Question

If the ratio of two numbers be $9:1$, then the ratio of geometric and harmonic means between them will be

• A. $1:9$
• B. $5:3$
• C. $3:5$
• D. $2:5$

Answer 1

The correct option is B

$5:3$

Explanation for correct option

Let the two number be $a$ and $b$

Given that the two number are in the ratio of $9:1$

$$\begin{gathered} \therefore \frac{a}{b}=\frac{9}{1} \\ \Rightarrow a=9b \end{gathered}$$

The geometric mean$\left(G.M\right)$ of the two number $a$ and $b$ is $=\sqrt{ab}=\sqrt{9b·b}=3b$

The harmonic mean$\left(H.M\right)$ of the two number $a$ and $b$ is $=\frac{2ab}{a+b}=\frac{2·9b·b}{9b+b}=\frac{18b^2}{10b}=\frac{9b}{5}$

Therefore the ratio between the geometric and harmonic means is

$$\begin{gathered} \Rightarrow \frac{G.M}{H.M}=\frac{3b}{{\displaystyle \frac{9b}{5}}} \\ \Rightarrow \frac{G.M}{H.M}=\frac{5}{3} \\ \therefore G.M:H.M=5:3 \end{gathered}$$

Hence, option B is correct.