Question

Find the value of $0.5737373....$ in fraction

• 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 $y=0.5737373.....$

$=0.5\overline{73}$

$=0.5+\frac{73}{{10}^3}+\frac{73}{{10}^5}+....$

$\Rightarrow$ $y=\frac{1}{2}+\frac{73}{{10}^3}\left(1+\frac{1}{{10}^2}+\frac{1}{{10}^4}+.....\right)$ …..(1)

Here, $1,\frac{1}{{10}^2},\frac{1}{{10}^4},....$ are in GP

Where, $a=1,r=\left(\frac{{10}^{-2}}{1}\right)$

$=\frac{1}{100}$

$\begin{array}{rcl}\therefore \left(1+\frac{1}{{10}^2}+\frac{1}{{10}^4}+.....\right)& =& \left[\frac{1}{\left(1-{\displaystyle \frac{1}{100}}\right)}\right]\\ & =& \frac{100}{99}\end{array}$ $\left[\mathbf{\because }{\mathit{S}}_{\mathbf{\infty }}\mathbf{=}\frac{\mathbf{a}}{\mathbf{1}\mathbf{-}\mathbf{r}}\right]$

Step 2. Put the value of $1,\frac{1}{{10}^2},\frac{1}{{10}^4},....$ in equation (1), we get

$y=\frac{1}{2}+\frac{73}{1000}\times \frac{100}{99}$

$\Rightarrow y=\frac{1}{2}+\frac{73}{990}$

$\Rightarrow y=\frac{568}{990}$

Hence, Option ‘C’ is Correct.