Question

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

Answer 1

Step $1:$ Using $A.M.$ and $G.M.$ of two numbers.
We know, $A.M.$ of two numbers $a$ and $b$ is given by
$A.M.=\frac{a+b}{2}$
$\Rightarrow \frac{a+b}{2}=10$
$\Rightarrow a+b=20\cdots \left(i\right)$

Also, $G.M.$ of two numbers $a$ and $b$ is given by
$GM=\sqrt{ab}$
$\Rightarrow \sqrt{ab}=8$
$\Rightarrow ab=64\cdots \left(ii\right)$

Step $2:$ Solving both equations to find $a$ and $b$.
From $\left(ii\right),$ Putting $a=\frac{64}{b}$ in equation $\left(i\right),$ we get
$\frac{64}{b}+b=20$
$\Rightarrow 64+b^2=20b$
$\Rightarrow b^2-20b+64=0$

Step $3:$ Solving quadratic equation by factorization method.
$b^2-20b+64-0$
$\Rightarrow b^2-4b-16b+64=0$
$\Rightarrow b(b-4)-16(b-4)=0$
$\Rightarrow (b-4)(b-16)=0$
$\Rightarrow b=4$ or $b=16$
Substituting $b=4$ in equation $\left(ii\right)$
$a=\frac{64}{4}=16$
Substituting $b=16$ in equation $\left(ii\right)$
$a=\frac{64}{16}=4$

Thus, the numbers $a$ and $b$ are $4,16$ or $16,4$ respectively.