Find the rational form of the number 0 . 123 - Vidyadip

Question

Find the rational form of the number $0 . \overline{123}$

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Answer

$\begin{aligned} & \text { Let } x=0.123123123 \ldots \Rightarrow x=0 . \overline{123} \ldots(1) \\ & \text { Multiplying } 1000 \text { on both rides } \\ & 1000 x=123.123123 \ldots \Rightarrow 1000 x=123 . \overline{123} \ldots(2) \\ & (2)-(1)=1000 x-x 123 . \overline{123}-0 . \overline{123} \\ & \Rightarrow 999 x=123 \\ & \Rightarrow x=\frac{123}{999} \\ & \Rightarrow x=\frac{41}{333} \text { Rational number }\end{aligned}$

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