If A and G are AM and GM between two positive numbers a and b are connected by the relation A+G=a-b then the numbers are in the ratio
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A = (a+b)/2
G = √(ab)
given is A+G = a–b,
substitute, we get (a+b)/2+√(ab) = a–b
, arrange and square both sides,
a²–10ab+9b² = 0,
divide by b², we get a quadratic in (a/b),
(a/b)² –10a/b + 9 = 0......
upon solving we get,
a/b = 1, 9.
G = √(ab)
given is A+G = a–b,
substitute, we get (a+b)/2+√(ab) = a–b
, arrange and square both sides,
a²–10ab+9b² = 0,
divide by b², we get a quadratic in (a/b),
(a/b)² –10a/b + 9 = 0......
upon solving we get,
a/b = 1, 9.
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