Answer
$\displaystyle \frac{112}{999}$
Work Step by Step
$ 0.\overline{112}=0.112112112\ldots$ =
$=0.112+0.000112+0.000000112+...$
$=\displaystyle \frac{112}{1000}+\frac{112}{1,000,000}+\frac{112}{1,000,000,000}+\cdots$
... an infinite geometric series, with
$a=\displaystyle \frac{112}{1000}$ and $r=\displaystyle \frac{1}{1000}$.
$|\displaystyle \frac{1}{1000}| < 1$, so the sum exists, $S=\displaystyle \frac{a}{1-r}$
$0.\displaystyle \overline{112}=\frac{\frac{112}{1000}}{1-\frac{1}{1000}}=\frac{\frac{112}{1000}}{\frac{999}{1000}}$
$=\displaystyle \frac{112}{999}$