Discussion
A fraction is greater than twice its reciprocal by 7/15. What is the fraction ?
Answer: B
The fraction is \(\frac{5}{3}\).
Test the condition \(x - \frac{2}{x} = \frac{7}{15}\):
\[ \begin{aligned} \frac{5}{3} - 2 \times \frac{3}{5} &= \frac{5}{3} - \frac{6}{5} \\ &= \frac{25 - 18}{15} = \frac{7}{15} \end{aligned} \]
For \(\frac{3}{5}\) the result is \(\frac{3}{5} - \frac{10}{3}\), negative; for \(\frac{4}{3}\) it is \(\frac{4}{3} - \frac{3}{2} \lt 0\); for \(\frac{3}{4}\) it is also negative. Since \(x\) must exceed twice its reciprocal, it must be greater than \(\sqrt{2}\), and only \(\frac{5}{3}\) is.
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