Question

Let the harmonic mean and geometric mean of two positive numbers be in the ratio $4:5$. Then the two numbers are in the ratio _________.

Answer 1

Let $a$ and $b$ be two positive numbers. Then, H.M. = $\frac{2ab}{a+b}$ and G.M. = $\sqrt{ab}$. According to question, H.M.; G.M. = $4:5$

$\therefore \frac{2ab}{(a+b)\sqrt{ab}}=\frac{4}{5}$

$\Rightarrow \frac{2\sqrt{ab}}{a+b}=\frac{4}{5}$

$\Rightarrow \frac{a+b+2\sqrt{ab}}{a+b-2\sqrt{ab}}=\frac{5+4}{5-4}$

$\Rightarrow (\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}}{)}^2=9$

$\Rightarrow \frac{(\sqrt{a}+\sqrt{b})}{\sqrt{a}-\sqrt{b}}=3,-3$

$\Rightarrow \frac{2\sqrt{a}}{2\sqrt{b}}=\frac{3+1}{3-1},\frac{-3+1}{-3-1}$

$\Rightarrow \frac{\sqrt{a}}{\sqrt{b}}=2,\frac{1}{2}\Rightarrow \frac{a}{b}=4,\frac{1}{4}$

$\Rightarrow a:b=4:1or1:4$