# 3.  Express the following in the form $\frac{p}{q}$ , where p and q are integers and q ≠ 0. (iii)     $0.\overline{001}$

G Gautam harsolia

Let $x = 0.\overline{001}= 0.001001....$             -(i)

Now, multiply by 1000 on both sides

$1000x= 1.001001...$

$\Rightarrow 1000x = 1 + x \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(using \ (i))$

$\Rightarrow 999x = 1$

$\Rightarrow x = \frac{1}{999}$

Therefore,  $\frac{p}{q}$  form of  $0.\overline{001}$  is  $\frac{1}{999}$

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