Chapter 8.4, Problem 23E

To determine

Find the fraction (in simplest form) of the given decimal.

Answer to Problem 23E

Thefraction (in simplest form) of the given decimal is $\frac{3200}{99}$ .

Explanation of Solution

Given:

The given decimal is $\mathrm{32.323232...}$ .

Calculation:

Write therepeating decimal as an infinite geometric series.

  $32+0.32+0.0032+0.000032\cdot \cdot \cdot$

The sum of an infinite geometric series with first term $a_1$ and common ratio $r$ is

  $S=\frac{a_1}{1-r}$ , where $r<1$ .

If $r\ge 1$ , then the series has no sum.

In the given series $a_1=32$ and $r=\frac{0.32}{32}=0.01<1$ .

  $\begin{array}{l}S=\frac{32}{1-0.01}\\ =\frac{32}{0.99}\\ =\frac{3200}{99}\end{array}$

Hence thefraction (in simplest form) of the given decimal is $\frac{3200}{99}$ .