jmc

First let's convert $0.4\overline{8}$ to a fraction. Let $p=0.4\overline{8}$ and multiply both sides of this equation by 10 to obtain $10p=4.8\overline{8}$. Subtracting the left-hand sides $10p$ and $p$ as well as the right-hand sides $4.8\overline{8}$ and $0.4\overline{8}$ of these two equations gives $9p=4.4$, which implies $p=44/90=22/45$.

Next, let's convert $0.\overline{37}$ to a fraction. Let $q=0.\overline{37}$ and multiply both sides of this equation by 100 to obtain $100q=37.\overline{37}.$ Subtracting the left-hand sides $100q$ and $q$ as well as the right-hand sides $37.\overline{37}$ and $0.\overline{37}$ of these two equations gives $99q=37$, which implies $q=\frac{37}{99}.$

We add $p$ and $q$ to get our answer: $$\begin{array}{rl}\frac{22}{45}+\frac{37}{99}& =\frac{22}{45}\cdot \frac{11}{11}+\frac{37}{99}\cdot \frac{5}{5}\\ & =\frac{242}{495}+\frac{185}{495}=\boxed{\frac{427}{495}}.\end{array}$$