Question

If A.M and G.M. of two positive numbers $a$ and $b$ are $10$ and $8$, respectively, find the numbers.

Answer 1

$A.M=\frac{a+b}{2}=10$

$\Rightarrow a+b=20$ ........$\left(1\right)$

$G.M=\sqrt{ab}=8$

$\Rightarrow ab=64$ ........$\left(2\right)$

From $\left(1\right)$ we have $b=20-a$

Substituting $b=20-a$ in $\left(2\right)$ we get

$a(20-a)=64$

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

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

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

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

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

$\therefore a=16,4$

$\Rightarrow b=20-16,20-4$ using $b=20-a$

$\therefore b=4,16$

Hence the numbers are $(a,b)=(16,4)$ or $(4,16)$