The idea of division | Multiplication and division | Arithmetic | Khan Academy

We've got 24 triangle
things right over here.

And what I want to
do in this video

is to divide it into
different numbers of groups.

So the first thing
I want to do is

I want to divide this 24
triangle things into 3 groups

and think about how many
do I end up per group.

So let's try that out.

So I'm going to divide
it into 3 equal groups.

So that is one equal
group right over there.

Then I have another equal
group right over here.

And then I have a third
equal group right over here.

So if I divide 24 into 3 equal
groups-- 1, 2, 3-- how many

are going to be in each group?

Well, we can count that.

We have 1, 2, 3, 4, 5,
6, 7, 8 in each group.

So we could say that 24
divided by 3 is equal to 8.

Now, you might say,
hey, this is very

similar to what we
saw in multiplication.

In multiplication, we said
if we have 3 groups of 8,

we could view that as
3 times 8 and get 24.

And you are exactly right.

Let me do those same colors--
we could also write that 3

times 8-- so if I
have 3 groups of 8,

that that is going
to be equal to 24.

So when we started in this
video, we had 24 things.

We want to divide it
into 3 equal groups.

We got 8 in each group, or you
could say 3 equal groups of 8

is equal to 24.

But there's even other ways
of thinking about this.

So let me clear this
up a little bit.

So let me clear that.

So in the first example, I
divided 24 into 3 equal groups.

But you could also
view 24 divided by 3

as dividing 24 into groups of 3.

So let's think about
what that looks like.

So if we divide it
into groups of 3,

then, for example,
this is a group of 3.

That is a group of 3.

This is a group of 3.

You might see where
this is going.

That's a group of 3.

That is another group of 3.

And we're going to think
about how many groups of 3

we're actually going to get.

So this is another group of 3.

And that's another group of 3.

So how many groups
of 3 did we get?

Let's see, we have 1, 2, 3,
4, 5, 6, 7, 8 groups of 3.

So another way of
viewing 24 divided by 3

is divide 24 into groups of 3.

And then you will
have 8 groups of 3.

And one way of
thinking about this--

if you want to express
the same thing in terms

of multiplication--
is if you have

8 groups of 3, that is also
going to be equal to 24.

Whether you have 3 groups
of 8 or 8 groups of 3,

either way, you're
going to have 24.

Now, let's make things
more interesting.

What I want you to think about
is, based on what we just saw,

what is 24 divided by 12?

And I encourage you
to pause the video,

draw out 24 triangles
like this, and try

to figure out what
24 divided by 12 is.

Well, I assume you've
paused the video.

And there's two ways to
think about 24 divided by 12.

You could say, well, let's
divide 24 into groups of 12

and think about how
many groups we have.

So we could do that.

So let's see.

This is one group of
12 right over here.

That's one group of 12, and then
here is another group of 12.

So how many groups
of 12 do we have?

Well, we have 2 groups of 12.

So we could say 24
divided by 12 is 2.

But another just as reasonable
way of doing this is you

could have said,
well, let me divide 24

into 12 groups instead
of groups of 12.

So if I want to divide
it into 12 groups,

12 equal groups--
well, let's see.

This is 1 equal group,
2 equal groups--

actually let me do it this way.

Well, let me do this-- 2 equal
groups, 3 4, 5, 6, 7, 8, 9, 10,

11, 12.

So once again, if you say, oh,
I'm going to divide 24 into 12

equal groups, how many do
you have in each group?

Well, you have 2.

So once again, 24
could be viewed

as 24 divided into
12 equal groups.

And how many do you
have in each group?

Or 24 divided into groups of
12, and how many groups would

you have?

And that's what we saw
in the last example.

So now, let's make things
even more interesting.

What I want you to think
about-- a couple of things.

I want you think about
what 24 divided by 6 is.

And I also want you to figure
out what 24 divided by--

let me use that
same color-- 4 is.

And once again, I encourage
you to pause the video,

draw these triangles,
and figure it out.

What is 24 divided by
6 and 24 divided by 4?

So let's tackle 24
divided by 6 first.

And let's try to divide
24 into 6 equal groups.

So let's see.

This could be 1 equal group,
2 equal groups-- in fact,

each group here is a group of 4.

And we have 6 rows.

So 3 equal groups, 4, 5, and 6.

And so if you divide
24 into 6 equal groups,

how many do you
have in each group?

Well, it's pretty
clear you have 4.

You have 4 In each group.

Another way we
could have thought

about that is we
could have said,

let me divide 24
into groups of 6.

So if you divided
24 into groups of 6,

you could have
viewed it like this.

So that's 1 group
of 6 right there.

That's another group
of 6 right over here.

That's another group of 6.

And I think you see how
many groups of 6 we have.

How many groups of 6 do we have?

We have 4.

We have 4 groups of 6.

Well, now let's think about
what 24 divided by 4 is.

Well, if I view 24 divided by
4 as taking 24 and dividing it

into 4 equal groups,
I've just drawn that.

I have 4 equal groups, and
in each group I have 6.

So notice 24 divided by 6 is 4.

24 divided by 4 is 6.

And that's because I could
view this as 4 groups of 6

or say that 4 times
6 is equal to 24.

Or you could just as
equivalently say that 6 times 4

is 24.

You could equivalently say
that 6 times 4 is equal to 24.