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: $\frac{4}{3}$
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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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