Answer
The required solution is $\frac{5}{11}$
Work Step by Step
Let us consider the given expression,
$\begin{align}
& 0.\overline{45}=0.454545\ldots \\
& \ \ \ \ \ \ =\frac{45}{100}+\frac{45}{10000}+\frac{45}{1000000}+\ldots
\end{align}$
And observe that $0.\overline{45}$ is an infinite geometric series with the first term $\frac{45}{100}$ and common ratio as $\frac{1}{100}$. As $r=\frac{1}{100}$, the condition that $\left| r \right|<1$ is met.
Therefore, we can use the formula to find the sum.
$\begin{align}
& 0.\overline{45}=\frac{{{a}_{1}}}{1-r} \\
& =\frac{\frac{45}{100}}{1-\frac{1}{100}} \\
& =\frac{45}{100}\times \frac{100}{99} \\
& =\frac{45}{99}
\end{align}$
Thus,
$0.\overline{45}=\frac{5}{11}$
