2 alGebra De mystif ieD
Fraction Multiplication
To illustrate concepts in fraction arithmetic, we will use pie charts. For example,
we represent the fraction^13 with the shaded region in Figure 1-1. That is,^13 is
one part out of three equal parts.
Let us now develop the rule for multiplying fractions, ab⋅=dc bdac. For example,
using this rule we can compute^23 ⋅^14 by multiplying the numerators, 2 and 1, and
the denominators, 3 and 4. Doing so, we obtain^23 ⋅=^142134 ⋅⋅ = 122 (we will concern
ourselves later with simplifying fractions). Let us see how to represent the
product^23 ⋅^14 on the pie chart. We can think of this fraction as “two-thirds of
one-fourth.” We begin with one-fourth represented by a pie chart in Figure 1-2.
Let us see what happens to the representation of one-fourth if we divide the
pie into twelve equal parts as in Figure 1-3.
FIGURE 1-1 FIGURE 1-2 FIGURE 1-
Now we see that the fraction^14 is the
same as 123. We can also see that when^14 is
itself divided into three equal pieces, each
piece represents one-twelfth, so two-thirds
of^14 is two-twelfths. (See Figure 1-4.) This
is why^2
3
1
4
⋅ is 122.
EXAMPLE
Perform the multiplication with the rule ab⋅cd = bdac.
2
3
4
5
⋅
According to the rule, we multiply the numerators, 2 and 4, and the
denominators, 3 and 5 to obtain^23 ⋅^45 = ^2435 ⋅⋅ 158. =
One-third
of one-fourth
FIGURE 1-
EXAMPLE
Perform the multiplication with the rule