Question

Find the rational number which is represented by the following decimal numbers : $0.11\overline{2233}$

Answer 1

Let $r=0.11\overline{2233}$, then $100r$ and $1000000r$ will be as follows:

$100r=100\times 0.11\overline{2233}=\mathrm{11.2233....}$

$1000000r=1000000\times 0.11\overline{2233}=\mathrm{112233.2233....}$

Now consider $1000000r-100r$ as shown below:

$1000000r-100r=112233.2233-11.2233=112222.00$

$\Rightarrow 999900r=112222$

$\Rightarrow r=\frac{112222}{999900}$

$\Rightarrow r=\frac{56111}{499950}$

Hence, the required rational number is $\frac{56111}{499950}$.