Page 80 - ArithBook5thEd ~ BCC

a shaded vertical rectangles is divided into d parts of which c are differently shaded. Hence there are
a · cdoubly shaded smaller rectangles, so that a · c is the new numerator. The rule is
a c a · c
of = .
b d b · d

We use this rule to define multiplication of fractions, replacing“of” by the multiplication symbol · (or
×):

a c a · c
· =
b d b · d

In words: The product of two fractions is the product of the numerators over the product of the
denominators.
3 5
Example 86. Find the product · .
8 7
Solution. Using the boxed rule,
3 5 3 · 5 15
· = = .
8 7 8 · 7 56

It is easy to see that fraction multiplication is just an expanded version of whole number multipli-
cation, since every whole number is a fraction with denominator 1. Thus the boxed rule applies when
one (or both!) of the multiplicands is a whole number.
Example 87. What is two thirds of five? That is, find the product 2 · 5.
3
Solution. Writing 5 as the fraction 5/1 and using the fraction multiplication rule, we get

2 5 2 · 5 10
· = = .
3 1 3 · 1 3

Example 88. One third of a twelve-member jury are women. How many women are on the jury?

Solution. One third of twelve equals 1 · 12 = 12 = 4. There are four women on the jury.
3 1 3
You may have noticed that in our very first example with rectangles,

3 2 6
· = ,
4 3 12
exactly half of the rectangle ends up doubly-shaded (6 is half of 12). So it is correct (and simpler) to
say that
3 2 1
· = .
4 3 2
Does this contradict our rule? No, because 1/2and 6/12 represent the same number. We’ll explain
this, and examine some other technical details regarding the representation of fractions, in the next
sections. Then we’ll continue with the arithmetic (division, addition, subtraction) of fractions.

Page 80