If the ratio of harmonic mean of two numbers to their geometric mean is 12 : 13, find the ratio of the numbers.
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Let a and b be two numbers.
Harmonic Mean, HM = 2ab(a+b)
geometric Mean, GM = √(ab)
2ab(a+b) : √(ab) = 12:13
(a+b)/√(ab) = 13/6
√(a/b) +√(b/a) = 13/6
Let √(a/b) = x
x + 1/x = 13/6
6x2 - 13x + 6 = 0
x = 2/3 or 3/2
a/b = x2
a/b = 4/9
Harmonic Mean, HM = 2ab(a+b)
geometric Mean, GM = √(ab)
2ab(a+b) : √(ab) = 12:13
(a+b)/√(ab) = 13/6
√(a/b) +√(b/a) = 13/6
Let √(a/b) = x
x + 1/x = 13/6
6x2 - 13x + 6 = 0
x = 2/3 or 3/2
a/b = x2
a/b = 4/9
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