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,