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Question
Find the rational numbers having the following decimal expansion:
\[0 . 6\overline8\]
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Solution
\[0 . 6\overline8\]
\[\text { Let } S = 0 . 6\overline8\]
\[ \Rightarrow S = 0 . 6 + 0 . 08 + 0 . 008 + 0 . 0008 + 0 . 00008 + . . . \infty \]
\[ \Rightarrow S = 0 . 6 + 0 . 08\left( 1 + {10}^{- 1} + {10}^{- 2} + {10}^{- 3} + . . . \infty \right)\]
\[\text { It is a G . P } . \]
\[ \therefore S = 0 . 6 + 0 . 08\left( \frac{1}{1 - {10}^{- 1}} \right)\]
\[ \Rightarrow S = 0 . 6 + \frac{0 . 8}{9}\]
\[ \Rightarrow S = \frac{6 . 2}{9}\]
\[ \Rightarrow S = \frac{62}{90} = \frac{31}{45}\]
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