Question

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

• A. $1:1$
• B. $2:1$
• C. $3:1$
• D. $4:1$

Answer 1

Answer: (D)

Harmonic mean of $a,b$ is $H=\frac{2ab}{a+b}$
Geometric mean $G=\sqrt{ab}$
Given: $\frac{H}{G}=\frac{4}{5}$,
so $\frac{2\sqrt{ab}}{a+b}=\frac{4}{5}$
or, $\frac{a+b}{2\sqrt{ab}}=\frac{5}{4}$
By componendo and dividendo
$\frac{(\sqrt{a}+\sqrt{b}{)}^2}{(\sqrt{a}-\sqrt{b}{)}^2}=\frac{9}{1}$
or $\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}}=\frac{3}{1}$
Again, by componendo and dividendo $\frac{2\sqrt{a}}{2\sqrt{b}}=\frac{3+1}{3-1}$
$\frac{\sqrt{a}}{\sqrt{b}}=2$
or $\frac{a}{b}=\frac{4}{1}$