Answer
$\displaystyle \frac{10,457}{4950}$
Work Step by Step
$2.11\overline{25}=2.11252525 \ldots = 2.11+ 0.00252525$
$=2.11+ 0.00\overline{25}$
$ 0.00\displaystyle \overline{25}=(\frac{25}{10,000}+\frac{25}{1,000,000}+\frac{25}{100,000,000}+\cdots )$
... an infinite geometric series.
$a=\displaystyle \frac{25}{10,000}$ , $r=\displaystyle \frac{1}{100}$.
$|\displaystyle \frac{1}{100}| < 1$, so the sum exists, $S=\displaystyle \frac{a}{1-r}$
$0.00\displaystyle \overline{25}=\frac{\frac{25}{10,000}}{1-\frac{1}{100}}=\frac{25}{9900}$,
so
$2.11\overline{25}= 2.11+0.00\overline{25}=$
$=\displaystyle \frac{211}{100}+\frac{25}{9900}=\frac{211\cdot 99+25}{9900}=\frac{20,914}{9900}$
$=\displaystyle \frac{10,457}{4950}$
