bottom face because they are both parallel and congruent.*0279

*This is called a prism; more specifically what is the base in the shape of?*0291

*What is the polygon?--this is a pentagon.*0299

*This is actually called a pentagonal prism because the base is in the shape of a pentagon; five sides.*0303

*Another example would be triangle; let's say... like that.*0313

*The bases are not going to be top and bottom.*0331

*In this case, it doesn't have to be top and bottom.*0333

*There is nowhere where it says the bases of the prism has to be top and bottom.*0335

*As long as the two faces that are opposite are both parallel and congruent.*0342

*Meaning they are facing the same direction; they are not going to ever intersect.*0351

*They are congruent; they have to be exactly the same.*0357

*This is also a prism because we do have two sides that are parallel and congruent.*0359

*This, the base is in the shape of a triangle.*0367

*This is a called a triangular prism; these are prisms.*0370

*We are pretty much only going to go over the rectangular prism.*0375

*But just so you understand what a prism is, these are just a few examples.*0379

*Again this is called base; prisms always have two bases.*0384

*We have to have two bases.*0393

*The rest of the sides, the rest of the faces, the ones that are not bases, are called lateral faces.*0396

*This right here, this right here, this side right here,*0407

*this backside right here, those are all lateral faces.*0412

*This is a lateral face, lateral face, lateral face.*0418

*This triangle is a lateral face; those are all lateral faces.*0423

*All the sides of a prism are either considered bases or lateral faces.*0428

*Regular prism; regular we know is when all the sides are the same.*0436

*It is regular; all the sides are the same; all the faces are the same.*0442

*That is what it means for a prism.*0446

*If I have a regular prism, that means all the faces are exactly the same.*0448

*They are all congruent.*0452

*If I have a rectangular prism where all the sides are the same,*0455

*then we know that each side has to be a square.*0466

*Each side of this is congruent.*0471

*This, the name for this, there is a specific name; it is a cube.*0474

*I am sure you heard of that before; cube.*0481

*A cube is a rectangular prism that is regular; regular prism.*0484

*We went over base; we went over faces; each side is a face.*0494

*This right here, each of these segments are called...*0503

*let me just draw this base to this base so you know that that is what I am talking about.*0511

*Each of these sides, these segments, they are called edges.*0517

*Edge there; this is an edge; this is an edge.*0525

*All of those are called edges; sides are faces; these are edges.*0529

*This right here is like the vertex; but the plural word for vertex is vertices.*0537

*All of these make up the vertices of a prism.*0548

*We have base or another face.*0554

*Because each one of these are faces, more specifically, this is a base.*0566

*But these are all considered faces, edges, and vertices.*0571

*The next type of solid is a cylinder; you have seen the shape before.*0579

*Maybe a can of soup; that is a cylinder.*0586

*Lot of things; a cup could be a cylinder.*0592

*A cylinder is when we have two bases that are congruent and parallel circles.*0596

*It is almost the same thing as a prism where we have two opposite faces being parallel and congruent.*0603

*With the bases, they are called bases.*0612

*But in this case, the two bases have to be circles.*0615

*If they are circles, then it is a cylinder.*0619

*We know that this is a base and this has to be the base.*0622

*Those are circles; the altitude is like the height.*0631

*A height is how tall this is; how tall is it standing; this is H.*0638

*If you lay it down sideways so it is like this, make sure that this has to be the height.*0646

*It is from base to base; that is considered the height.*0655

*Again cylinder is a solid where we have two circles as the bases and this is the height.*0661

*The altitude is the height.*0672

*Our first example is classifying each prism by the shape of its bases.*0678

*This first one, it is almost the same as the one that we went over.*0686

*The base for this, it has a few different pairs that we can label as bases.*0694

*But you can't label all of them as bases.*0702

*Only two faces can be the bases; two faces can be bases.*0704

*If you want, you can label the top and the bottom as bases.*0710

*Or if you want, you can label the front and the back.*0714

*Or the left and the right sides; as long as it is only two sides.*0716

*But make sure it is not top and like left.*0722

*They have to be opposite sides; it has to be congruent.*0725

*I want to label my top and my bottom as my bases.*0730

*Figure out this shape of the base; it is a rectangle.*0738

*It is in the shape of a rectangle; this would be a rectangular prism.*0747

*The next one, here if you look at this side right here, the left side,*0760

*left side and right side, see how they are intersecting right here.*0769

*They can't be called the bases.*0773

*Even though they are the same, they are congruent,*0774

*they are not parallel because they are intersecting.*0778

*The bottom side, this rectangle right here, is not parallel and congruent with any other side.*0782

*So it has to be this triangle here; this triangle, this front and this back.*0790

*Those would be the two bases; again parallel and congruent.*0801

*That is in the shape of a triangle; the base in the shape of a triangle.*0809

*This is called triangular prism.*0815

*Then the third one, here even though...*0826

*This one is a little bit tricky because we do have opposite sides being parallel and congruent.*0835

*But they can't be... the lateral faces... I forgot to mention this to you guys.*0840

*But lateral faces, meaning the sides that are not bases, they have to be rectangular.*0845

*This top and this bottom are the only sides that are not rectangular.*0853

*They are not rectangles; they have to be the bases.*0858

*If you are having a difficult time identifying what sides are the bases,*0870

*just look for the sides that are not rectangles.*0876

*Look for the sides that are not rectangles.*0881

*You should look at this side, the top side and the bottom side, because they are not rectangles.*0885

*Let's say those two are not parallel and congruent.*0890

*Then it wouldn't be considered a prism; it is not a prism.*0894

*This with this bottom side are the bases.*0900

*This shape, one, two, three, four, five, six.*0909

*Remember a polygon with six sides is a hexagon.*0914

*This is a hexagonal; I just put a ?al; hexagonal prism.*0919

*That is the name for this right here; they are all prisms.*0929

*This is a rectangular prism; triangular prism; hexagonal prism; based on the shape of its bases.*0935

*Next example, name two different edges, bases, and vertices of the prism.*0945

*Remember what edges are, bases are, and vertices are.*0953

*Edges are like the edges of this prism, the segments.*0958

*Two different edges; AC and DE or ED.*0966

*It doesn't matter because either way you go from here to here.*0981

*Or here to here, it is the same thing.*0984

*I can say BE; any *