Find the two numbers whose A.M. is 25 and G.M. is 20. - Vidyadip

Question

Find the two numbers whose A.M. is 25 and G.M. is 20.

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Answer

Given,
A.M. = 25
G.M. = 20
Now,
$\text{A}.\text{M}.=\frac{\text{a}+\text{b}}{2}=25$
And, $\text{G}.\text{M}=\sqrt{\text{ab}}=20$
$\text{a}+\text{b}=50,\text{ab}=400$
$(\text{a}-\text{b})=\sqrt{(\text{a}+\text{b})^2-4\text{ab}}$
$=\sqrt{(50)^2-16000}$
$=\sqrt{2500-1600}$
$=\pm30$
$\text{a}-\text{b}=\pm30\\ {\text{a}+\text{b}=50}\\ \overline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \ \ \text{2a} = \ \ \ 80$
$\text{a}=40$
Also, $-2\text{b}=-20$
$\text{b}=10$
$\therefore$ The numbers are 40, 10.

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