jmc

Remember that $\frac{1}{3}=0.\overline{3}.$ We can multiply both the numerator and denominator by $10$ to help simplify the fraction: $$\begin{array}{rl}\frac{0.\overline{3}}{0.8\overline{3}}\cdot \frac{10}{10}& =\frac{0.\overline{3}\cdot 10}{0.8\overline{3}\cdot 10}=\frac{3.\overline{3}}{8.\overline{3}}\\ & =\frac{3+\frac{1}{3}}{8+\frac{1}{3}}=\frac{\frac{10}{3}}{\frac{25}{3}}\\ & =\frac{\text{cancelto}210}{\cancel{3}}\cdot \frac{\cancel{3}}{\text{cancelto}525}=\boxed{\frac{2}{5}}.\end{array}$$