Question

Find the rational number which is represented by the following decimal numbers : $2.00\overline{12}$

Answer 1

Let $r=2.00\overline{12}$, then $100r$ and $10000r$ will be as follows:

$100r=100\times 2.00\overline{12}=200.1212$

$1000r=1000\times 2.00\overline{12}=20012.1212$

Now consider $10000r-100r$ as shown below:

$1000r-100r=20012.1212-200.1212=19812.00\Rightarrow 9900r=19812\Rightarrow r=\frac{19812}{9900}\Rightarrow r=\frac{1651}{825}$

Hence, the required rational number is $\frac{1651}{825}$.