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# Introduction to ratios

## 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 10 horses and I have 5 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 10 horses for every 5 dogs. So I could say the ratio of horses to dogs is 10:5. Or I could also write that as a fraction. I could say the ratio of horses to dogs is 10/5. Or I could just write it out. I could say it is 10 to 5. 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. That's the number of horses. And that's all the number of horses. 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 10 horses for every 5 dogs. But what does that mean? That means I have 5 horses for every 1 dog. Sorry. Not 5 horses for every 1 dog. It means I have 2 horses for every 1 dog, right? If for 5 dogs I have 10, that means that for every 1 of these dogs, there are 2 horses. For every one of-- every 2 of these horses, there's 1 dog. I just kind of reasoned through that. So this is-- I have 2 horses for every 1 dog. But how do you get there? How do you get from 10:5 to 2:1? 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 5. So you divide both of them by 5 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 2/1 or 2 to 1. 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 5 dogs, I have 10 horses. Or if I divide both of these by 5, for every 1 dog, I have 5 horses. So the ratio of dogs to horses is 5:10 or 1:5. Or you could write it this way. 1 to-- I can write it-- Let me write it down here. 1/5. Or I could write 1 to 5. 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 20 apples. Let's say I have 40 oranges. And let's say that I have 60 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 60 strawberries-- for every 60 strawberries, I have 40 oranges and I have 20 apples. And this would be legitimate. You could say the ratio of apples to oranges to strawberries are 20:40-- Sorry. 20:40:60. 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 20. If we divide all of them by 20, we can then say for every 1 apple, I now have-- you divide this guy by 20 --I have 2 oranges, and I have 3 strawberries. So the ratio of apples to oranges to strawberries is 1:2:3. And I got that, in every case, by just dividing these guys all by 20. I divided by 20. 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 3 strawberries, I have 2 oranges and I have 1 apple. So then it would be 3:2:1. 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 2/3. Which I could've also written as 2:3 just like that. So for every 2 boys, I have 3 girls or for every 3 girls, I have 2 boys. And let's say that there are 40 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 2:3. Hate to be so stereotypical, but it doesn't hurt. 2:3. The ratio of boys to girls is 2:3. So this stands for every 3 girls, there's 2 boys. For every 2 boys, there's 3 girls. But what does it also say? It also says for every 5 students, there are what? There are 2 boys and 3 girls. Now why is this statement helpful? Well how many groups of 5 students do I have? I have 40 students in my class right there, right? I have 40 students in my class. And for every 5 students, there are 2 boys and 3 girls. So how many groups of 5 students do I have? So I have a total of 40 students. Let me do it in this purple color. I have 40 students and then there are 5 students per group. And I figured out that group just by looking at the ratio. For every 5 students, I have 2 boys and 3 girls. How many groups of 5 students do I have? So that means that I have 8 groups-- 40 divided by 5 --I have 8 groups of 5 students. Now we're wondering how many girls there are. So each group is going to have 3 girls. So how many girls do I have? I have 8 groups, each of them have 3 girls. So I have 8 groups times 3 girls per group is equal to 24 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 2 boys. There's 8 groups. There's 16 boys. Or you could say there's 40 students. 24 of them are girls. 40 minus 24 is 16. So either way you get to 16 boys. And if you want to pick up a fast way to do it. It would be identical. You'd say look, 2 plus 3 is 5. For every 5 students, 2 boys, 3 girls. How many groups are there? You say 40 divided by 5 is equal to 8 groups. Every group has 3 girls. So you do 8 times 3 is equal to 24 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 2:5:10. 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 51 animals. And I want to know how many chickens do I have. Well we do the same idea. For every 2 sheep, I have 5 chickens and I have 10 pigs. That tells me for every 17 animals-- So every group of 17 animals, what do I have? And where did I get 17 from? I just added 2 plus 5 plus 10. For every 17 animals, I'm going to have-- Let me pick a new color. I'm going to have 2 sheep, 5 chickens, and 10 pigs. Now, how many groups of 17 animals do I have? I have a total of 51 animals. So if there's 17 animals per group, 51 animals divided by 17 animals per group. I have 3 groups of 17 animals. Now I want to know how many chickens. Every group has 5 chickens. We already know that. And I have 3 groups. So I have 3 groups. Every group has 5 chickens. So I'm going to have 3 times 5 chickens, which is equal to 15 chickens. Not too bad. All I did is add these up and say for every 17 animals, I've got 5 chickens. I've got 3 groups of 17. So for each of those groups, I have 5 chickens. 3 times 5 is 15 chicken. You could use the same process to figure out the number sheep or pigs you might have.