Question
Express the following decimals in the form $\frac{\text{p}}{\text{q}}:$ $0.\overline{37}$
✓
Answer
Let $\text{x}=0.\overline{37}$ Now, $\text{x}=0.3737\ ...(\text{i})$ Multiplying equation $(i)$ by $10$.
$\therefore \ 10\text{x}=3.737\ ...(\text{ii}) $ Multiplying equation $(ii)$ by $10$.
$100\text{x}=37.3737\ ...(\text{iii}) $
Subtracting equation $(i)$ by $(iii)$
$\therefore\ 100\text{x}-\text{x}=37$
$\Rightarrow99\text{x}=37$
$\Rightarrow\text{x}=\frac{37}{99}$
Hence, $0.\overline{37}=\frac{37}{99}$
$\therefore \ 10\text{x}=3.737\ ...(\text{ii}) $ Multiplying equation $(ii)$ by $10$.
$100\text{x}=37.3737\ ...(\text{iii}) $
Subtracting equation $(i)$ by $(iii)$
$\therefore\ 100\text{x}-\text{x}=37$
$\Rightarrow99\text{x}=37$
$\Rightarrow\text{x}=\frac{37}{99}$
Hence, $0.\overline{37}=\frac{37}{99}$
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