### Instruction: Express as a fraction in lowest terms: $0.\overline{1} + 0.\overline{01}$.### Response: I notice that both decimals have repeating digits, so I can use a trick to convert them to fractions.
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.