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# Introduction to ratios

What a ratio is. Simple ratio problems. Created by Sal Khan.

Video transcript

Let's try to learn a thing or two about ratios. So ratios are just expressions that compare quantities. So that might just be a fancy way-- let me just --of saying something that you may or may not understand. So let me give you actual examples. If I have ten horses and I have five dogs. And someone were to come to me and say Sal, what is the ratio is of horses to dogs? So I want to know how many horses do I have for some number of dogs. So I could say I have ten horses for every five dogs. So I could say the ratio of horses to dogs is ten:five. Or I could also write that as a fraction. I could say the ratio of horses to dogs is ten / five. Or I could just write it out. I could say it is ten to five. These all are saying the same thing. And the thing I write first, or the thing that I write on top, is the number of horses. So this is the number of horses right there. If I wanted to talk about the number of dogs, this is the number of dogs, that's the number of dogs, or that's the number of dogs. I'm just-- These are all just expressions that are comparing two quantities. Now, I just said I have ten horses for every five dogs. But what does that mean? That means I have five horses for every one dog. Sorry. Not five horses for every one dog. It means I have two horses for every one dog, right? If for five dogs I have ten, that means that for every one of these dogs, there are two horses. For every one of-- every two of these horses, there's one dog. I just kind of reasoned through that. So this is-- I have two horses for every one dog. But how do you get there? How do you get from ten:five to two:one? Well you can think of what's the biggest number that divides into both of these numbers? What's their greatest common divisor? I have a whole video on that. The biggest number that divides into both of these guys is five. So you divide both of them by five and you can kind of get this ratio into a reduced form. And if I write it here, it would be the same thing as two / one or two to one. And so what's interesting about ratios, it isn't literally, or doesn't always have to be literally, the number of horses and the numbers of dogs you have. What a ratio tells you is how many horses do I have for every dog. Or how many dogs do I have for every horse. Now, just to make things clear, what if someone asked me what is the ratio of dogs to horses? So what's the difference in these two statements? Here I said horses to dogs. Here I'm saying dogs to horses. So, since I've switched the ratio-- What I'm looking for-- I'm looking for the ratio of dogs to horses, I switch the numbers. So dogs-- For every five dogs, I have ten horses. Or if I divide both of these by five, for every one dog, I have five horses. So the ratio of dogs to horses is five:ten or one:five. Or you could write it this way. one to-- I can write it-- Let me write it down here. one / five. Or I could write one to five. And the general convention-- This wouldn't be necessarily incorrect. That's not wrong. But the general convention is to get your ratio or your fraction, if you want to call it that, into the simplest form or into this reduced form right there. Let's just do a couple of other examples. Let's say I have twenty apples. Let's say I have forty oranges. And let's say that I have sixty strawberries. Now what is the ratio of apples to oranges to strawberries? I could write it like this. I could write what is the ratio of-- I'll write it like this --apples:oranges:strawberries? Well I can start off by literally saying, well for sixty strawberries-- for every sixty strawberries, I have forty oranges and I have twenty apples. And this would be legitimate. You could say the ratio of apples to oranges to strawberries are twenty:forty-- Sorry. twenty:forty:sixty. And that wouldn't be wrong. But we saw before, we could put into reduced form. So we think of what's the largest number that divides into all three of these? We can't just do it into two of these now because now my ratio has three actual quantities. Well the largest number that divides into all of these guys is twenty. If we divide all of them by twenty, we can then say for every one apple, I now have-- you divide this guy by twenty --I have two oranges, and I have three strawberries. So the ratio of apples to oranges to strawberries is one:two:three. And I got that, in every case, by just dividing these guys all by twenty. I divided by twenty. I think you get the general idea. If someone were to ask you what's the ratio of-- Let me just write it down because it never hurts to have a little bit more clarification. If someone wanted to know the ratio of strawberries to oranges-- Let me get into my orange color. Strawberries to oranges to apples. I thought I was going to do that in yellow. To apples. What is this ratio going to be? Well for every three strawberries, I have two oranges and I have one apple. So then it would be three:two:one. The general idea is whatever order someone asks you for the different items, you put-- the ratio is going to be in that same exact order. Now, in all of the examples so far I gave you the number of quantity-- the quantity of things we had and I-- we figured out the ratio. What if it went the other way? What if I told you a ratio? What if I said the ratio of boys to girls in a classroom is-- Let's say the ratio of boys to girls is two / three. Which I could've also written as two:three just like that. So for every two boys, I have three girls or for every three girls, I have two boys. And let's say that there are forty students in the classroom. And then someone were to ask you how many girls are there? How many girls are in the classroom? So this seems a little bit more convoluted than what we did before. We know the total number of students and we know the ratio. But how many girls are in the room? So let's think about it this way. The fact that the ratio of boys to girls-- I'll write it like this. Boys-- Maybe I'll be stereotypical with the colors. The ratio of boys to girls is equal to two:three. Hate to be so stereotypical, but it doesn't hurt. two:three. The ratio of boys to girls is two:three. So this stands for every three girls, there's two boys. For every two boys, there's three girls. But what does it also say? It also says for every five students, there are what? There are two boys and three girls. Now why is this statement helpful? Well how many groups of five students do I have? I have forty students in my class right there, right? I have forty students in my class. And for every five students, there are two boys and three girls. So how many groups of five students do I have? So I have a total of forty students. Let me do it in this purple color. I have forty students and then there are five students per group. And I figured out that group just by looking at the ratio. For every five students, I have two boys and three girls. How many groups of five students do I have? So that means that I have eight groups-- forty divided by five --I have eight groups of five students. Now we're wondering how many girls there are. So each group is going to have three girls. So how many girls do I have? I have eight groups, each of them have three girls. So I have eight groups times three girls per group is equal to twenty-four girls in the classroom. And you could do the same exercise with boys. How many boys are there? There's a couple of ways you could do it. You could say for every group, there are two boys. There's eight groups. There's sixteen boys. Or you could say there's forty students. twenty-four of them are girls. forty minus twenty-four is sixteen. So either way you get to sixteen boys. And if you want to pick up a fast way to do it. It would be identical. You'd say look, two plus three is five. For every five students, two boys, three girls. How many groups are there? You say forty divided by five is equal to eight groups. Every group has three girls. So you do eight times three is equal to twenty-four girls. Let's do one that's a little bit harder than that. Let's do one where I say that the ratio of let's say-- Well let's go back to the farm example. The ratio of sheep-- I'll do sheep in white. The ratio of sheep to-- I don't know --chickens to-- I don't know. What's another farm animal? --to pigs. The ratio of sheep to chicken to pigs-- Maybe I should just say chicken right there. The ratio of sheep to chicken to pigs-- Or chickens. I should say chickens. Is-- Let's say the ratio is two:five:ten. And notice, I can't reduce this anymore. There's no number that divides into all of these. So this is the ratio if sheep to chickens to pigs. And let's say that I have a total of fifty-one animals. And I want to know how many chickens do I have. Well we do the same idea. For every two sheep, I have five chickens and I have ten pigs. That tells me for every seventeen animals-- So every group of seventeen animals, what do I have? And where did I get seventeen from? I just added two plus five plus ten. For every seventeen animals, I'm going to have-- Let me pick a new color. I'm going to have two sheep, five chickens, and ten pigs. Now, how many groups of seventeen animals do I have? I have a total of fifty-one animals. So if there's seventeen animals per group, fifty-one animals divided by seventeen animals per group. I have three groups of seventeen animals. Now I want to know how many chickens. Every group has five chickens. We already know that. And I have three groups. So I have three groups. Every group has five chickens. So I'm going to have three times five chickens, which is equal to fifteen chickens. Not too bad. All I did is add these up and say for every seventeen animals, I've got five chickens. I've got three groups of seventeen. So for each of those groups, I have five chickens. three times five is fifteen chicken. You could use the same process to figure out the number sheep or pigs you might have.