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

Question

Find the rational number having the following decimal expansions:$0.\overline{68}$

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Answer

The rational number can be written as:
$0.\overline{68}=0.6+0.08+0.008+0.0008+\ \dots\infty$
$=\frac35+8[0.01+0.001+0.0001+0.00001+\ \dots\infty]$
$=\frac{3}{5}+8\Big[\frac{1}{100}+\frac{1}{100}+\dots\infty\Big]$
This is an infinite G.P. with first term $\frac{1}{100}$ and common ratio $\frac{1}{10}$
$=\frac{3}{5}+8\times\frac{1}{100}\times\frac{1}{1-\frac{1}{10}}$
$=\frac{3}{5}+\frac{4}{45}$
$=\frac{31}{45} $

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