Question

The A.M. of two numbers exceeds their G.M. by $15$ and H.M. by $27$. Find the numbers.

Answer 1

Given,

$AM=\frac{a+b}{2}$

$HM=\frac{2ab}{a+b}$

$GM=\sqrt{\frac{a}{b}}$

AM exceeds GM by $15$

$\frac{a+b}{2}=\sqrt{\frac{a}{b}}+15$

AM exceeds HM by $27$

$\frac{a+b}{2}=\frac{2ab}{a+b}+27$

$\Rightarrow \sqrt{\frac{a}{b}}+15=\frac{2ab}{a+b}+27$

solving the above equation, we get,

$a=120,b=30$