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For two positive numbers, the ratio of their harmonic mean to geometric mean is 12:13. What is the ratio of the two numbers?

1. 1:2
2. 2:3
3. 3:4
4. None of these

Correct answer: None of these

Solution

H/G = 2sqrt(ab)/(a+b) = 2t/(t^2+1) with t = sqrt(a/b). Setting this to 12/13 gives 6t^2 - 13t + 6 = 0, so t = 3/2 (or 2/3) and a:b = t^2 = 9:4. Since 9:4 is not listed, the answer is None of these.

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