Chapter 3.2, Problem 27PPS

To determine

To find: the fraction of $-0.\overline{09}$ .

Answer to Problem 27PPS

The fraction is $-\frac{1}{11}$ .

Explanation of Solution

Given:

The decimal is $-0.\overline{09}$ .

Calculation:

Let,

  $x=\mathrm{0.090909...}\mathrm{......}\text{}\left(1\right)$

Multiply by 100 in above equation as follows:

  $\begin{array}{c}100x=100\times \mathrm{0.090909...}\\ 100x=\mathrm{9.090909...}\mathrm{.......}\left(2\right)\end{array}$

Subtract the equation (1) from (2) as follows:

  $\begin{array}{c}100x-x=\left(\mathrm{9.090909...}\right)-\left(\mathrm{0.090909...}\right)\\ 99x=9\\ x=\frac{\cancel{9}\times 1}{\cancel{9}\times 11}\\ =\frac{1}{11}\end{array}$

For $-0.\overline{09}$ ,

  $x^{\prime }=-\frac{1}{11}$

Thus, the fraction is $-\frac{1}{11}$ .