### Instruction: Compute the product of $0.\overline{123}$ and $9$, and write your result as a fraction in simplified form.### Response: To compute the product of $0.\overline{123}$ and $9$, I need to first convert $0.\overline{123}$ to a fraction.
I know that a repeating decimal can be written as a fraction by using a system of equations.
Let me call $x = 0.\overline{123}$.
Then, if I multiply both sides by 1000, I get $1000x = 123.\overline{123}$.
Subtracting $x$ from both sides, I get $999x = 123$.
Dividing both sides by 999, I get $x = \frac{123}{999}$.
This fraction can be simplified by dividing the numerator and denominator by 3.
So, $x = \frac{41}{333}$.
Now, I can multiply this fraction by 9 to get the product.
I get $9x = 9 \times \frac{41}{333} = \frac{369}{333}$.
This fraction can also be simplified by dividing the numerator and denominator by 3.
So, $9x = \frac{123}{111}$.