Question

If sum of A.M. and H.M between two positive numbers is 25 and their GM is 12, then sum of numbers is.

• A. $9$
• B. $18$
• C. $32$
• D. $18$ or $32$

Answer 1

The correct option is C $32$

Let $a$ and $b$ are the numbers.
A.M = $\frac{a+b}{2}$
H.M = $\frac{2ab}{a+b}$
G.M = $\sqrt{ab}$
But given, $AM+HM=25$ and $GM=12$
$\Rightarrow \frac{a+b}{2}+\frac{2ab}{a+b}=25$ and $ab=144$
Let $a+b=t$
$\Rightarrow \frac{t}{2}+\frac{288}{t}=25$
$\Rightarrow t^2-50t+576=0$
Solving above quadratic gives
$t=-18$ or $t=32$
As $a,b$ are positive numbers.
$\therefore t=a+b=32$
Hence, option C.