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Expressing Decimals as Rational Numbers

Question

Express the following to the form $\frac{p}{q}$ , where $p$ and $q$ are integers and $q \neq 0$ .
$0. ¯ ¯¯¯¯¯¯ ¯ 001$ .

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Solution

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$x = 0.001$
$x = 0.001001001$
$100 x = .100100100 . . .$
$100000 x = 100.100100$
$99900 x = 100$
$x = \frac{100}{99900}$
$x = \frac{11}{999}$

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