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Question

Represent the following mixed infinite decimal periodic fractions as common fractions:
$\frac{x + 1}{1 + x + x} : \frac{1}{x ^{2} - x}$

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Solution

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$\frac{x + 1}{1 + x + x} : \frac{1}{x ^{2} - x}$

$= \frac{( x + 1 ) ( x - 1 )}{( 1 + x + x ) ( x - 1 )} : \frac{1}{x ( x ^{2 - \frac{1}{2}} - 1 )}$

$= \frac{( x ) ^{2} - 1}{( x ) ^{3} - 1} \times x (x^{\frac{3}{2}} - 1)$

$= \frac{( x - 1 )}{x ^{\frac{3}{2}} - 1} x (x^{\frac{3}{2}} - 1)$

$= x (x - 1)$

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