### Instruction: What is $\frac{0.\overline{3}}{0.8\overline{3}}$? Express your answer as a common fraction in lowest terms.### Response: This problem involves dividing two repeating decimals.
One way to approach this is to convert both decimals to fractions and then divide them.
To convert a repeating decimal to a fraction, I need to use a trick where I multiply by a power of 10 to shift the decimal point and then subtract the original number.
For example, to convert 0.\overline{3} to a fraction, I can let $x = 0.\overline{3}$ and then multiply both sides by 10 to get $10x = 3.\overline{3}$.
Then, I can subtract $x$ from $10x$ to eliminate the repeating part and get $9x = 3$.
Solving for $x$, I get $x = \frac{1}{3}$.
So, 0.\overline{3} is equivalent to $\frac{1}{3}$.
Similarly, to convert 0.8\overline{3} to a fraction, I can let $y = 0.8\overline{3}$ and then multiply both sides by 10 to get $10y = 8.\overline{3}$.
Then, I can subtract $y$ from $10y$ to get $9y = 7.5$.
Solving for $y$, I get $y = \frac{7.5}{9} = \frac{15}{18} = \frac{5}{6}$.
So, 0.8\overline{3} is equivalent to $\frac{5}{6}$.
Now, I can divide $\frac{1}{3}$ by $\frac{5}{6}$ by multiplying by the reciprocal of the divisor.
That is, $\frac{1}{3} \div \frac{5}{6} = \frac{1}{3} \times \frac{6}{5} = \frac{6}{15} = \frac{2}{5}$.
So, the final answer is $\frac{2}{5}$.