Question

If the ratio of the H.M. and GM. between two numbers a and bis 4 : 5, then a: b is

Options

• 1 : 2 or 2 : 1

• 1 : 3 or 3 : 1

• 1 : 4 or 4 : 1

• 2 : 3 or 3 : 2

Answer 1

1 : 4 or 4 : 1

Explanation:

G.M = `sqrt(ab)` and H.M = `(2ab)/(a + b)`

Given, `(H.M)/(G.M) = 4/5`

⇒ `(2ab)/(sqrt(ab)(a + b)) = 4/5`

⇒ `(4a^2b^2)/(ab(a + b)^2) = 16/25`

⇒ `25ab = 4(a^2 + b^2 + 2ab)`

⇒ `4a^2 + 4b^2 - 17ab` = 0

⇒ `4(a/b)^2 - 17(a/b) + 4` = 0

⇒ `4(a/b)^2 - 16(a/b) - (a/b) + 4` = 0

⇒ `[4(a/b) - 1][a/b - 4]` = 0

⇒ `a/b = 1/4` or `a/b = 4/1`