Answer
$\frac{112}{999}$
Work Step by Step
We express the number as a sum of fractions:
$0.\overline{112}=0.11211211\ldots=\frac{112}{1000}+\frac{112}{1,000,000}+\frac{112}{1,000,000,000}+...$
We know that this represents an infinite geometric series with $a=\frac{112}{1000}$ and $r= \frac{1}{1000}$.
We know the sum of an infinite geometric series is:
$S_{\infty}=\frac{a}{1-r}$
$S_{\infty}=\frac{\frac{112}{1000}}{1-\frac{1}{1000}}=\frac{112}{999}$
