Question
Find the rational number having the following decimal expansions:$3.\overline{52}$
✓
Answer
$3.\overline{52}=3+0.52222\ \dots$$=3+0.5+0.02+0.002+0.0002+\ \dots$
$=3.5+\frac{2}{10^2}+\frac{2}{10^3}+\frac{2}{10^4}+\ \dots$
$=3.5+\frac{2}{10^2}\Big(1+\frac{1}{10}+\frac{1}{10^2}+\ \dots\Big)$
$=\frac{35}{10}+\frac{2}{100}\Bigg(\frac{1}{1-\frac{1}{10}}\Bigg)$
$=\frac{35}{10}+\frac{2}{100}\times\Big(\frac{10}{9}\Big)$
$=\frac{35}{10}+\frac{2}{90}$
$=\frac{315+2}{90}$
$3.\overline{52}=\frac{317}{90}$
$=3.5+\frac{2}{10^2}+\frac{2}{10^3}+\frac{2}{10^4}+\ \dots$
$=3.5+\frac{2}{10^2}\Big(1+\frac{1}{10}+\frac{1}{10^2}+\ \dots\Big)$
$=\frac{35}{10}+\frac{2}{100}\Bigg(\frac{1}{1-\frac{1}{10}}\Bigg)$
$=\frac{35}{10}+\frac{2}{100}\times\Big(\frac{10}{9}\Big)$
$=\frac{35}{10}+\frac{2}{90}$
$=\frac{315+2}{90}$
$3.\overline{52}=\frac{317}{90}$
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