Find the two numbers whose A.M. is 25 and GM is 20.

Solution:

Let A.M. and G.M. between the two numbers $a$ and $b$ be $A$ and $G$, respectively.

$A=25$

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

$\Rightarrow a+b=50$     ...(i)

Also $G=20$

$\Rightarrow \sqrt{a b}=20$

$\Rightarrow a b=400$    ....(ii)

Now, putting the value of $a$ in $($ ii $)$ :

$\Rightarrow(50-b) b=400$

$\Rightarrow b^{2}-50 b+400=0$

$\Rightarrow b^{2}-10 b-40 b+400=0$

$\Rightarrow b(b-10)-40(b-10)=0$

$\Rightarrow(b-10)(b-40)=0$

$\Rightarrow \mathrm{b}=10,40$

If $b=10$, then, $a=400$.

And, if $b=40$, then $a=10 .$

Thus, the two numbers are 10 and $40 .$