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1. If the difference between the reciprocal of a positive proper fraction and the fraction itself be $\frac{9}{20}$ , then the fraction is :

A. $\frac{3}{5}$

B. $\frac{4}{5}$

C. $\frac{5}{4}$

D. $\frac{3}{10}$

Solution:

Let the fraction be

\frac{a}{1}

Then,

\begin{aligned} \Leftrightarrow \frac{1}{a} - a = \frac{9}{20} \\ \Leftrightarrow \frac{1 - a^{2}}{a} = \frac{9}{20} \\ \Leftrightarrow 20 - 20 a^{2} = 9 a \\ \Leftrightarrow 20 a^{2} + 9 a - 20 = 0 \\ \Leftrightarrow 20 a^{2} + 25 a - 16 a - 20 = 0 \\ \Leftrightarrow 5 a (4 a + 5) - 4 (4 a + 5) = 0 \\ \Leftrightarrow (4 a + 5) (5 a - 4) = 0 \\ \Leftrightarrow a = \frac{4}{5} [∵ a \neq - \frac{5}{4}] \end{aligned}

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