Find the rational number having the following decimal expansions:… - Vidyadip

Question

Find the rational number having the following decimal expansions:$0.\overline{3}$

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Answer

$0.\overline{3}=0.3333\ \dots$
$=0.3+0.03+0.003+\ \dots$
$=\frac{3}{10}+\frac{3}{10^2}+\frac{3}{10^3}+\ \dots$
$=\frac{3}{10}\Big(1+\frac{1}{10}+\frac{1}{10^2}+\ \dots\Big)$
$=\frac{3}{10}\Bigg(\frac{1}{1-\frac{1}{10}}\Bigg)$
$=\frac{3}{10}\times\frac{10}{9}$
$=\frac39$
$0.\overline{3}=\frac13$

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