# Dividing Fractions

**0.1 Dividing Fractions
**

In this section, we will discuss the two interpretations of division for fractions,

and we will see why the standard "invert and multiply" procedure for dividing

fractions gives answers to fraction division problems that agree with what

we expect from the meaning of division.

**The Two Interpretations of Division for Fractions**

Let's review the meaning of division for whole numbers, and see how to

interpret division for fractions.

**The "how many groups?" interpretation**

With the "how many groups?" interpretation of division, 12 ÷ 3 means the

number of groups we can make when we divide 12 objects into groups with

3 objects in each group. In other words, 12 ÷ 3 tells us how many groups of

3 we can make from 12.

Similarly, with the "how many groups?" interpretation of division,

means the number of groups we can make when we divide
of an object

into groups with of an object in each group.
In other words, tells us

how many groups of we can make from
. For example, suppose you are

making popcorn balls and each popcorn ball requires
of a cup of popcorn.

If you have of a cup of popcorn, then how
many popcorn balls can

you make? In this case you want to divide of
a cup of popcorn into groups

(balls) with
of a cup of popcorn in each
group. According to the "how

many groups?" interpretation of division, you can make

popcorn balls.

**The "how many in one (each) group?" interpretation**

With the "how many in each group?" interpretation of division, 12 ÷ 3 means

the number of objects in each group when we distribute 12 objects equally

among 3 groups. In other words, 12 ÷3 is the number of objects in one group

if we use 12 objects to evenly fill 3 groups. When we work with fractions, it

often helps to think of "how many in each group?" division story problems as

asking "how many are in one whole group?", and it helps to think of filling

groups or part of a group. So in the context of fractions, we will usually

refer to the "how many in each group?" interpretation as "how many in one

group?".

With the "how many in one group?" interpretation of division,

is the number of objects in one group when we distribute
of an object

equally among of a group. A clearer way to
say this is: is the

number of objects (or fraction of an object) in one whole group when
of

an object fills of a group. For example,
suppose you pour of a
pint of

blueberries into a container and this fills
of the container. How many pints

of blueberries will it take to fill the whole container? In this case,
of a

pint of blueberries fills (i.e., is distributed equally among)
of a group (a

container). So according to the "how many in one group?" interpretation

of division, the number of pints of blueberries in one whole group (one full

container) is

One way to better understand fraction division story
problems is to think

about replacing the fractions in the problem with whole numbers. For example,

if you have 3 pints of blueberries and they fill 2 containers, then how

many pints of blueberries are in each container? We solve this problem by

dividing 3 ÷ 2, according to the "how many in each group?" interpretation.

Therefore if we replace the 3 pints with
of a pint, and the 2
containers with

of a container, we solve the problem in the
same way as before: 3 ÷ 2 now

becomes .

Here is another way to think about the problem. Because
of the container

is filled, and because this amount is
of a pint, therefore
of the

number of pints in a full container is
of a pint. In other
words:

×number of pints in
full container =

Therefore

number of pints in full container =

**Dividing by Versus
Dividing in **

In mathematics, language is used much more precisely and carefully than in

everyday conversation. This is one source of difficulty in learning mathematics.

For example, consider the two phrases:

dividing by ,

dividing in .

You may feel that these two phrases mean the same thing, however,
mathematically,

they do not. To divide a number, say 5, by
means to calculate

Remember that we read A÷ B as A divided by
B. We would divide

5 by if we
wanted to know how many half cups of our are in 5 cups of

our, for example. (Notice that there are 10 half-cups of our in 5 cups of

our, not .)

On the other hand, to divide a number in half means to find half of that

number. So to divide 5 in half means to find
of 5. One half of 5
means

So dividing in
is the same as
dividing by 2.

**The "Invert and Multiply" Procedure for Fraction Di-vision**

Although division with fractions can be difficult to interpret, the procedure

for dividing fractions is quite easy. To divide fractions, such as

and

we can use the familiar "invert and multiply" method in
which we invert the

divisor and multiply by it:

and

**reciprocal**

Another way to describe this "invert and multiply" method
for dividing

fractions is in terms of the reciprocal of the divisor. The **reciprocal **of
a

fraction is the fraction
. In order to divide fractions, we should
multiply

by the reciprocal of the divisor. So in general,