Converting Repeating Decimals into Fractions

Solution 1: We are given that $0.11111 \ldots = \frac{1}{9}$ . Dividing throughout by 11, we see that $0.010101\ldots = \frac{1}{9 \times 11 } = \frac{1}{99}$ .
Thus, $0.121212 \ldots = 0.111111\ldots + 0.010101\ldots = \frac{1}{9} + \frac{1}{99} = \frac{12}{99} = \frac{4}{33}$ .

Solution 2: Let $S = 0.121212 \ldots$ . We have

$\begin{array} { r lll } S & = 0& . & 121212 \ldots \\ 100S & = 12 & . & 121212\ldots \\ \end{array}$

Taking the difference of these two lines, we get that $99 S = 12$ , so $S = \frac{12}{99} = \frac{4}{33}$ .