Question

The AM of two numbers exceeds the GM by $\frac{3}{2}$ and the GM exceeds the HM by $\frac{6}{5}$, then find the sum of those numbers

Answer 1

Let the numbers be $a$ and $b$

Now using the relation $G^2=AH$
$\Rightarrow G^2=(G+\frac{3}{2})(G-\frac{6}{5})$
$\Rightarrow G^2=G^2+\frac{3}{10}G-\frac{9}{5}$
$\Rightarrow G=6$ and $A=\frac{15}{2}$
i.e. $ab=36$

Also $a+b=15$
Hence, the two numbers are $3$ and $12$.