Dividing fractions

Dividing fractions

Welcome to the presentation
on dividing fractions.

Let's get started.

So before I give you the
intuition-- actually, I might

do that in a different module--
I'm just going to show you the

mechanics of how you
divide a fraction.

And it turns out that
it's actually not much

more difficult than
multiplying fractions.

If I were to ask you, 1/2
divided by 1/2, whenever you

divide by a fraction, or
actually, when you divide by

any number, it's the same thing
as multiplying by its inverse.

So 1/2 divided by 1/2 is
equal to 1/2 times 2/1.

We just inverted--
inverse-- the second 1/2.

And we know from the
multiplication module, 1/2

times 2/1, well, that's
just equal to 2/2,

or it's equal to 1.

And that makes sense because,
actually, any number divided

by itself is equal to 1.

1/2 divided by 1/2 is 1, just
like 5 divided by 5 is 1, just

like 100 divided by 100 is 1.

And this isn't a new principal.

Actually, you were
always doing it.

But isn't this also the
same thing as 2 times the

inverse of 2, which is 1?

I'll show it to you.

Actually, let me give you a
couple more examples to show

that dividing fractions really
isn't a new concept, this whole

notion of multiplying
by the inverse.

If I were to tell you
what is 12 divided by 4?

Well, we know the answer to
this, but I'm going to show

you that this is the same
thing as 12 times 1/4.

12/1 times 1/4 4 is
12/4, which is 3.

And 12/4 is really just another
way of writing 12 divided by 4,

so it's kind of a long way of
getting to the same point.

But I just wanted to show you
that what we're doing in this

module is nothing new than
what we've always been doing

when we divide by a number.

Division is the same thing.

Dividing by a number is the
same thing as multiplying by

the inverse of that number.

And just as a review, an
inverse, if I have a number

A, the inverse-- inv, short
for inverse-- is 1 over A.

So the inverse of 2/3 is 3/2,
or the inverse of 5, because 5

is the same thing as 5/1,
so the inverse is 1/5.

So let's do some fraction
division problems.

What is 2/3 divided by 5/6?

Well, we know that this is the
same thing as 2/3 times 6/5,

and that's equal to 12/15.

We can divide the numerator and
denominator by 3, that's 4/5.

What is 7/8 divided by 1/4?

Well, that's the same
thing as 7/8 times 4/1.

Remember, I just
flipped this 1/4.

Divide by 1/4 is the same
thing as multiplying by 4/1.

That's all you've got to do.

And then we could use a little
shortcut we learned in the

multiplication module.

8 divided by 4 is 2.

4 divided by 4 is 1.

So that equals 7/2.

Or if you wanted to write that
as a mixed number, this is, of

course, an improper fraction.

Improper fractions have
a numerator larger

than the denominator.

If you wanted to write that as
a mixed number, 2 goes into 7

three times with a remainder
of 1, so that's 3 and a half.

You can write it either way.

I tend to keep it this
way because it's

easier to deal with.

Let's do a ton of more
problems, or at least as many

more as we can do in the
next four or five minutes.

What's negative 2/3
divided by 5/2?

Once again, that's the same
thing as minus 2/3-- whoops--

as minus 2/3 times what?

It's times the inverse of
5/2, which is 2/5, and

that equals minus 4/15.

What is 3/2 divided by 1/6?

Well, that's just the same
thing as 3/2 times 6/1,

I think you might
be getting it now.

Let's see, let's
do a couple more.

And, of course, you can always
pause, and look at this whole

presentation again, so you can
get confused all over again.

Let's see, let's do minus
5/7 divided by 10/3.

Well, this is the same thing
as minus 5/7 times 3/10.

I just multiplied
by the inverse.

That's all I keep doing
over and over again.

Minus 5 times 3.

Minus 15.

7 times 10 is 70.

If we divide the numerator
and the denominator by

5, we get minus 3/14.

We could have also
just done it here.

We could have done 5, 2,
and we would have gotten

minus 3/14 as well.

Let's do one or two
more problems.

I think you kind of
get it, though.

Let's say 1/2
divided by minus 3.

Ah-ha!

So what happens when you take a
fraction and you divide it by

a whole number or an integer?

Well, we know any whole number
can be written as a fraction.

This is the same thing as
1/2 divided by minus 3/1.

And dividing by a fraction is
the same thing as multiplying

by it's inverse.

So the inverse of negative 3/1
is negative 1/3, and this

equals negative 1/6.

Let's do it the other way.

What if I had negative
3 divided by 1/2?

Same thing.

Negative 3 is the same thing as
minus 3/1 divided by 1/2, which

is the same thing as minus 3/1
times 2/1, which is equal to

minus 6/1, which
equals minus 6.

Now, let me give you a
little bit of intuition

of why this works.

Let's say I said 2
divided by 1/3.

Well, we know that this
is equal to 2/1 times

3/1, which equals 6.

So how does 2, 1/3,
and 6 relate?

Well, let's look
at it this way.

If I had two pieces of pizza.

I have two pieces of pizza.

Here's my two pieces
of pizza right.

Two right here.

So I have two pieces of pizza,
and I'm going to divide

them into thirds.

So I'm going to divide
each pizza into a third.

I'll draw the little
Mercedes sign.

So I'm dividing each pizza
into a third, right?

How many pieces do I have?

Let's see, 1, 2, 3, 4, 5, 6.

I have 6 pieces.

So you might want to sit and
ponder that for a little bit,

but I think it might make a
little bit of sense to you.

Let's do one more just
to tire your brain.

If I had negative 7/2 divided
by 4/9-- let's pick a negative

4/9-- well, that's the same
thing as minus 7/2 times

minus 9/4, right?

I just multiplied by the
inverse of negative 4/9.

9 times 7 is equal to--
negative 7 times negative

9 is positive 63,
and 2 times 4 is 8.

Hopefully, I think you have a
good idea of how to divide by

a fraction now, and you can
try out the dividing

fractions modules.

Have fun!