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Question
Find the rational number whose decimal expansion is \[0 . 423\].
Solution
\[\text { Let the rational number S be } 0 . 4\overline{23} . \]
\[ \because S = 0 . 4\overline{23}= 0 . 4 + 0 . 023 + 0 . 00023 + 0 . 0000023 + . . . \infty \]
\[ \Rightarrow S = 0 . 4 + 0 . 023\left[ 1 + {10}^{- 2} + {10}^{- 4} + . . . \infty \right]\]
\[\text { Clearly, S is a geometric series withthe first term, a, being1 and the common ratio, r, being } {10}^{- 2} . \]
\[ \therefore S = 0 . 4 + 0 . 023\left[ \frac{1}{1 - {10}^{- 2}} \right]\]
\[ \Rightarrow S = 0 . 4 + \frac{2 . 3}{99}\]
\[ \Rightarrow S = \frac{419}{990}\]
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