Question

Express the recurring decimal $0.1\overline{576}$ as a rational number using concept of infinite geometric series.

• A. $\frac{175}{1110}$
• B. $\frac{1576}{9999}$
• C. $\frac{576}{999}$
• D. $\frac{157}{1110}$

Answer 1

The correct option is A $\frac{175}{1110}$

$0.1\overline{576}=\mathrm{0.1576576576...}$

$=0.1+0.0576+0.0000576+0.0000000576+...$

$=0.1+0.0576[1+{10}^{-3}+{10}^{-6}+...]$

$=0.1+0.576\left[\frac{1}{1-{10}^{-3}}\right]=0.1+0.576\left[\frac{{10}^3}{999}\right]=\frac{175}{1110}$