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## Arithmetic

### Course: Arithmetic > Unit 15

Lesson 1: Fractions as division# Understanding fractions as division

Sal shows how a/b and a÷b are equivalent. That is, the fraction bar and the division symbol mean the same thing. Created by Sal Khan.

## Want to join the conversation?

- so we can do division or multiplecarion(17 votes)
- yes you can but it also depends on what it is asking you to do on a problem.(2 votes)

- If you upvote me to 100 I will donate 150$ to khan Academy.(17 votes)
- can somebody get this to 10 likes plzzzz(16 votes)
- Why? Also no sorry. Next time give a good reason to, but if you give me a reason that sounds legitimate THAT I will upvote.(0 votes)

- just anotha day of khan making things complicated FOR NO REASON💯🔥🗣🙏(14 votes)
- honestly i just put the vid on 2x speed and it still counts💅(1 vote)

- If you still don't understand how to divide fractions. I will show you. There are 3 steps.

So say you were doing 1/3 divided by 6 or 6 divided by a third.

We will do 1/3 divided by 6 first.

Step 1: Keep the number 1/3

Step 2: Switch the division symbol to multiplication

Step 3: Find the reciprocal of 6. And its 1/6 so 1/3 times 1/6 is 1/18. Sometimes you need to simplify.

Now 6 divided by 1/3

Keep 6.

Turn to muliplication

Switch 1/3 to 3/1

its easier as 6/1x3/1 and that's 18/1 which can be reduced to 18

Hope this helped(9 votes)- Bruh what I want to know is how to divide a fraction with a fraction that has a different denominator.(1 vote)

- Same can you get me a moon badge you need to give me 10 likes(5 votes)
- Any tips? I would love that please! I still cant do it myself.(3 votes)
- Swap the numerator and the denominator, (the top and bottom numbers in a fraction) on the second fraction. WHATEVER YOU DO, DO NOT DO IT FOR THE FIRST FRACTION. For example, 1/2 x 3/9, so it would be re-written as 1/2 x 9/3, and then you just multiply it, which would be 9/6, and then, you divide the numerator by the denominator, so just think, "how many times can 6 go into 9? And how much is left over?" So, if you were to do that, the answer would be 1 3/9, which, by the way, was you second fraction that you swapped. And, if you know how to, then you would simplify, and your final answer would be 1 1/3. (Oh, and by the way, the word for swapping the numerator and the denominator is called recipricals. So if you ever see that word in a test or a math question, then you know what it is.(4 votes)

- when it says 5 millimeter can fit in a bottle of perfume how do you answer it when it says how many 2 millimeters can fit in there it is too R one how do you write that so that i dont get the answer wrong again and again.(3 votes)
- You’re right that one way we could write the answer to that problem is 2 R 1, but this lesson is focusing on
*fractions*in division. How can we represent the answer with a fraction instead?

We could use an improper fraction, 5/2. We could also use a mixed number, 2 1/2.

Does that help?(4 votes)

## Video transcript

When we were first exposed to
multiplication and division, we saw that they had an
inverse relationship. Or another way of
thinking about it is that they can
undo each other. So for example, if I had 2 times
4, one interpretation of this is I could have
four groups of 2. So that is one group of 2, two
groups of 2, three groups of 2, and four groups of 2. And we learned many, many
videos ago that this, of course, is going to be equal to 8. Well, we could express a very
similar idea with division. We could start with 8 things. So let's start with one, two,
three, four, five, six, seven, eight things. So now we're going
to start with the 8. And we could say, well,
let's try to divide that into four groups,
four equal groups. Well, that's one equal
group, two equal groups, three equal groups,
and four equal groups. And we see when we
start with 8 divide it into four equal
groups, each group is going to have
2 objects in it. So you probably see
the relationship. 2 times 4 is 8. 8 divided by 4 is 2. And actually, if we did 8
divided by 2, we would get 4. And this is generally true. If I have something
times something else is equal to whatever their product
is, if you take the product and divide by one of
those two numbers, you'll get the other one. And that idea
applies to fractions. It actually makes a lot
of sense with fractions. So for example, let's say
that we started off with 1/3 and we wanted to
multiply that times 3. Well, there's a couple of
ways we could visualize it. Actually, let me just
draw a diagram here. So let's say that this
block represents a whole, and let me shade in a 1/3 of it. So that's 1/3. We're going to multiply by 3. So we're going to
have 3 of these 1/3's. Or another way of
thinking about it, it's going to be 1/3 plus
another 1/3 plus another 1/3. That's our first 1/3, our
second 1/3, and our third 1/3. And we get the whole. This is 3/3, or 1. So this is going
to be equal to 1. So you use the exact same idea. If 1/3 times 3 is
equal to 1, then that means that 1 divided by
3 must be equal to 1/3. And this comes straight
out of how we first even thought about fractions. The first way that we ever
thought about fractions was, well, let's
start with a whole. And that whole would be our 1. And let's divide it into 3
equal sections, the same way that we divided this
8 into 4 equal groups. So if you divide this
into 3 equal sections, the size of each
of those sections is going to be exactly 1/3. Now, this leads to an
interesting question that might be popping
in your brain. Notice, we have 1
is the numerator, 3 is the denominator,
and we just said that this is equal
to the numerator divided by the denominator. 1 over 3 is the same
thing as 1 divided by 3. Is this always true
for a fraction? Well, let's just do the
same thought experiment, but let's do it with
a different fraction. Let's take 3/4 and
multiply it by 4. So multiply it by 4. So once again, let's see
if I could draw 1/4 here. Let me do this in a new color. So let's say that this block
right over here is a whole. We'll divide it into
four equal sections. So now I've divided
it into fourths. And let me copy and paste it
so I can use it multiple times. So copy. All right. Now, 3/4, that's
going to be-- we can assume-- I didn't
draw it perfectly. Actually, I could draw it a
little bit better than that just to make the four equal
sections actually look equal. So that looks like a
little bit better of a job. I'm trying to make them
four equal sections. Let me copy that one. So let me use it for later. Now, 3/4. This is four equal sections, and
3/4 represents three of them-- one, two, three. But now we're going
to multiply it by 4. So we're going to
have 3/4 four times. So we're going to need
some more wholes here. So let's throw in another whole. So this is one 3/4. Now let me do the next
3/4 in another color. So that's a 1/4, that's a
second 1/4, that's a third 1/4. That's another 3/4. And now let's do-- so we've
done two 3/4 just now. Let me make it clear. This is the first 3/4,
and then this plus this is the second 3/4. Now let's do a third 3/4. And we're going to have to use
another whole right over here. And I will do that
in this color. So my third 3/4,
so here's a 1/4, here's my second 1/4,
here's a third 1/4. So in green, I have another 3/4. And now we need four 3/4. So let's do that in a color I
have not used yet, maybe white. So that's a 1/4, that's two
1/4, and that is three 1/4. So notice, now I have now I have
one 3/4, two 3/4, three 3/4, and four 3/4. And what did I do when
I got those four 3/4? Well, it's pretty clear. This is turned into 3 wholes. So this is equal to 3 wholes. Well, if 3/4 times
4 is equal to 3, that means that 3 divided
by 4 is equal to 3/4. So the same idea again. 3 over 4 is the same
thing as 3 divided by 4. And in general, this is true. The fraction symbol here can
be interpreted as division. And looking at this
diagram right here, it made complete sense. If you started with
3 wholes, and you want to divide it into 4 equal
groups, one group, two groups, three groups, four
groups, each group is going to have 3/4 in it.