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### Using Proportions and Ratios

- Ratio: A comparison of two quantities
- a to b
- a : b
- a/b
- Proportion: An equation with two equal ratios

### Using Proportions and Ratios

- m*15 = 5*3

- Assume the other two sides are x and y.
- [4/x] = [5/y] = [6/9]
- [4/x] = [6/9]
- 6x = 4*9
- x = 6
- [5/y] = [6/9]
- 6y = 5*9
- y = 7.5.

- 2*2(x + 4) = 3x + 5
- 4x + 8 = 3x + 5

With 3 dollars, we can get 2 bottles of coke. Find how many bottles we can get with 18 dollars.

- [dollars/bottles], [3/2] = [18/x]
- 3x = 2*18

Each box has 10 pencils, find how many pencils in 5 boxes.

- [boxes/pencils], [1/10] = [5/x]
- x = 5*10

- (3x + 2)*2 = 5*(x + 8)
- 6x + 4 = 5x + 40

- 6x + 7x + 8x = 180
- 21x = 180
- x = [60/7]
- 6*[60/7] = [360/7]
- 7*[60/7] = 60

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Using Proportions and Ratios

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Ratio 0:05
- Definition and Examples of Writing Ratio
- Proportion 2:05
- Definition of Proportion
- Examples of Proportion
- Using Ratio 5:53
- Example: Ratio
- Extra Example 1: Find Three Ratios Equivalent to 2/5 9:28
- Extra Example 2: Proportion and Cross Products 10:32
- Extra Example 3: Express Each Ratio as a Fraction 13:18
- Extra Example 4: Fin the Measure of a 3:4:5 Triangle 17:26

### Geometry Online Course

### Transcription: Using Proportions and Ratios

*Welcome back to Educator.com.*0000

*For this next lesson, we are going to talk about ratios and proportions.*0002

*First, what is a ratio? A ratio is a comparison between two things, most likely two parts or two quantities--two of something.*0006

*If the first thing is a, and the second thing is b, we have three ways that we can write ratios.*0018

*The first way is by saying a:b, and it is going to be read with the word "to", "a to b," "x to y."*0026

*You can write it like this; this is how it is mainly written--ratios are mainly written like this, a:b.*0038

*It could also be written as a fraction; most likely, instead of writing it like this, you would probably write it like this: a/b.*0046

*Fractions could be a ratio between a and b.*0053

*If you have, let's say, boys to girls, you can write it "boys to girls" like that, or you can write boys to girls.*0061

*You are comparing the number of boys and the number of girls.*0080

*So, if I say that there is a classroom with 30 students...let's say that 13 are boys and 17 are girls;*0085

*then you are going to write 0797; and make sure...if I ask for the ratio of boys to girls,*0096

*you have to give the number of boys before the number of girls.*0105

*You have to give the numbers in the order of the ratio: boys to girls is 13 to 17.*0110

*Or if it asks for the ratio of girls to boys, then you would have to say 17 to 13.*0117

*Then, a proportion would be two equal ratios--if you have a ratio equaling a ratio, then that becomes a proportion.*0126

*An equation with two equal ratios--we know that an equation is anything with an equals sign,*0136

*so, since we have a ratio, equals sign, ratio, that becomes an equation, which is a proportion.*0141

*So, here is one ratio, here: 2:3; if I have an equivalent ratio (equivalent just means anything that is the same, equal to)--*0149

*an equivalent ratio to that could be, we could say, 4/6; that is equivalent: 2:3 is equal to 4:6.*0162

*I can also say it is equal to 8/12, and so on; these are all equivalent ratios.*0174

*If I just have two of these ratios, then that becomes a proportion.*0186

*Now, to solve a proportion, if we have an unknown value in one of these--*0191

*because, since we know that the ratio is equal to the ratio, we can solve it using cross-products;*0195

*now, here, if I have a ratio a:b equal to the ratio c:d, then a and d--those two are called extremes.*0202

*a and d are called the extremes, and b and c are called the means.*0222

*The number up here, from the first ratio, and the number down here, the denominator of the second ratio, are the extremes.*0233

*And then, this denominator and this numerator are called the means.*0241

*So, for cross-products, you are going to multiply the extremes and make it equal to the product of the means.*0248

*It becomes ad = bc; so it is a times d, equal to b times c.*0258

*If you look at this example here, if we have 2/3 equal to 4/6; now, there is nothing for us to solve, because we don't have any unknown values.*0275

*But just to check, just to do our work here, just to see that they are equivalent ratios,*0290

*I am going to cross-multiply (these are the extremes): 2 times 6 is 12; I am going to make it equal to 3 times 4*0296

*(and that is a 3; it looks like an 8); 3 times 4 is 12.*0307

*So, see how they equal each other; and we know that this is a correct proportion, because cross-products work.*0313

*Now, if you have an unknown value--let's say that we didn't know that this was 4--*0322

*it equals x/6; then again, you solve the cross-products: 2 times 6 is 12, equal to 3 times x, which is 3x.*0329

*How do you solve for x? Make sure you divide the 3; then we know that 4 is equal to x, and we already know that that is 4, because it is there.*0339

*But that is how you would solve for unknown values.*0349

*Now, let's use ratio; this is kind of a problem that a lot of students struggle with.*0355

*When you have a triangle, three angles of a triangle--the ratio is 4 to 6 to 8; there are three of them there.*0362

*It is OK; you can have 3; you can have 4; 4 to 6 to 8--that is the ratio--that is the relationship between the three angles.*0380

*It doesn't mean that this is 4 degrees, this is 6 degrees, and this is 8 degrees.*0387

*No, just the ratio between them is 4 to 6 to 8; and we have to find the measure of each angle.*0392

*Now, since we know that, no matter what this angle is, that number that I multiplied by 4...*0399

*So then, we just know that it is 4 times something; 4 times something is going to be this angle right here.*0410

*Since the ratio of this angle is 4, this one is 6, and this one is 8, no matter what these angles are, they have to keep that same relationship.*0419

*That means that, for every 4 of this, there is 6 of that, and there is 8 of that.*0428

*So, let's say I am going to make it x; that means that this angle is going to be 4x, because I have to multiply a value for this.*0434

*So, 4 times something is going to be this angle; then, 6 times...it has to be that same number; and this is also going to be 8x.*0446

*4x...and we know that the three angles of a triangle add up to 180, so this is going to be 4x + 6x + 8x = 180,*0458

*because we know that, again, whatever this angle is, whatever that angle is, and that angle--we know that they have to add up to 180.*0472

*And it is not 4 degrees; it is going to have to be a larger number.*0479

*We know that it is going to be larger than 4, and then again, for every 4 of this, there is going to be 6 of that and 8 of that.*0487

*This x is like that; it is like that word "that"; so what is it that these numbers are being multiplied by?*0497

*We are going to solve for x here: 4x + 6x is 10x, plus 8x is 18x; that is equal to 180; divide the 18; x is 10.*0506

*This angle measure is 4 times 10, which is 40 degrees; then this one is 6 times 10, which is 60 degrees; and this one would be 80 degrees.*0526

*It is like whatever these angles are--when you simplify them all the way, if you divide all of them by the same number, you are going to get 4:6:8.*0540

*So then, you are just finding that number that was divided from each of those.*0552

*x is 10, and it is asking for the measure of each angle, so we have that: here is 40, 60, and 80 degrees.*0557

*Another example, the first example, actually: Find three ratios equivalent to 2/5.*0569

*Now, this is an easy one, because all we have to do is find any ratio; there are going to be plenty; we just have to find three that are equivalent.*0575

*2/5: now, we can multiply each of these; we have to multiply it by the same factor.*0587

*So, let's say 2 times 2; that is 4; 5 times 2 is 10; so 2/5 is equivalent to 4/10.*0595

*There is one; how about another one? 6/15--there is one.*0607

*And then, we have an 8: 2 times 4 is 8; 5 times 4 is 20; there are three equivalent ratios.*0617

*Solve each proportion by using cross-products: here we have an unknown x.*0633

*We are going to use cross-products; again, these are called the extremes; these are called the means.*0640

*If you have a hard time remembering them, this is a; that is the first one that you would refer to, so this is a.*0648

*And then, the one that involves the a is extremes; and then, this is the second one, because it is bc; that would be the means.*0657

*Since, if we say that this is a, then we would probably refer to this one first, and e comes before m in the alphabet;*0671

*this one right here would be the one that you call out first.*0688

*You are not going to call this ratio first; so from here, this is the numerator;*0692

*so then, the one that is involving that one is the extremes.*0697

*E is before m in the alphabet; so it is extremes, and then these two are the means.*0702

*Then, you multiply those two; it becomes 17x =...that is 77; to solve for x, I divide by 17, so x equals 77/17.*0709

*And that doesn't simplify, so I can leave it like that.*0729

*The next one: here, I am going to do the same thing; I multiply; I am going to use cross-products.*0733

*This way--that becomes 8 times (x + 4) equals 7 times x, which is 7x.*0744

*I am going to do 8x + 32 = 7x; then here, if I subtract the 8x, then I get 32 = -x.*0755

*And then, x, therefore, is equal to -32, so that would be the value of x.*0769

*8x + 32 = 7x, and then you are just moving the 8x over, and x becomes -32.*0778

*And that is all you have to find; you don't have to solve for anything else--just find x.*0788

*That is it for this example: the next one: express each ratio as a fraction in simplest form.*0797

*The first one: 2 inches on a map represent 100 miles.*0805

*You know that, if you have a map in front of you--like a map of a city or something--*0814

*and then you look, and then each inch, or every two inches, would actually be 100 miles in real life; "find the ratio involving one inch"--*0820

*well, this is a pretty simple problem: 2 inches is 100 miles; 1 inch would be how many miles?*0833

*It is just half of it; but it is just so that you can represent it using ratios.*0841

*The ratio for #1: Remember, we talked about part-to-part, or something to something else, a to b.*0852

*So, what would be your ratio that you are going to use to represent #1?*0859

*Well, we can say inches to miles; that would be the ratio that we are going to be using.*0863

*You could do miles to inches, if you want to, but that is going to change your proportion,*0875

*because, remember: with a ratio, if you say "boys to girls," then you have to name boys first, because you said "boys" first.*0878

*If you say "girls to boys," then you have to mention girls first.*0885

*Inches to miles: we can make this into a proportion by doing the number of inches (that is 2), over...*0890

*what is the number of miles?...2 inches to 100 miles; that is the ratio.*0899

*Now, I need an equivalent ratio: the one inch goes on top; that is inches that we mentioned first--1 inch to how many miles?*0906

*This one would be x; that is how you would set up a proportion for that problem.*0916

*This is just that ratio that you are basing it on.*0922

*Make sure that the top number represents inches, and the bottom number represents miles.*0925

*And then, you are going to cross-multiply 2x = 100, and then divide it to x = 50.*0930

*The same thing works for #2: With 10 gallons of gas, Sarah can drive 280 miles; find how many miles she can travel on 2 gallons.*0942

*We have a ratio between gallons and miles.*0957

*Now again, you can do miles over gallons, or you can do gallons over miles; it doesn't matter.*0963

*Let's do gallons over miles; that is going to be the ratio that we are going to base this problem on.*0970

*Then, what is the first ratio? 10 gallons of gas to what--how many miles?*0978

*Per 10 gallons, you can travel 280 miles; that would be your ratio that represents gallons to miles.*0986

*Then, find how many miles you can travel on 2 gallons.*0993

*We had to write the 2 on the top, because that is the gallons; and then we are going to find...2 gallons to how many miles?--that is x.*0996

*So then, cross-multiply: 10x is equal to 560; I am going to divide the 10; x is equal to 56.*1010

*So then, if 10 gallons equals 280 miles, then 2 gallons would be 56 miles, based on the same ratio.*1026

*And the fourth example is very similar to what we did earlier, finding the ratio.*1047

*Before, it was three angles; now we are going to look at the ratio of three sides.*1057

*It works the same way: again, the ratio would be 3:4:5.*1061

*Here (it doesn't mean that this is the length of the side), it is as if...let's say that this side was actually 6.*1072

*The ratio is 3, but let's say this side's length is 6--then what would this length be?*1085

*It has to be 8, and then this has to be 10.*1094

*Or if this was, let's say, 9, this would have to be 12, and this would have to be 15.*1099

*No matter what these sides are, it has to be the same ratio--it has to keep that ratio,*1109

*meaning that if you are going to multiply this number to this, then that same number*1116

*has to be multiplied to all three, to keep that same ratio.*1123

*So again, that number is going to be x: here, 3x + 4x + 5x...and in this case, it is not like the angles of a triangle,*1128

*where we know that all three angles add up to 180.*1144

*In this case, they have to tell us that the perimeter (because that is this side plus this side plus that side) is 60.*1148

*We add all of this up: here is 9; here is 12; 12x is equal to 60.*1156

*When you divide the 12, x is equal to 5; so if x is 5, that means if the ratio is 3:4:5, this shortest side is going to be 3(5), which is 15.*1165

*And then, it is 4 times 5, which is 20, and 5 times 5, which is 25.*1185

*So then, these three are the actual lengths of the sides.*1192

*If you add them up, they are going to add up to 60, which is perimeter; and it keeps the same ratio of 3:4:5.*1196

*That is it for this lesson on ratio; we will see you next time.*1205

*Thank you for watching Educator.com.*1209

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