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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 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.