Express as a fraction in lowest terms: $0.\overline{1} + 0.\overline{01}$
For the first one, $0.\overline{1}$, I let $x = 0.\overline{1}$ and multiply both sides by 10 to get $10x = 1.\overline{1}$. Then I subtract $x$ from both sides to eliminate the decimal part and get $9x = 1$. Solving for $x$, I get $x = \frac{1}{9}$. So $0.\overline{1} = \frac{1}{9}$. For the second one, $0.\overline{01}$, I do a similar thing but multiply by 100 instead of 10. I let $y = 0.\overline{01}$ and multiply both sides by 100 to get $100y = 1.\overline{01}$. Then I subtract $y$ from both sides to get $99y = 1$. Solving for $y$, I get $y = \frac{1}{99}$. So $0.\overline{01} = \frac{1}{99}$. Now I need to add these two fractions. To do that, I need a common denominator. The lowest common denominator of 9 and 99 is 99, since 9 is a factor of 99. So I multiply the first fraction by $\frac{11}{11}$ to get $\frac{11}{99}$. Then I add it to the second fraction and get $\frac{11}{99} + \frac{1}{99} = \frac{12}{99}$. This fraction can be simplified by dividing both the numerator and the denominator by 3. So I get $\frac{4}{33}$ as the final answer. 