If you need to learn the basics of math or are simply looking to brush up on your skills, Professor Mary Pyo’s Basic Math course is the perfect solution. Targeting 5th to 6th grade math, Mary begins with easy to understand explanations of concepts. She then follows up with many examples that include many topics seen in the actual classroom. Mary begins with Expressions and Equations then moves onto topics such as Fractions, Ratios, Geometry, Graphing, and Probability. Professor Pyo received her Master’s in Educational Curriculum and Instruction, her specialized credentials in Foundational mathematics, and has been teaching for over 10 years.

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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 expression0136

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 280267

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 base0129

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 be0167

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 exponent0178

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 see0188

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 place0128

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 50174

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 number0547

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

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 them0318

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 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 wrong0562

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

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 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 earns0170

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 number0129

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 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 20427

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 this0588

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 see0639

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 be0847

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 number0164

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 fraction0248

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

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

The next one, median; remember the median, the keyword is middle.0588

Be careful here, the most common mistake for this one0595

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 see0937

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

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

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

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

If a point is on let's say right here, that doesn't fall under any quadrants.0720

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

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

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

The trick here is to figure out whether my x-coordinates are going to be positive or negative0830

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

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 shirt0027

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 M0217

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 event0350

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 order0396

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 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 up0587

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 outcomes0600

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 one0763

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 60805

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 events0002

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

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 for0058

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 multiply0289

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

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 affected0553

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 A0581

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 be0666

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 be0681

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 multiply1185

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

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 yellow1456

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 happening1505

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 means1537

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

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 afterwards1822

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 blue1842

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 151922

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 then1945

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

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 that0704

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

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 much0351

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 like0509

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 thing0586

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

But again we still have to understand that I can find the probability0636

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 subtract0985

the percent of the probability that she is going to pass the test0994

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 marble1110

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 3/81178

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 52 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 22 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 32 times 22.0412

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

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 here0200

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 40265

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 490525

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

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

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 method0699

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 different0380

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 1/20731

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 smallest0752

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 10069

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 number0178

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 fraction0283

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 number0693

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

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

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 dollar0377

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 answer0080

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 top0530

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

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

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 denominator1250

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

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

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

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 multiples0531

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

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 fractions0983

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 back0020

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

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 here0213

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

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

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

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 that0787

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

From here, you can just multiply your numerators.0934

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 sure1146

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

If you want to change it to a mixed number, you can go back a few lessons1201

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

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

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

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 top0233

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

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 that0987

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

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

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 feet0605

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 kilometers0918

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 numbers0053

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 bigger0441

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

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 positive0785

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

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 there0834

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 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 problem0019

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

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

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

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 plus0667

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 plus0677

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 that0007

if one of the two numbers is negative, then your answer is going to be negative.0014

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

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

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

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

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 that0059

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, 32 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 32; 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; 32 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

23; 23, be careful, 23 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

If any of these problems were confusing to you, if you forgot how to do some of these0646

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 something0538

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 itself0204

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

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 math0237

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

Then my answer becomes a negative.0506

If I have two negatives signs, whether it is a negative times a negative0513

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

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 together0810

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 steps1084

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

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 divide0324

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

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

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 to0114

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

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 both1057

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 41564

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 then1809

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

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 them0036

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 fractions0117

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 different0428

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

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 have0022

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 top0039

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 bottom0077

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 miles0109

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

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 do0510

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

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 want0060

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 probability0371

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 bottom0630

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 girls0658

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 it0345

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

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 decimal1381

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

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

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 be0067

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 point0140

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 10306

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 right0337

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 point0466

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 equation0565

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 equation0603

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 spaces0889

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

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 left0091

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 because0349

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

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 problems0929

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 numbers1009

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 out1020

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

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 again1820

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

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 left0307

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 years0491

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 year0659

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 bigger0904

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

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 bank1004

and that 16 dollars that the bank paid her.1009

In all, to find the total amount that she has, you can just do1013

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 account1052

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 dollars1070

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 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 point1472

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 tax0002

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 that0074

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

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 you0207

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 end0284

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 dollars0424

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 have0538

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 dollars0613

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 to0688

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 decimal0749

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 tax0838

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 sure0991

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 lines0119

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

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 degrees0356

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 angles0690

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 are0851

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

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 angle0045

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

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

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 B0190

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 together0263

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 C0317

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 1800528

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 out0708

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 same0754

because the distance from here to here and the distance from here to here0759

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 other0879

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 arrow0105

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 one0126

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 comparing0515

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

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

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

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 52 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 had0472

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 away0646

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

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 back0085

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

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

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

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 circle0064

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 πr2; r2.0090

Be careful; this is not r times 2.0097

It is an exponent; that means it is r times itself that many times.0100

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 r2 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; 42; 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 operations0182

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... 42 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 πr2, 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 102.0420

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

102 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 right0504

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, πr2; π is 3.14; the radius is 2; 22.0683

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

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

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 one of these edges, you can name.0987

Faces... I am going to use a different color; faces.0994

Faces, any one of these faces; I can say CDBE; this is a rectangle.1005

I can say rectangle CDEB; or I can say triangle ACB.1018

If you recognize that this is also the base.1039

But remember all these sides are just called faces.1043

Face is another word for saying sides; face.1046

The last one, vertices; vertices, think of vertex, those points right there.1053

I can say point B; point E; those are vertices.1065

The third example, name the solid for each object.1080

We are going to see what shape, what the name of the solid is for each of these.1086

The first one, a can of soup.1092

We know a can of soup looks like this; this will be cylinder.1094

A shoebox, we know that a shoebox is in the shape of a prism.1105

More specifically, it would be a rectangular prism.1111

Camping tent; this one...1116

I am just going to draw it out for you guys so you guys can see a little bit better.1119

Yes, there is different versions of camping tents.1123

But this main one like this if you can see that... horrible at drawing... like that.1125

This would be... look at the bases; the bases are triangles.1145

This is a triangular prism.1152

A roll of paper towels; paper towels, it looks like this.1161

It has that, a little bit longer; it has the hole in the middle; like that.1167

This one would be cylinder.1178

The fourth example, we are going to write true or false for each statement.1189

The first one, a cylinder has congruent bases.1193

Cylinder, we know it must have congruent bases.1199

Or else it is not going to be called a cylinder.1202

The bases are congruent circles; so this is true.1207

A triangular prism has three faces; triangular prism, like this; does it have three faces?1217

Just the front and the back, this right here, just the bases alone, there is two.1233

But then you have to think that there is this side; that is three.1242

The other side which is this, this other side, that is four.1248

Then the bottom side, that is five; each one of those is faces.1256

Does it have three face?--no; this one is false.1263

A cube is a rectangular prism.1271

Remember a cube is a regular prism where all the sides are the same.1275

There is a cube; is it a rectangular prism?1281

Are the bases in the shape of rectangles?1285

It is actually in the shape of squares.1290

But isn't a square a type of rectangle?1292

So this one is true; cube is a rectangular prism.1295

That is it for this lesson; thank you for watching Educator.com.1306

Welcome back to Educator.com.0000

For the next lesson, we are going to go over the volume of a rectangular prism.0002

Volume unlike area is looking for the measurements of the space inside.0010

We talked about area and surface area.0021

Area is always just the space that it is covering.0027

But volume, it has to do with a solid, three-dimensional solid,0031

and all the space that it is covering inside.0035

If I were to take this rectangular prism, this box, and fill it with something,0037

fill it with sand or fill it with water, that would measure the volume.0043

In the volume of a prism, whether it is a rectangular prism or a triangular prism,0050

any type of prism, it is going to be this formula here: the area of the base times the height.0055

For rectangular prism, we have different pairs that we can label as the base.0064

We can label the top and the bottom as the base.0077

Remember for prisms, the base has to be parallel and congruent.0080

There is two bases.0086

It is going to be the opposite sides that are parallel and congruent.0089

Rectangular prism has a three different pairs of sides that are opposite, parallel, and congruent.0093

It is really up to you which sides you want to label as the base.0103

If I say that the top and the bottom, let's just call these the bases.0115

This top and this bottom are the bases0120

because they are opposite sides and they are parallel and congruent.0127

We are going to find the area of the base; then multiply that to the height.0132

If the area of the base is the length times the width,0141

from here, if we call this the length, we call this the width,0144

it is going to be this measure times this times the height.0149

Let's say that this right here has measures of 5; let's say this is 5.0162

The area of the base, length times the width, let's call that the area of the base.0170

That is going to be 25; let's say that the height is also 5.0174

25 times 5; that is going to be the height.0182

The volume of this is going to be 25 times 5 which is 135.0187

Once you find the volume, we know area is units squared.0195

Volume is going to be units cubed.0199

Anytime you are dealing with volume, it is always going to be units cubed.0205

If I said centimeters, 5 centimeters, then it is going to be 125 centimeters cubed.0209

Let's do a few examples; the first one, find the volume of the rectangular prism.0218

Because it is a rectangular prism, we know it is just length times the width times the height.0223

Those three measures multiply together.0229

If you want to think of it as the area of the base times the height, you can call this the base.0233

We are going to find the length times the width times the height.0243

Those three measure multiplied together is the volume.0247

Length times the width times the height.0254

We are going to say 10 meters times 4 meters times 5 meters.0261

10 times 4 is 40; 40 times 5; this 4 times 5 is 20.0274

20 and then I am going to include that 0; 40 times 5 is 200.0286

Volume is meters cubed; that is the volume of this rectangular prism.0292

Find the volume of a cube.0302

We know a cube is a special type of rectangular prism0304

and that all the sides, all the faces, are congruent.0307

All six sides are congruent.0312

Here this is 2 kilometers; this is 2 kilometers; each face is a square.0317

This is 2; then this is going to be 2.0325

We know that this is also going to be 2.0329

The volume is 2 times 2 times 2 which is...0332

2 times 2 is 4; 4 times 2 is 8.0341

The volume of this cube will be 8; we see that it is kilometers cubed.0345

For the third example, we are going to find the volume of the solid.0357

If you look here, we have two rectangular prisms and they are stacked on each other.0361

Whenever you have two different solids like this, we are going to find the volume of each one.0367

Then we can add them together.0373

It is like the volume of this bottom rectangular prism plus the volume of the top prism.0374

Let's say prism one is the one on the bottom.0383

Prism number one, volume is going to be this measure, 4 times 10.0389

Let's say that is the base.0399

I am going to color that red for the base.0402

Area of the base, 4 times 10; then times that right there.0405

4 times 10 times the other measure of 10.0414

We know 4 times 10 is 40; 40 times 10...0423

Remember whenever we multiply number to 10, we can just0432

take this number and then add this 0 to that same number.0438

40 times 10 is 400; that is meters cubed.0443

This prism here, prism number two, we can label this top one as the base.0450

The area of that... if this is 6, this side and this side are the same.0461

This side with this side are congruent.0469

If this is 6 meters, then this is also going to be 6 meters.0472

The area of the base is going to be 6 times 6.0477

The height is 2 meters; this is 36 times 2 which is 72 meters cubed.0485

I have the volume of both prisms.0505

Now I am going to add them together to find the volume of the whole solid, whole thing.0508

400 meters cubed, that is the volume of the first one.0515

Plus 72 meters cubed is going to be 472 meters cubed.0520

That is the volume of this whole thing.0531

That is it for this lesson; thank you for watching Educator.com.0536

Welcome back to Educator.com.0001

For the next lesson, we are going to go over volume of a triangular prism.0002

Volume remember is the measure of all the space inside the prism or the solid.0010

Whenever you take a solid, a three-dimensional object,0021

and you fill it with something, you are measuring volume, all the space inside.0025

Just like for the rectangular prism, to find the volume of a triangular prism,0033

you are going to find the area of the base and multiply that to the height.0039

For a triangular prism, we know that the bases are triangles0044

which is why it is called the triangular prism.0049

When we find the area of the base, we are finding the area of the triangle.0052

The height, be careful with the height because... the triangle has a height.0058

But the height that we have to use here, it is not the height for the triangle.0065

It is the height of the prism; it is the height between the two bases.0070

Let me explain that a little bit more.0076

For a volume of a prism, it is always area of the base,0082

whatever the base is, times the height of the solid, of the prism.0086

Since we are dealing with triangular prisms, the area of the base is going to be the area of the triangle.0093

Here is the base.0102

We know that the area of a triangle is 1/2 base times height.0105

That is the area of a triangle.0112

Be careful; like I said, this height is not the same as this height.0115

The height here, this is just talking about the triangle; only the triangle.0119

The height here is going to be the height of the prism.0125

We know that this a base and this is a base.0128

The height is going to measure the distance between the two bases, this base here to the back base.0133

This right here is going to measure the height here.0143

Again area of the base which is the triangle, 1/2 base times height, times the height of the prism.0149

That is the volume of this triangular prism.0157

Here if you look at our example, this is triangular prism.0164

It might be a little bit difficult to see.0171

But just think of this as the top; that is the peak.0172

That is the top part; this bottom, that is the ground.0178

This triangular prism, the base would be the triangles.0189

Again just to review over what makes a prism,0198

remember that we have two bases that are both parallel and congruent0203

which will be these two triangles on opposite sides here.0210

There is two sides; there is two bases always for every prism.0213

They are always going to be the non-rectangle.0217

If you have a prism and you have couple of different pairs that can be opposite, parallel, and congruent,0220

you always look for the ones that are not rectangles because the bases are those.0231

All the rest of the sides, all the lateral faces, the sides that are not bases,0239

the lateral faces are going to be rectangles.0244

If you look here, see that is the rectangle.0246

This is a rectangle; and then the bottom side will be a rectangle.0249

Again those are my bases.0256

To find the volume, we are going to find the area of the base.0260

For the area of the base, I can label that as capital B.0267

If I do a lowercase B, then that is talking about just a base, just one side.0271

That is considered the base.0276

But a capital B represents the area of a base like a side.0277

That is called the base; capital B is for area of the base.0282

Then times the height.0287

You can also think of the volume of a prism in this formula right there.0288

Again triangular prism, our base is going to be the triangle.0295

1/2 base times height of the triangle times the height of the prism.0301

I am going to just put P here for prism.0308

That is the height of the prism.0312

That is just to show what type of height it is.0313

For my triangle, my base and my height; remember that between these two, base and the height,0320

whenever you find the area of a triangle, the base and the height have to be perpendicular.0329

For this, one of these will be considered the base.0333

The other one will be considered the height.0337

6 and 8 will be my base and my height.0339

Just remember that the base and the height, they have to be perpendicular to each other.0343

1/2 times 6 times 8... I don't like the...0350

You can think of it that way; 1/2 base times height.0360

The height of the prism again is going to be the measure between the two bases.0363

This is going to be the height of the prism; that is 10.0370

1/2 times 6; 1/2 times 6; think of this as 6/1.0378

Half of 6 is 3; you can also do 1/2 times 6.0386

You are going to multiply whenever you do fractions.0390

1 times the 6; you are going to multiply the top numbers together.0392

Then multiply the bottom numbers together.0396

1 times 6 is 6; over... 2 times 1 is 2; 6/2 is 3.0398

3 times 8 is 24; times that by 10.0405

Remember whenever you multiply a number times 10,0415

you are just going to add a 0 at the end of it like that.0417

Our units is centimeters.0424

Remember whatever you do with volume, it is always units cubed.0426

Meaning to the 3rd power; centimeters cubed, 3rd power.0430

That is the volume of this triangular prism.0436

The next example, again we have volume being the area of the base0443

Capital B for area of the base times the height of the prism.0449

The reason why I am just adding it like this is because0456

this is the formula for any type of prism, not just triangular prism.0458

But it could be hexagonal prism; it could be rectangular prism.0462

Any type of prism, this will be the volume.0470

Just whatever the base is, you are going to find the area of that base.0473

Then multiply to the height of the prism.0477

Here triangular prism, this is the area of the trian