Question

If the ratio of $H.M$. to the $G.M$. of two numbers is $6:10$ then the ratio of the numbers is

• A. $5:3$
• B. $3:1$
• C. $1:9$
• D. None of these

Answer 1

Answer: (C)

Let the number are $a\&b$
$\therefore \frac{H.M.ofa\&b}{G.M.ofa\&b}=\frac{6}{10}$
or $\frac{2ab}{a+b}\times \frac{1}{\sqrt{ab}}=\frac{6}{10}$
or $10\sqrt{ab}=3\left(a+b\right)$ or $3a-10\sqrt{ab}+3b=0$
or $\left(3\sqrt{a-\sqrt{b}}\right)\left(\sqrt{a-3\sqrt{b}}\right)=0$
$\therefore$ Either $\frac{a}{b}=\frac{1}{9}$ or $\frac{a}{b}=\frac{9}{1}$
But $\frac{a}{b}=\frac{1}{9}$ is true $\because \frac{H.M.}{G.M.}<1$
or $a:b=1:9$