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Q.
If the ratio of $H.M$. to the $G.M$. of two numbers is $6 :10$ then the ratio of the numbers is
Sequences and Series
Solution:
Let the number are $a \&b$
$\therefore \frac{H.M. of a \&b}{G. M. of a \&b} = \frac{6}{10}$
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 $a: b = 1: 9$