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3.10.2 Exercises

Find the reciprocals of the following numbers. Change improper fractions to mixed or whole numbers.
3
1.
7
2. 11
14
3.
5
1
4.
25
5
5.
6

3.10.3 Division is Multiplication by the Reciprocal of the Divisor

Consider again the division problem
3 2
÷ .
4 3
Using the LCD, this is equivalent to
9 8
÷
12 12
We want to know how many times 8 things of a certain size (1/12) go into 9 things of the same size.
The actual denominator doesn’t really matter, only that it is the same for both fractions. In this form
9
it seems obvious that the answer is the quotient of the numerators, .
8
Now, how were the numbers 9 and 8 obtained from the original fractions? Well, in writing equivalent
fractions with the LCD, we computed the numerators as follows: 9 =3 · 3and 8 = 4 · 2. We have
shown that
3 2 3 · 3 3 3
÷ = = · .
4 3 4 · 2 4 2
The original division turned into multiplication by the reciprocal of the divisor,and the LCD wasn’t
actually needed.
We have derived a simple rule:

a c a d
÷ = ·
b d b c

In words: the quotient of two fractions is the product of the dividend and the reciprocal of the divisor.
5
Example 117. Perform the division 5 ÷ .
12 6
5
Solution. We convert the division into multiplication by the reciprocal of .
6
5 5 5 6
÷ = ·
12 6 12 5
30
=
60
1
= (cancelling 30).
2

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