Express $1.\overline{03}$ as a reduced fraction, given that $0.\overline{01}$ is $\frac{1}{99}$.
I notice that the decimal $1.\overline{03}$ is the same as $1 + 0.\overline{03}$. To find $0.\overline{03}$ as a fraction, I can use a similar technique as for $0.\overline{01}$. I multiply both sides of the equation $0.\overline{03} = x$ by 100 to get $3.\overline{03} = 100x$. Then I subtract the original equation from this one to get $3 = 99x$. Solving for $x$, I get $x = \frac{3}{99} = \frac{1}{33}$. Therefore, $0.\overline{03} = \frac{1}{33}$. Adding this to 1, I get $1.\overline{03} = 1 + \frac{1}{33} = \frac{34}{33}$. This is already in reduced form, so I have my answer. 