Chapter 7.4, Problem 30E

Solution

To calculate : The fractional value of repeating decimal $0.090909\cdots$ in the lowest terms.

The fractional value of repeating decimal $0.090909\cdots$ in the lowest terms. is $\frac{1}{11}$ .

Given information :

The repeating decimal $0.090909\cdots$ .

Formula used :

The sum of infinite geometric series with first term $a_1$ and common ration $r$ is calculated using the formula $S=\frac{a_1}{1-r}$ .

Here, $\left|r\right|<1$ . If $\left|r\right|\ge 1$ , then the series has no sum.

Calculation :

Consider the repeating decimal $0.090909\cdots$ ,

The repeating decimal can be written in the form of infinite geometric series as follows.

  $0.090909\cdots =0.09+0.09\left(0.01\right)+0.09{\left(0.01\right)}^2+\cdots$

For the given series the value of $a_1$ is $0.09$ and $r$ is $0.01$ .

As the common ratio $\left|0.01\right|<1$ , the sum of the series exists.

Then the sum of infinite geometric series is calculated using the formula $S=\frac{a_1}{1-r}$ .

  $\begin{array}{c}S=\frac{0.09}{1-0.01}\\ =\frac{0.09}{0.99}\\ =\frac{9}{99}\\ =\frac{1}{11}\end{array}$

Thus, the fractional value of repeating decimal $0.090909\cdots$ in the lowest terms. is $\frac{1}{11}$ .