# Dividing Fractions

**Explaining Why "Invert and Multiply" is Valid by Re-
lating Division to Multiplication**

The procedure for dividing fractions is easy enough to carry out, but why is it

a valid method? Before we answer this question in general, consider a special

case. Recall that every whole number is equal to a fraction (for example,

). Therefore we can apply the "invert and multiply" procedure to

whole numbers as well as to fractions. According to this procedure,

Notice that this result, that
, agrees with our findings earlier in

this chapter: that we can describe fractions in terms of division, namely that

In general, why is the "invert and multiply" procedure a valid way to

divide fractions? One way to explain this is to relate fraction division to

fraction multiplication. Recall that every division problem is equivalent to a

multiplication problem (actually two multiplication problems):

is equivalent to

(or B · ? = A). So

is equivalent to

Now remember that we want to explain why the "invert and
multiply" rule

for fraction division is valid. This rule says that
ought to be equal to

Let's check that this fraction works in the place of the ?
in Equation 1. In

other words, let's check that if we multiply
times
, then we really do
get

:

Therefore the answer we get from the "invert and multiply"
procedure really

is the answer to the original division problem
. Notice that the
line

of reasoning above applies in the same way when other fractions replace the

fractions
and
used above.

It will still be valuable to explore fraction division further, interpreting

fraction division directly rather than through multiplication.

**Class Activity 0A: Explaining "Invert and Multiply" by
Relating Division to Multiplication
Using the "How Many Groups?" Interpretation to Ex-
plain Why "Invert And Multiply" Is Valid**

Above, we explained why the "invert and multiply" procedure for dividing

fractions is valid by considering fraction division in terms of fraction multiplication.

Now we will explain why the "invert and multiply" procedure is

valid by working with the "how many groups?" interpretation of division .

Consider the division problem

The following is a story problem for this division
problem:

How many
cups of water are in
of a cup of water?

Or, said another way:

How many times will we need to pour
cup of water into a

container that holds
cup of water in order to fill the container?

From the diagram in Figure 1 we can say right away that the answer to this

problem is "one and a little more" because one half cup clearly fits in two

thirds of a cup, but then a little more is still needed to fill the two thirds
of

a cup. But what is this "little more"? Remember the original question: we

want to know how many
cups of water are in
of a cup of water. So the

answer should be of the form "so and so many
cups of water." This means

that we need to express this "little more" as a fraction of
cup of water.

How can we do that? By subdividing both the
and the
into common

parts, namely by using common denominators.

Figure 1: How Many 1/2 Cups of Water Are in 2/3 Cup?

When we give
and
the common
denominator of 6, then, as on the

right of Figure 1, the
cup of water is made out of 3 parts (3 sixths of a

cup of water), and the
cup of water is made out of 4 parts (4 sixths of a

cup of water), so the "little more" we were discussing above is just one of

those parts. Since
cup is 3 parts, and
the "little more" is 1 part, the "little

more" is
of the
cup of water.
This explains why
there's a

whole
cup plus another
of the
cup in
of a cup of water.

To recap: we are considering the fraction division problem
in terms

of the story problem "how many
cups of water are in
of a cup of water?"

If we give
and
the common
denominator of 6, then we can rephrase

the problem as "how many
of a cup are in
of a cup?" But in terms of

Figure ??, this is equivalent to the problem "how many 3s are in 4?" which

is the problem 4 ÷ 3, whose answer is
. Notice that
is exactly the

same answer we get from the "invert and multiply" procedure for fraction

division:

So the "invert and multiply" procedure gives the same
answer to
that

we arrive at by using the "how many groups?" interpretation of division.

The same line of reasoning will work for any fraction division problem

Thinking logically, as above, and interpreting
as "how many
cups

of water are in
cups of water?", we can conclude that

The final expression,
is the answer provided by the "invert and multiply"

procedure for dividing fractions. Therefore we know that the "invert and

multiply" procedure gives answers to division problems that agree with what

we expect from the meaning of division.

Class Activity 0B: "How Many Groups?" Fraction Di-

vision Problems

Using the "How Many in One Group?" Interpretation

to Explain Why "Invert And Multiply" Is Valid

Above, we saw how to use the "how many groups?" interpretation of division

to explain why the "invert and multiply" procedure for fraction division is

valid. We can also use the "how many in one group?" interpretation for

the same purpose. This interpretation, although perhaps more difficult to

understand, has the advantage of showing us directly why we can multiply

by the reciprocal of the divisor in order to divide fractions.

Consider the following "how many in one group?" story problem for

You used
of can of paint to
paint
of a wall. How many cans

of paint will it take to paint the whole wall?

This is a "how many in one group?" problem because we can think of the

paint as " filling"
of the wall. We can also see that this is a
division problem

by writing the corresponding number sentence:

·(amount to paint the whole wall) =

Therefore

amount to paint the whole wall =

We will now see why it makes sense to solve this problem
by multiplying

by the reciprocal of
, namely by
. Let's focus on the
wall to be painted,

as shown in Figure 2. Think of dividing the wall into 5 equal sections, 3 of

the 1/2 can of paint is divided equally

the 1/2 can of paint is divided equally among 3 parts |

the amount of paint for the full wall is 5 times the amount in one part |

Figure 2: The Amount of Paint Needed for the Whole Wall is
of the
Can

Used to Cover
of the Wall

which you painted with the
can of paint. If you
used
a can of paint to

paint 3 sections, then each of the 3 sections required
or
cans

of paint. To determine how much paint you will need for the whole wall,

multiply the amount you need for one section by 5. So you can determine

the amount of paint you need for the whole wall by multiplying the
can of

paint by
and then multiplying
that result by 5, as summarized in Table 1.

But to multiply a number by
and then multiply it
by 5 is the same as

multiplying the number by
. Therefore we can
determine the number of

cans of paint you need for the whole wall by multiplying
by
:

This is exactly the "invert and multiply" procedure for
dividing
It

shows that you will need
of a can of paint for the whole wall.

use can paint for of the wall |

use can paint for of the wall |

use can paint for 1 whole wall |

in one step: |

use can paint for of the wall |

Table 1: Determining How Much Paint to Use for a Whole
Wall if
Can of

Paint Covers
of the Wall

The argument above works when other fractions replace
and
, thereby

explaining why

In other words, to divide fractions, multiply the dividend
by the reciprocal

of the divisor.