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Show that \(0.3333\cdots =0.\overline 3\) 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 \(0.3333\cdots =0.\overline 3\) can be expressed in the form \(\frac {p}{q}\) where \(p\) and \(q\) are integers and  \(q\neq 0.\)

Answer

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Solution

As we do not know what \(0 . \overline{3}\) is, let us call it  \(x\)and so
\(x=0.3333 \ldots\)
\(\Rightarrow 10 x=10 \times(0.333 \dots)=3.333 \dots\)
So, \(3.3333 \ldots=3+x,\) since \(x=0.3333 \ldots\)
Therefore, \(10 x=3+x\)
Solving for \(x,\) we get
\(9 x=3\)
\(\Rightarrow x=\dfrac{1}{3}\)
Hence, \(0 . \overline{3}=\dfrac13\).
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