If AM and HM of two numbers are $27$ and $12$ respectively, then their GM is:

9

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18

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24

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36

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Solution

The correct option is A $18$
Let $a$ and $b$ be two numbers, then
$A M = \frac{a + b}{2}$
$\Rightarrow 27 = \frac{a + b}{2}$
$\Rightarrow a + b = 54$
and $H M = \frac{2 a b}{a + b}$
$\Rightarrow 12 = \frac{2 a b}{54}$
$\Rightarrow a b = 324$
$∴ G M = \sqrt{a b} = \sqrt{324} = 18$

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