jmc

We can split $0.5\overline{10}$ into $0.5+0.\overline{01}$.

First we convert $0.\overline{01}$ to a fraction by setting $0.\overline{01}=x$. Multiplying both sides by 100, we get $100x=1.\overline{01}$. We subtract those two equations to get: $$\begin{array}{rl}100\cdot x-x& =1.\overline{01}-0.\overline{01}⟹\\ 99\cdot x& =1⟹\\ x& =\frac{1}{99}.\end{array}$$We add 1/99 to $0.5=1/2$ to get $0.5\overline{01}=\frac{1}{2}+\frac{1}{99}=\boxed{\frac{101}{198}}$.