jmc

In general, to express the number $0.\overline{n}$ as a fraction, we call it $x$ and subtract it from $10x$: \begin{array}{r r c r@{}l} &10x &=& n&.nnnnn\ldots \\ - &x &=& 0&.nnnnn\ldots \\ \hline &9x &=& n & \end{array} This shows that $0.\overline{n}=\frac{n}{9}$.

Hence, our original problem reduces to computing $\frac{8}{9}+\frac{2}{9}=\boxed{\frac{10}{9}}$.