jmc

To express the number $0.\overline{12}$ as a fraction, we call it $x$ and subtract it from $100x$: \begin{array}{r r c r@{}l} &100x &=& 12&.121212\ldots \\ - &x &=& 0&.121212\ldots \\ \hline &99x &=& 12 & \end{array}This shows that $0.\overline{12}=\frac{12}{99}$.

But that isn't in lowest terms, since $12$ and $99$ share a common factor of $3$. We can reduce $\frac{12}{99}$ to $\frac{4}{33}$, which is in lowest terms. The sum of the numerator and denominator is $4+33=\boxed{37}$.