Express the following recurring decimals as rational numbers. 2.3… - Vidyadip

Question

Express the following recurring decimals as rational numbers.
$2.3 \overline{5}$

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Answer

$2.3 \overline{5}=2.3555 \ldots=2.3+0.05+0.005+0.0005+\ldots$
The terms $0.05,0.005,0.0005,…$ are in G.P.
$\therefore a=0.05, r=\frac{0.005}{0.05}=0.1,|r|=|0.1|<1$
$\therefore$ Sum to infinity exists.
$\therefore$ Sum to infinity
$=2.3+\frac{a}{1-r}$
$=2.3+\frac{0.05}{1-0.1}$
$=2.3+\frac{0.05}{0.9}$
$=\frac{23}{10}+\frac{5}{90}$
$=\frac{212}{90}=\frac{106}{45}$

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