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The geometric mean and harmonic mean of two non negative observations are 10 and 8 respectively. Then what is the arithmetic mean of the observations equal to?
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Verified Answer
The correct answer is:
12,5
Let 'a' and 'b' be two non-negative numbers. $\mathrm{GM}=\sqrt{\mathrm{ab}}=10$
$\Rightarrow \quad a b=100$
and $\mathrm{H.M.}=\frac{2 \mathrm{ab}}{\mathrm{a}+\mathrm{b}}=8$
$\Rightarrow \frac{200}{a+b}=8$
$\Rightarrow \quad a+b=25$
Consider $(a-b)^{2}=(a+b)^{2}-4 a b=625-400=225$
$\Rightarrow a-b=15$
and $a+b=25$
$\Rightarrow 2 a=40 \Rightarrow a=20$ and $b=5$
A.M. $=\frac{20+5}{2}=12.5$
$\Rightarrow \quad a b=100$
and $\mathrm{H.M.}=\frac{2 \mathrm{ab}}{\mathrm{a}+\mathrm{b}}=8$
$\Rightarrow \frac{200}{a+b}=8$
$\Rightarrow \quad a+b=25$
Consider $(a-b)^{2}=(a+b)^{2}-4 a b=625-400=225$
$\Rightarrow a-b=15$
and $a+b=25$
$\Rightarrow 2 a=40 \Rightarrow a=20$ and $b=5$
A.M. $=\frac{20+5}{2}=12.5$
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