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.