Question

If AM between two numbers exceeds their GM by $\frac{5}{2}$ and the GM exceeds their HM by 2, the ratio of numbers is

• A. 1:4
• B. 4:1
• C. 1:2
• D. 2:1

Answer 1

Answer: (A), (B)

The correct options are
A

1:4

B

4:1

Let A be A.M., G be G.M. and H be HM between 2 numbers a & b

Given A = G + $\frac{5}{2}$ --------------(1)

G = H + 2 $\Rightarrow$ H = G - 2 -------------(2)

We know that $G^2$ = AH

$G^2$ = (G + $\frac{5}{2}$)(G-2) [from (1) & (2)]

$\text{/}{G}^{\text{/}2}=\text{/}{G}^{\text{/}2}$ -2G + $\frac{5}{2}$G - 5

-4G + 5G = 10

G = 10

$\Rightarrow$ ab = 100 ----------(3)

From (1), A = $\frac{a+b}{2}$ = $\frac{25}{2}$

a+b = 25 ----------(4)

Solving (3) & (4), we get

a + $\frac{100}{a}$ = 25

$a^2$ - 25a + 100 = 0

$a^2$ - 20a - 5a + 100 = 0

a(a - 20) - 5(a -20) = 0

a = 5 (or) a = 20

$\Rightarrow$ b = 20 (or) b = 5

So, the ratio of numbers is 1:4 (or) 4:1.