### Instruction: Express as a common fraction the reciprocal of $0.\overline7$.### Response: I notice that $0.\overline7$ is a repeating decimal, which means it can be written as a fraction using a trick.
I let $x = 0.\overline7$, and then I multiply both sides by 10 to get $10x = 7.\overline7$.
Now I subtract $x$ from $10x$ to eliminate the repeating part, and I get $9x = 7$.
Solving for $x$, I divide both sides by 9 and I get $x = \frac{7}{9}$.
This means that $0.\overline7 = \frac{7}{9}$, and its reciprocal is $\frac{9}{7}$.