Final answer:
To express $0.\overline{1} + 0.\overline{01}$ as a fraction in lowest terms, we convert each repeating decimal to a fraction, then sum and simplify to get $\frac{4}{33}$.
Step-by-step explanation:
We need to find the sum of two repeating decimals and express it as a fraction in the lowest terms.
First, let's find the fraction form of $0.\overline{1}$, which is a repeating decimal. Let x = $0.\overline{1}$. Multiplying both sides of the equation by 10, we get 10x = $1.\overline{1}$. Subtracting the original equation from this gives us 9x = 1, hence x = $\frac{1}{9}$.
Now, let's tackle $0.\overline{01}$. Similarly, let y = $0.\overline{01}$. Multiplying both sides by 100, we get 100y = $1.\overline{01}$. Subtracting our original equation from this one gives us 99y = 1, therefore y = $\frac{1}{99}$.
Finally, to find the sum of x and y, we add their fraction forms: $\frac{1}{9} + \frac{1}{99}$ = $\frac{11}{99} + \frac{1}{99}$ = $\frac{12}{99}$. Simplifying this fraction by dividing numerator and denominator by their greatest common divisor, 3, we get $\frac{4}{33}$, which is in its lowest terms.