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Show that \(0.2353535... =0.2\overline{35}\) can be expressed in the form \(\frac {p}{q},\) where \(p\) and \(q\) are integers and \(q\neq 0.\)
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Question

Show that \(0.2353535... =0.2\overline{35}\) can be expressed in the form \(\frac {p}{q},\) where \(p\) and \(q\) are integers and \(q\neq 0.\)

Answer

We have, \(x=0.2\overline{35}\).                     ...(i)
Multiplying eqn. (i) by \(10\), we get
\(10x=2.\overline{35}\)                             ...(ii)
Multiplying eqn. (ii) by \(1000\), we get
\(1000x=235.\overline{35}\)                   ...(iii)
Subtracting eqn. (ii) from eqn. (iii), we get
\(1000x-10x=235.\overline{35}-2.\overline{35}\)
\(\Rightarrow 990x=233\)
\(\Rightarrow x=\frac{233}{990}\).
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