Question

$0.5737373\dots .=$

• A. $\frac{284}{497}$
• B. $\frac{284}{495}$
• C. $\frac{568}{990}$
• D. $\frac{567}{990}$

Answer 1

The correct option is C

$\frac{568}{990}$

Explanation for the correct option:

Step 1. Let $x=0.5737373\dots .$

$$\begin{gathered} =0.5+\left(\frac{73}{{10}^3}\right)+\left(\frac{73}{{10}^5}\right)+\dots \\ =\frac{1}{2}+\left(\frac{73}{{10}^3}\right)\left(1+\frac{1}{{10}^2}+\frac{1}{{10}^4}+...\right)...\left(1\right) \end{gathered}$$

Here, $1+\frac{1}{{10}^2}+\frac{1}{{10}^4}+...$ is a GP

where, $a=1,r=\frac{1}{{10}^2}$

$\Rightarrow$$1+\frac{1}{{10}^2}+\frac{1}{{10}^4}+...=\frac{1}{\left(1-{\displaystyle \frac{1}{100}}\right)}$ $\left[\mathbf{\because }{\mathit{S}}_{\mathbf{\infty }}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{a}}{\mathbf{1}\mathbf{-}\mathbf{r}}\right]$

$=\frac{100}{99}$

Step 2. Equation (1) becomes

$$\begin{gathered} =\frac{1}{2}+\left(\frac{73}{{10}^3}\right)\left(\frac{100}{99}\right) \\ =\frac{1}{2}+\left(\frac{73}{10}\right)\left(\frac{1}{99}\right) \\ =\frac{1}{2}+\left(\frac{73}{990}\right) \\ =\frac{\mathbf{568}}{\mathbf{990}} \end{gathered}$$

Hence, Option ‘C’ is Correct.