Question : A fraction is greater than twice its reciprocal by $\frac{7}{15}$. What is the fraction?
Option 1: $\frac{3}{5}$
Option 2: $\frac{5}{3}$
Option 3: $\frac{3}{4}$
Option 4: $\frac{4}{3}$
Correct Answer: $\frac{5}{3}$
Solution :
Let the fraction be $x$.
According to the question,
$x-\frac{2}{x}=\frac{7}{15}$
$\Rightarrow 15x^2-7x-30=0$
$\Rightarrow 15x^2-25x+18x-30=0$
$\Rightarrow 5x (3x-5)+6(3x-5)=0$
$\Rightarrow (3x-5)(5x+6)=0$
$\Rightarrow (3x-5)=0$ or $(5x+6)=0$
$\Rightarrow x=\frac{5}{3},\frac{-6}{5}$
Hence, the correct answer is $\frac{5}{3}$.
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