Question

The harmonic mean of two numbers is $4$. Their arithmetic mean $A$ and geometric mean $G$ satisfy the relation $2A+G^2=27$. Find the numbers.

Answer 1

Given $HM=4$

$\because$ $AM\times HM={\left(GM\right)}^2$

$\therefore$ $4\times A=G^2=4A$

But it is given $2A+G^2=27$

$\therefore$ $2A+4A=27$ $\Rightarrow$ $6A=27$

$\Rightarrow$ $A=\frac{9}{2}$

and $G^2=18$

$\therefore$ $\frac{a+b}{2}=\frac{9}{2}$ and $ab=18$

$\Rightarrow$ $a+b=9$ and $ab=18$

$\Rightarrow$ $a=6,b=3$ or $a=3,b=6$

Hence, two numbers are $(3,6)$