Answer
$$ \frac{4}{33} $$
Work Step by Step
Since
\begin{align*}
0.121212 &= 0.12+0.0012+0.000012\\
&=0.12(1)+0.12(10^{-2})+ 0.12(10^{-4})+\cdots \\
&=0.12[1+\frac{1}{10^2}+\frac{1}{10^4}+\cdots ]
\end{align*}
Since
$$1+\frac{1}{10^2}+\frac{1}{10^4}+\cdots $$
which is a geometric series and has the sum
$$\frac{1}{1-\frac{1}{100}}= \frac{100}{99} $$
Hence
$$0.12121212= 0.12 \frac{100}{99}= \frac{12}{99} =\frac{4}{33}$$
