Question

The harmonic mean of two numbers is 4. The arithmetic mean A & the geometric mean G satisfy the relation $2A+G^2=27$. Find the largest of those numbers?.

• A. $6$
• B. $4$
• C. $8$
• D. $5$

Answer 1

The correct option is A $6$

Let the numbers be $a$ and $b$

Then $4=\frac{2ab}{a+b}$

$A=\frac{a+b}{2}$

$G^2=ab$

$\therefore 4=\frac{G^2}{A}\Rightarrow 4A=G^2$

$2A+G^2=27\Rightarrow 6A=27\Rightarrow A=\frac{9}{2}$

$\Rightarrow a+b=9$ ...(1)

$G^2=4A=18\Rightarrow ab=18$

$\therefore {(a-b)}^2={(a+b)}^2-4ab={\left(9\right)}^2-4\left(18\right)=9$

$a-b=\pm 3$ ...(2)

From (1) and (2)

$a=6,b=3$ and $a=3,b=6$