Question

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

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

Answer 1

Answer: (A), (B)

The correct options are
A $1:4$
B $4:1$

Let the numbers are $a$ and $b,$ so
$\text{H.M.}=\frac{2ab}{a+b}$
$\text{G.M.}=\sqrt{ab}$
Given, $\text{H.M.}:\text{G.M.}=4:5$
$\Rightarrow \frac{\frac{2ab}{a+b}}{\sqrt{ab}}=\frac{4}{5}$
$\Rightarrow \frac{2\sqrt{ab}}{a+b}=\frac{4}{5}$
$\Rightarrow 2(a+b)=5\sqrt{ab}$

Squaring both the sides
$4(a^2+2ab+b^2)=25ab$
$\Rightarrow 4a^2-16ab+4b^2=0$
$\Rightarrow (a-4b)(4a-b)=0$
$\Rightarrow \frac{a}{b}=4,\frac{1}{4}$
Hence, the correct answer is option (A) and (B).