Question

If the harmonic mean between two quantities is to their geometric mean as $24$ to $25$,then the quantities are in the ratio

• A. $3:16$
• B. $4:9$
• C. $4:3$
• D. $16:9$

Answer 1

The correct option is D $16:9$

Let a and b be two numbers

Harmonic mean, $HM=2ab(a+b)$

Geometric mean, $GM=\sqrt{ab}$

$2ab(a+b):\sqrt{ab}=24:25$

$\frac{(a+b}{)}\sqrt{ab}=\frac{25}{12}$

$\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}=\frac{25}{12}$

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

$x+\frac{1}{x}=\frac{25}{12}$

$12x^2-25x+12=0$

$x=\frac{4}{3}or\frac{3}{4}$

$\frac{a}{b}=x^2$

$\frac{a}{b}=\frac{16}{9}$