What is $\frac{0.\overline{3}}{0.8\overline{3}}$? Express your answer as a common fraction in lowest terms.
Remember that $\frac{1}{3} = 0.\overline{3}.$ We can multiply both the numerator and denominator by $10$ to help simplify the fraction: \begin{align*}
\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}} \\
&=\dfrac{3+\frac{1}{3}}{8+\frac{1}{3}} =\dfrac{\frac{10}{3}}{\frac{25}{3}} \\
&=\frac{\cancelto{2}{10}}{\cancel{3}} \cdot \frac{\cancel{3}}{\cancelto{5}{25}\hspace{3mm}} =\boxed{\frac{2}{5}}.
\end{align*}