### Instruction: What is $\frac{0.\overline{72}}{0.\overline{27}}$? Express your answer as a common fraction in lowest terms.### Response: This problem involves repeating decimals and division, which can be tricky.
One way to simplify the problem is to convert the repeating decimals to fractions using a clever trick.
For example, to convert 0.\overline{72} to a fraction, I can let x = 0.\overline{72} and multiply both sides by 100 to get 100x = 72.\overline{72}.
Then I can subtract x from both sides to eliminate the decimal part and get 99x = 72.
Dividing both sides by 99, I get x = \frac{72}{99}.
I can reduce this fraction by dividing both the numerator and the denominator by 9, and get x = \frac{8}{11}.
Similarly, to convert 0.\overline{27} to a fraction, I can let y = 0.\overline{27} and multiply both sides by 100 to get 100y = 27.\overline{27}.
Then I can subtract y from both sides to eliminate the decimal part and get 99y = 27.
Dividing both sides by 99, I get y = \frac{27}{99}.
I can reduce this fraction by dividing both the numerator and the denominator by 9, and get y = \frac{3}{11}.
Now I have simplified the problem to $\frac{x}{y} = \frac{\frac{8}{11}}{\frac{3}{11}}$.
To divide fractions, I can flip the second fraction and multiply, and get $\frac{x}{y} = \frac{8}{11} \times \frac{11}{3}$.
I can cancel out the common factor of 11 and get $\frac{x}{y} = \frac{8}{3}$.
This is the answer in simplest terms.