Question

For two positive real number's $a$ and $b$, If the A.M. exceeds their G.M. by $2$ and the G.M. exceeds their H.M. by $\frac{8}{5}$, then the value of $a+b$ is

• A. $20$
• B. $30$
• C. $40$
• D. $50$

Answer 1

The correct option is A $20$

Let the A.M., G.M. and H.M. be $A,G$ and $H$ respectively,
$A-G=2\cdots \left(1\right)$
$G-H=\frac{8}{5}\cdots \left(2\right)$
We know that,
$G^2=AH$
Using equation $\left(1\right)$ and $\left(2\right)$,
$\Rightarrow G^2=(G+2)(G-\frac{8}{5})\Rightarrow G^2=G^2+(2-\frac{8}{5})G-\frac{16}{5}\Rightarrow \frac{2G}{5}=\frac{16}{5}\Rightarrow G=8$
Now,
$A=\frac{a+b}{2}=2+G\therefore a+b=20$