Question
Express the following decimals in the form $\frac{\text{p}}{\text{q}}:$ $0.\overline{621}$
✓
Answer
Let $\text{x}=0.\overline{621}$
Now, $\text{x}=0.621621\ ...(\text{i})$ Multiplying equation $(i) $ by $1000,$
$\therefore1000\text{x}=621.621621\ ...(\text{ii})$ Subtracting equation $(i)$ by $(ii)$
$\therefore\ 1000\text{x}-\text{x}=621$
$\Rightarrow999\text{x}=621$
$\Rightarrow\text{x}=\frac{621}{999}=\frac{69}{111}=\frac{23}{37}$
Hence, $0.\overline{621}=\frac{23}{37}$
Now, $\text{x}=0.621621\ ...(\text{i})$ Multiplying equation $(i) $ by $1000,$
$\therefore1000\text{x}=621.621621\ ...(\text{ii})$ Subtracting equation $(i)$ by $(ii)$
$\therefore\ 1000\text{x}-\text{x}=621$
$\Rightarrow999\text{x}=621$
$\Rightarrow\text{x}=\frac{621}{999}=\frac{69}{111}=\frac{23}{37}$
Hence, $0.\overline{621}=\frac{23}{37}$
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