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

Question

Find the rational number having the following decimal expansions:$0.\overline{231}$

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Answer

$0.\overline{231}=0.231231231\ \dots$$=0.231+0.000231+0.000000231+\ $
$=\frac{231}{10^3}+\frac{231}{10^6}+\frac{231}{10^9}+\ \dots$
$=\frac{231}{10^3}\Big(1+\frac{1}{10^3}+\frac{1}{10^6}+\ \dots\Big)$
$=\frac{231}{1000}\Bigg(\frac{1}{1-\frac{1}{1000}}\Bigg)$
$0.\overline{231}=\frac{231}{999}$

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