WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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For this next lesson, we are going to talk about ratios and proportions.
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First, what is a ratio? A **ratio** is a comparison between two things, most likely two parts or two quantities--two of something.
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If the first thing is a, and the second thing is b, we have three ways that we can write ratios.
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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."
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You can write it like this; this is how it is mainly written--ratios are mainly written like this, a:b.
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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.
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Fractions could be a ratio between a and b.
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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.
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You are comparing the number of boys and the number of girls.
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So, if I say that there is a classroom with 30 students...let's say that 13 are boys and 17 are girls;
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then you are going to write 0797; and make sure...if I ask for the ratio of boys to girls,
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you have to give the number of boys before the number of girls.
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You have to give the numbers in the order of the ratio: boys to girls is 13 to 17.
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Or if it asks for the ratio of girls to boys, then you would have to say 17 to 13.
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Then, a proportion would be two equal ratios--if you have a ratio equaling a ratio, then that becomes a proportion.
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An equation with two equal ratios--we know that an equation is anything with an equals sign,
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so, since we have a ratio, equals sign, ratio, that becomes an equation, which is a proportion.
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So, here is one ratio, here: 2:3; if I have an equivalent ratio (equivalent just means anything that is the same, equal to)--
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an equivalent ratio to that could be, we could say, 4/6; that is equivalent: 2:3 is equal to 4:6.
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I can also say it is equal to 8/12, and so on; these are all equivalent ratios.
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If I just have two of these ratios, then that becomes a proportion.
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Now, to solve a proportion, if we have an unknown value in one of these--
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because, since we know that the ratio is equal to the ratio, we can solve it using cross-products;
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now, here, if I have a ratio a:b equal to the ratio c:d, then a and d--those two are called extremes.
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a and d are called the extremes, and b and c are called the means.
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The number up here, from the first ratio, and the number down here, the denominator of the second ratio, are the extremes.
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And then, this denominator and this numerator are called the means.
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So, for cross-products, you are going to multiply the extremes and make it equal to the product of the means.
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It becomes ad = bc; so it is a times d, equal to b times c.
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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.
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But just to check, just to do our work here, just to see that they are equivalent ratios,
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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
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(and that is a 3; it looks like an 8); 3 times 4 is 12.
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So, see how they equal each other; and we know that this is a correct proportion, because cross-products work.
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Now, if you have an unknown value--let's say that we didn't know that this was 4--
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it equals x/6; then again, you solve the cross-products: 2 times 6 is 12, equal to 3 times x, which is 3x.
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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.
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But that is how you would solve for unknown values.
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Now, let's use ratio; this is kind of a problem that a lot of students struggle with.
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When you have a triangle, three angles of a triangle--the ratio is 4 to 6 to 8; there are three of them there.
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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.
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It doesn't mean that this is 4 degrees, this is 6 degrees, and this is 8 degrees.
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No, just the ratio between them is 4 to 6 to 8; and we have to find the measure of each angle.
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Now, since we know that, no matter what this angle is, that number that I multiplied by 4...
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So then, we just know that it is 4 times something; 4 times something is going to be this angle right here.
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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.
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That means that, for every 4 of this, there is 6 of that, and there is 8 of that.
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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.
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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.
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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,
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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.
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And it is not 4 degrees; it is going to have to be a larger number.
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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.
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This x is like that; it is like that word "that"; so what is it that these numbers are being multiplied by?
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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.
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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.
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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.
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So then, you are just finding that number that was divided from each of those.
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x is 10, and it is asking for the measure of each angle, so we have that: here is 40, 60, and 80 degrees.
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Another example, the first example, actually: Find three ratios equivalent to 2/5.
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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.
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2/5: now, we can multiply each of these; we have to multiply it by the same factor.
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So, let's say 2 times 2; that is 4; 5 times 2 is 10; so 2/5 is equivalent to 4/10.
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There is one; how about another one? 6/15--there is one.
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And then, we have an 8: 2 times 4 is 8; 5 times 4 is 20; there are three equivalent ratios.
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Solve each proportion by using cross-products: here we have an unknown x.
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We are going to use cross-products; again, these are called the extremes; these are called the means.
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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.
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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.
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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;
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this one right here would be the one that you call out first.
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You are not going to call this ratio first; so from here, this is the numerator;
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so then, the one that is involving that one is the extremes.
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E is before m in the alphabet; so it is extremes, and then these two are the means.
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Then, you multiply those two; it becomes 17x =...that is 77; to solve for x, I divide by 17, so x equals 77/17.
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And that doesn't simplify, so I can leave it like that.
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The next one: here, I am going to do the same thing; I multiply; I am going to use cross-products.
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This way--that becomes 8 times (x + 4) equals 7 times x, which is 7x.
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I am going to do 8x + 32 = 7x; then here, if I subtract the 8x, then I get 32 = -x.
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And then, x, therefore, is equal to -32, so that would be the value of x.
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8x + 32 = 7x, and then you are just moving the 8x over, and x becomes -32.
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And that is all you have to find; you don't have to solve for anything else--just find x.
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That is it for this example: the next one: express each ratio as a fraction in simplest form.
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The first one: 2 inches on a map represent 100 miles.
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You know that, if you have a map in front of you--like a map of a city or something--
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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"--
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well, this is a pretty simple problem: 2 inches is 100 miles; 1 inch would be how many miles?
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It is just half of it; but it is just so that you can represent it using ratios.
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The ratio for #1: Remember, we talked about part-to-part, or something to something else, a to b.
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So, what would be your ratio that you are going to use to represent #1?
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Well, we can say inches to miles; that would be the ratio that we are going to be using.
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You could do miles to inches, if you want to, but that is going to change your proportion,
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because, remember: with a ratio, if you say "boys to girls," then you have to name boys first, because you said "boys" first.
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If you say "girls to boys," then you have to mention girls first.
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Inches to miles: we can make this into a proportion by doing the number of inches (that is 2), over...
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what is the number of miles?...2 inches to 100 miles; that is the ratio.
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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?
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This one would be x; that is how you would set up a proportion for that problem.
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This is just that ratio that you are basing it on.
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Make sure that the top number represents inches, and the bottom number represents miles.
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And then, you are going to cross-multiply 2x = 100, and then divide it to x = 50.
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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.
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We have a ratio between gallons and miles.
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Now again, you can do miles over gallons, or you can do gallons over miles; it doesn't matter.
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Let's do gallons over miles; that is going to be the ratio that we are going to base this problem on.
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Then, what is the first ratio? 10 gallons of gas to what--how many miles?
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Per 10 gallons, you can travel 280 miles; that would be your ratio that represents gallons to miles.
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Then, find how many miles you can travel on 2 gallons.
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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.
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So then, cross-multiply: 10x is equal to 560; I am going to divide the 10; x is equal to 56.
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So then, if 10 gallons equals 280 miles, then 2 gallons would be 56 miles, based on the same ratio.
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And the fourth example is very similar to what we did earlier, finding the ratio.
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Before, it was three angles; now we are going to look at the ratio of three sides.
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It works the same way: again, the ratio would be 3:4:5.
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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.
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The ratio is 3, but let's say this side's length is 6--then what would this length be?
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It has to be 8, and then this has to be 10.
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Or if this was, let's say, 9, this would have to be 12, and this would have to be 15.
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No matter what these sides are, it has to be the same ratio--it has to keep that ratio,
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meaning that if you are going to multiply this number to this, then that same number
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has to be multiplied to all three, to keep that same ratio.
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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,
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where we know that all three angles add up to 180.
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In this case, they have to tell us that the perimeter (because that is this side plus this side plus that side) is 60.
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We add all of this up: here is 9; here is 12; 12x is equal to 60.
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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.
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And then, it is 4 times 5, which is 20, and 5 times 5, which is 25.
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So then, these three are the actual lengths of the sides.
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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.
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That is it for this lesson on ratio; we will see you next time.
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Thank you for watching Educator.com.