lean_workbook_plus_45482

Let $\frac{m}{n}=a$ . We're given that $a=\frac{7+\frac{1}{a}}{65-\frac{1}{a}}=\frac{7a+1}{65a-1}$ . Cross-multiplying, we get $65a^2-a=7a+1\implies 65a^2-8a-1=0$ . From the quadratic formula, we have $a=-\frac{1}{13}$ or $\frac{1}{5}$ . It's given that $a$ is positive, so we must have $a=\frac{1}{5}$ .