Algebra Demystified 2nd Ed

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