Question

The ratio of the harmonic mean to the geometric mean of two positive numbers is $\frac{12}{13}$. The numbers are in the ratio.

• A. $\frac{12}{13}$
• B. $\frac{13}{12}$
• C. $\frac{4}{3}$
• D. $\frac{4}{9}$

Answer 1

The correct option is D $\frac{4}{9}$

Let the two numbers be a and b, then
Harmonic mean is H $=\frac{2ab}{a+b}$
Geometric mean is G $=\sqrt{ab}$
Given $\frac{H}{G}=\frac{12}{13}$
Therefore, 13H = 12G
Squaring on both sides, we get a quadratic equation in $a/b$
$36\frac{a^2}{b^2}-97\frac{a}{b}+36=0$
$\frac{a}{b}=\frac{4}{9}$ and $\frac{1}{2}$
Hence, option 'D' is correct.