Let G be the geometric mean of two positive numbers a and b, and M be the arithmetic mean of 1/a and 1/b. If 1/M : G is 4 : 5, then a : b can be:

G = √ab, M = (a+b)/(2ab)
1/M : G = 4 : 5
(2ab)/(a+b) : √ab = 4 : 5
(2ab)/(a+b) = (4√ab)/5
10ab = 4√ab(a+b)
25a²b² = 4ab(a+b)²
25ab = 4(a+b)²
25ab = 4(a²+b²+2ab)
25ab = 4a²+4b²+8ab
4a²-17ab+4b²=0
Let x = a/b
4x²-17x+4=0
(4x-1)(x-4) = 0
x = 1/4 or x = 4
a/b = 1/4 or a/b = 4
Therefore, a : b can be 1 : 4 or 4 : 1