Question

the arithmetic mean of two quantities is 5 and geometric mean is 4 .Find their harmonic mean and also the quantities

Answer 1

Let a and b be two positive two positive numbers.


Then, H. M. = 2ab/a + b. and G. M. = √ab


ATQ HM : GM = 4 : 5


∴ 2ab/(a + b) √ab = 4/5


2√ab / a + b = 4/5 a + b + 2√ab/a + b 2√ab = 5 + 4/ 5 – 4


(√a + √b)2 /(√a - √b)2 = 9/1 √a + √b/√a - √b 3, -3


2√a/2√b = 3 + 1/3 – 1, -3 + 1/ -3 – 1


√a/√b = 2, 1/2 a/b = 4, 1/4 a : b = 4 : 1 or 1 : 4


ALTERNATE SOLUTION:


Left for two + ve no. ‘s a and b, a/b = m


Then G = √ab = b√m and H = 2ab/a + b = 2nb√m/b + bm


∴ H/G = 4/5 2√m /m + 1 = 4/5 5√m = 2m + 2


2m - 5√m + 2 = 0 (√m – 2) (√m – 1/2) = 0


m = 4 or 1/4 ∴ req. ratio = 4 : 1 or 1 : 4.