Question

If the ratio of harmonic mean between two quantities to their geometric means is $12:13$, prove that the quantities are in the ratio of $4:9$.

Answer 1

Let $a$ and $b$

$\therefore HM=\frac{2ab}{a+b}$ and $GM=\sqrt{ab}$

$HM:GM=\left(\frac{2ab}{a+b}\right):\sqrt{ab}=12:13$

$⟹\frac{a+b}{\sqrt{ab}}=\frac{13}{6}$

$⟹\sqrt{a/b}+\sqrt{b/a}=13/6$

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

$⟹x+\frac{1}{x}=\frac{13}{6}$

$⟹6x^2-13x+6=0$

$⟹x=\frac{2}{3}$

$⟹x=\sqrt{a/b}$

$⟹x^2={\left(\frac{2}{3}\right)}^2=\frac{4}{9}$

$\therefore a:b=4:9$

Hence proved.