Question

Find the rational number which is represented by the following decimal numbers : $2013.\overline{13}$

Answer 1

Let $r=2013.\overline{13}$, then $100r$ will be as follows:

$100r=100\times 2013.\overline{13}=\mathrm{201313.13....}$

Now consider $100r-r$ as shown below:

$100r-r=201313.1313-2013.1313=199300.00\Rightarrow 99r=199300\Rightarrow r=\frac{199300}{99}$

Hence, the required rational number is $\frac{199300}{99}$.