The geometric mean of A and B is 50% more than A. Then the value of B is how…
2024
The geometric mean of A and B is 50% more than A. Then the value of B is how much percentage more than A?
- A.
50%
- B.
75%
- C.
100%
- D.
125%
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Correct answer: D
The geometric mean (GM) of two positive numbers A and B is defined as GM = √(A×B); squaring this identity, GM2 = A×B, always holds regardless of the specific values of A and B.
The geometric mean is given as 50% more than A, so GM = 1.5·A.
Apply the identity: GM2 = A×B ⇒ (1.5·A)2 = A×B ⇒ 2.25·A2 = A×B.
Divide both sides by A (A > 0): B = 2.25·A.
Convert to a percentage increase: ((B − A)/A) × 100 = ((2.25A − A)/A) × 100 = 125%.
Verify with A = 100: B = 225, and √(100×225) = √22500 = 150, which is exactly 50% more than 100 — consistent with the given condition.
So B is 125% more than A.