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Find the Rational Number Whose Decimal Expansion is 0 . 423 . - Mathematics

Find the rational number whose decimal expansion is \[0 . 423\].

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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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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 20 Geometric Progression
Exercise 20.4 | Q 7 | Page 40
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