Answer
$\frac{122}{99}$
Work Step by Step
$1.2323\ldots $
This can be written as,
$1.2323\ldots =1+0.2323\ldots $
And,
$0.232323\ldots =0.23+0.0023+0.000023+\cdots $
This is an infinite geometric series:
${{a}_{1}}=0.23$ and ${{a}_{2}}=0.0023$.
So, the value of $\left| r \right|$ is,
$\begin{align}
& \left| r \right|=\left| \frac{0.0023}{0.23} \right| \\
& =\left| 0.01 \right| \\
& =0.01
\end{align}$
Find the limit of the infinite geometric series by using the formula
${{S}_{\infty }}=\frac{{{a}_{1}}}{1-r}$.
$\begin{align}
& {{S}_{\infty }}=\frac{{{a}_{1}}}{1-r} \\
& =\frac{0.23}{1-0.01} \\
& =\frac{0.23}{0.99} \\
& =\frac{23}{99}
\end{align}$
The fraction notation of the decimal number $0.2323\ldots $ is $\frac{23}{99}$.
And the fraction notation of the decimal number $1.2323\ldots $ is,
$1+\frac{23}{99}=\frac{122}{99}$
Thus, the fraction notation of the decimal number $1.2323\ldots $ is $\frac{122}{99}$.
