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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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## QuestionMathsClass 9

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

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
$$1000x-10x=235.\overline{35}-2.\overline{35}$$
$$\Rightarrow 990x=233$$
$$\Rightarrow x=\frac{233}{990}$$.