Question

If the ratio of harmonic mean of two numbers to their geometric mean is 12 : 13, find the ratio of the numbers.

• A. $\frac{4}{9}$ or $\frac{9}{4}$
• B. $\frac{2}{3}$ or $\frac{3}{2}$
• C. $\frac{2}{5}$ or $\frac{5}{2}$
• D. None of these

Answer 1

The correct option is A $\frac{4}{9}$ or $\frac{9}{4}$

Harmonic mean of two numbers = $\frac{2ab}{a+b}$
Geometric mean of two numbers = $\sqrt{ab}$
$\frac{2\sqrt{ab}}{a+b}=\frac{12}{13}$
$\frac{\sqrt{\frac{a}{b}}}{\frac{a}{b}+1}=\frac{6}{13}$
Solve using options. Clearly, option (a) satisfies. Hence, option (a) is the answer.