If the arithmetic mean of two numbers, a and b, a > b > 0, is five times their geometric mean, then what is (a+b)/(a-b) equal to?

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1 Answer
Mar 18, 2018

(a+b)/(a-b) = (5sqrt(6))/12

Explanation:

From the information we are given:

1/2(a+b) = 5sqrt(ab)

Multiply both sides by 2 to get:

a+b = 10sqrt(ab)

Square both sides to get:

(a+b)^2 = a^2+2ab+b^2 = 100ab

Subtract 4ab from both ends to get:

(a-b)^2 = a^2-2ab+b^2 = 96ab

So:

((a+b)/(a-b))^2 = (100ab)/(96ab) = 25/24 = (5^2 * 6)/12^2 = ((5sqrt(6))/12)^2

So since a > b we can take the positive square root to find:

(a+b)/(a-b) = (5sqrt(6))/12