Visualizing fraction multiplication

Let's think a little
bit about what

it means to multiply fractions.

Say I want to multiply
1/2 times 1/4.

Well, one way to
think about this

is we could view
this as 1/2 of a 1/4.

And what do I mean there?

Well let me take a whole,
let me take a whole here,

and let me divide
it into fourths.

So let me divide
it into fourths,

so I'll divided into
4 equal sections.

And so 1/4 would be 1 of
these 4 equal sections.

But we want to take 1/2 of that.

So how do we take half of that?

Well, we could divide this
into 2 equal sections,

and then just take 1 of them.

So divide it into
2 equal sections,

and then take 1 of them.

So we're taking this pink area,
this whole pink area is 1/4,

and now we're going
to take 1/2 of it.

We're now going
to take 1/2 of it.

So that's this yellow
square right over here.

But what fraction of the whole
does this yellow represent?

Well, it now represents 1
out of 1, 2, 3, 4, 5, 6, 7,

8 equal sections.

So this right over here, this
represents 1/8 of the whole.

And so we see conceptually
that 1/2 times 1/4,

it completely makes sense,
that 1/2 of 1/4 should be 1/8.

And it hopefully makes
sense that you get this 8

by multiplying
the 2 times the 4.

You started with
4 equal sections,

but then you divided each
of those 4 equal sections

into 2 equal sections.

So then you have 8
total equal sections

that you split your whole into.

Let's do another
example, but now let's

multiply two
fractions that don't

have 1's in the numerator.

So let's multiply, let's
multiply 2/3 times 4/5.

And I encourage you
now to pause the video

and do something very
similar to what I just did.

Try to represent 4/5
of a whole and then try

to represent 2/3 of that
4/5 and see what fraction

of the whole you actually have.

So pause now.

So let's think about this.

Let's represent 4/5.

So if I have a
whole like this, let

me try to divide it
into 5 equal sections.

5 equal sections, so let's say
that is 1 equal section, that

is 2 equal sections, that is
3, 4, and 5-- I can do a better

job than this.

This is always the hard part.

I'm trying my best to
make them look, at least,

like equal sections--
2, 3, 4, and 5.

I think you get the point here.

I'm trying to make
them equal sections.

And we want 4/5.

So we want 4 of these
5 equal sections.

So this would be 1 of the 5
equal sections, 2 of them,

3 of them, and then 4 of them.

So that right over there is 4/5.

Now we can view this
as 2/3 of the 4/5.

So how can we think about that?

Well, we could take this section
and divide it into thirds.

So let's do that.

Divide it into thirds.

So we're going divide it
into 3 equal sections.

So that's 1/3, and then 2/3.

So we took each of
the 5 equal sections,

and we divided them
into 3 equal sections.

Now what's going to
be 2/3 of the 4/5?

Well, that's going to be
this part right over here.

So let me make this clear.

This is 1/3 of the 4/5.

And then this would
be 2/3 of the 4/5.

So this right over here, would
be 2/3 of the 4/5, or 2/3 times

4/5.

But what fraction of the
whole does that represent?

Well, how many total, how many
total equal sections do we now

have?

Well, we have 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14,

15.

So we have 15 equal sections.

I'm using a new color.

We have 15 equal sections,
and that make sense.

We started with
5 equal sections,

but then we divided each of
those into 3 equal sections.

So now we have 5 times
3 total equal sections.

And then how many of
those are now colored in?

Well, we see it's 2 times 4.

1, 2, 3, 4, 5, 6, 7, 8.

How many of them are in the
2/3 of the 4/5, I should say.

And there's 8 of them, 8
of the 15 equals sections.

And so there you have it.

It should hopefully
now make visual sense,

or it makes conceptual
sense, that 2/3 times 4/5--

you can obviously compute
it by just multiplying

the numerators, 2 times 4 is 8.

And then multiplying the
denominators, 3 times 5 is 15--

but hopefully this now makes
conceptual sense as 2/3 of 4/5.