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# Understanding division of fractions

Using a number line, we'll explain the rule of "invert and multiply" when dividing two fractions. Created by Sal Khan.

Video transcript

Let's think about what
it means to take 8/3 and divide it by 1/3. So let me draw a
number line here. So there is my number line. This is 0. This is 1. And this is 2. Maybe this is 3 right over here. And let me plot 8/3. So to do that, I just need
to break up each whole into thirds. So let's see. That's 1/3, 2/3, 3/3,
4/3, 5/3, 6/3, 7/3, 8/3. So right over here. And then of course,
9/3 would get us to 3. So this right over here is 8/3. Now, one way to think
about 8/3 divided by 3 is what if we take this length. And we say, how
many jumps would it take to get there, if we're
doing it in jumps of 1/3? Or essentially, we're
breaking this up. If we were to break up
8/3 into sections of 1/3, how many sections would I have,
or how many jumps would I have? Well, let's think about that. If we're trying to
take jumps of 1/3, we're going to have to go 1,
2, 3, 4, 5, 6, 7, 8 jumps. So we could view
this as-- let me do this in a different color. I'll do it in this orange. So we took these 8
jumps right over here. So we could view 8/3 divided
by 1/3 as being equal to 8. Now, why does this
actually make sense? Well, when you're dividing
things into thirds, for every whole, you're
now going to have 3 jumps. So whatever value
you're trying to get to, you're going to have that
number times 3 jumps. So another way of thinking about
it is that 8/3 divided by 1/3 is the same thing
as 8/3 times 3. And we could either
write it like this. We could write
times 3 like that. Or, if we want to
write 3 as a fraction, we know that 3 is the
same thing as 3/1. And we already know how
to multiply fractions. Multiply the numerators. 8 times 3. So you have 8-- let me
do that that same color. You have 8 times 3 in the
numerator now, 8 times 3. And then you have 3 times
1 in the denominator. Which would give you 24/3, which
is the same thing as 24 divided by 3, which once
again is equal to 8. Now let's see if this
still makes sense. Instead of dividing by 1/3,
if we were to divide by 2/3. So let's think about what
8/3 divided by 2/3 is. Well, once again, this is
like asking the question, if we wanted to break up
this section from 0 to 8/3 into sections of
2/3, or jumps of 2/3, how many sections, or how many
jumps, would I have to make? Well, think about it. 1 jump-- we'll do this
in a different color. We could make 1 jump. No, that's the same
color as my 8/3. We could do 1 jump. My computer is doing
something strange. We could do 1 jump, 2
jumps, 3 jumps, and 4 jumps. So we see 8/3 divided
by 2/3 is equal to 4. Now, does this make sense in
this world right over here? Well, if we take 8/3 and we
do the same thing, saying hey, look, dividing by a
fraction is the same thing as multiplying by a reciprocal. Well, let's multiply by 3/2. Let's multiply by the
reciprocal of 2/3. So we swap the numerator
and the denominator. So we multiply it times 3/2. And then what do we get? In the numerator, once again,
we get 8 times 3, which is 24. And in the denominator, we
get 3 times 2, which is 6. So now we get 24 divided
by 6 is equal to 4. Now, does it make sense
that we got half the answer? If you think about the
difference between what we did here and
what we did here, these are almost the same,
except here we really just didn't divide. Or you could say you divided by
1, while here you divided by 2. Well, does that make sense? Well, sure. Because here you
jumped twice as far. So you had to take half
the number of steps. And so in the first
example, you saw why it makes sense
to multiply by 3. When you divide by a
fraction, for every whole, you're making 3 jumps. So that's why when you
divide by this fraction, or whatever is in
the denominator, you multiply by it. And now when the numerator
is greater than 1, every jump you're
going twice as far as you did in this first
one right over here. And so you would have to
do half as many jumps. Hopefully that makes sense. It's easy to think
about just mechanically how to divide fractions. Taking 8/3 divided by 1/3 is
the same thing as 8/3 times 3/1. Or 8/3 divided by 2/3 is the
same thing as 8/3 times 3/2. But hopefully this video
gives you a little bit more of an intuition of
why this is the case.