Geometry 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 40296

(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 number1116

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