Question

The A.M. between two positive numbers exceeds the GM by $5$, and Gm exceed the H.M. by $4$. Then the numbers are

• A. $10,40$
• B. $10,20$
• C. $20,40$
• D. $10,50$

Answer 1

The correct option is A $10,40$

Given

A.M $=5+$GM -$\left(1\right)$

G.M $=4+$H.M - $\left(2\right)$

We know

$G.M^2=AM\cdot HM$

$G{M}^2=(5+GM)(GM-4)$

$G{M}^2=5GM-20+G{M}^2-4GM$

$\therefore GM=20$

$AM=25$ (from $\left(1\right)$)

$\sqrt{ab}=20$, $\frac{a+b}{2}=25\Rightarrow a+b=50$

$\Rightarrow ab=400$ …$\left(3\right)$

$\Rightarrow (a-b{)}^2=(a+b{)}^2-4ab$

$={50}^2-4\times 400$

$=2500-1600$

$=900$

$\therefore a-b=\sqrt{900}=30$ …$\left(4\right)$

from $\left(3\right)$ & $\left(4\right)$

$a+b=50$

$a-b=30$

______________

$2a=80$

$a=40$ & $b=10$

$\therefore$ The numbers are $(40,10)$.