Question

If AM and GM of two positive numbers a and b are 10 and 8 respectively, find the numbers.

Answer 1

AM = 10

$\therefore \frac{a+b}{2}=10$

$\Rightarrow a+b=20$ ......... (i)

Also, G = 8

$\therefore \sqrt{ab}=8$

$\Rightarrow ab=8^2$

$\Rightarrow ab=64$ ............(ii)

Using (i) and (ii);

$\Rightarrow a(20-a)=64$

$\Rightarrow a^2-20a+64=0$

$\Rightarrow a^2-16a-4a+64=0$

$\Rightarrow a(a-16)-4(a-16)=0$

$\Rightarrow (a-16)(a-4)=0$

$\Rightarrow a=4,16$

If a = 4, then b = 16

And, if a = 16, then b = 4