Question

If the harmonic mean between two positive numbers is to their Geometric mean as $12:13$, the numbers are in the ratio.

• A. $12:13$
• B. $1/12:1/13$
• C. $4:9$
• D. $1/4:1/9$

Answer 1

The correct option is C $4:9$

Let the two positives numbers be $a,b$.

Harmonic mean $=\frac{2ab}{a+b}$

Geometric mean $=\sqrt{ab}$

According to the question,

$\frac{\frac{2ab}{a+b}}{\sqrt{ab}}=\frac{12}{13}$

$\frac{2\sqrt{ab}}{a+b}=\frac{12}{13}$

$\frac{\sqrt{ab}}{a+b}=\frac{6}{13}$

$\frac{a+b}{\sqrt{ab}}=\frac{13}{6}$

$\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}=\frac{13}{6}$ $.........\left(1\right)$

Let $x=\sqrt{\frac{a}{b}}$

Therefore,

$x+\frac{1}{x}=\frac{13}{6}$

$6x^2-13x+6=0$

On solving above equation, we get

$x=3/2$ or $x=2/3$

So, the numbers are

$a:b=4:9$

Hence, this is the answer.