Chapter 8.4, Problem 19E

To determine

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

Answer to Problem 19E

Thefraction (in simplest form) of the given decimal is $\frac{2}{9}$ .

Explanation of Solution

Given:

The given decimal is $0.222$ .

Calculation:

Write therepeating decimal as an infinite geometric series.

  $0.2+0.02+0.002+\cdot \cdot \cdot$

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 $a_1=0.2$ and $r=\frac{0.02}{0.2}=0.1<1$ .

  $\begin{array}{l}S=\frac{0.2}{1-0.1}\\ =\frac{0.2}{0.9}\\ =\frac{2}{9}\end{array}$

Hence thefraction (in simplest form) of the given decimal is $\frac{2}{9}$ .