jmc

We can rewrite $0.1\overline{23}$ as $0.1$ + $0.0\overline{23}$. The first decimal is simply $\frac{1}{10}$. Let the second decimal be $x$. Multiplying by 100, we have $100x=2.3\overline{23}$, which gives $99x=2.3⟹x=\frac{23}{990}$. Therefore, $0.1\overline{23}=\frac{1}{10}+\frac{23}{990}=\frac{61}{495}$. Thus $a+b=61+495=\boxed{556}$.