Question

The A.M. of two numbers exceeds the G.M. by $\frac{3}{2}$ and the G.M. exceeds the H.M. by $\frac{6}{5};$ find the larger number.

Answer 1

Let the numbers be $a$ and $b,$ now using the relation
$G^2=AH=(G+\frac{3}{2})(G-\frac{6}{5})=G^2+\frac{3}{10}G-\frac{9}{5}$
$\Rightarrow$ $G=6$
i.e. $ab=36$
also $a+b=15$
Hence the two numbers are $3$ and $12.$