Question

If the A.M. of the two numbers exceeds their G.M. by $10$ and their H.M. by $16$, find the numbers.

Answer 1

$A=G+10$

$A=H+16$

$G+10=H+16$

$G-6=H$

$G^2=AH$

$G^2=(G+10)(G-6)$

$G^2=G^2-6G+10G-60$

$G=15=\sqrt{ab}$

$a=\frac{225}{b}$

$A=25=\frac{a+b}{2}$

$a+b=50$

$b^2+225=50b$

$b=45$ or $b=5$

$a=5$ or $a=45$