A sum
I have two numbers.
The
geometric mean
of these two numbers is larger than the smaller number by 12.
The
arithmetic mean
of these numbers is smaller than the larger number by 24.
Find the sum of these two numbers.
Courtesy: AMTI 2015
The answer is 60.
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1 solution
Let the larger number be a , and the smaller number be b
Translate the two statements above into equations:
a b = b + 1 2 ⟹ Eq.(1)
2 a + b = a − 2 4 a + b = 2 a − 4 8
a = b + 4 8 ⟹ Eq.(2)
Substitute Eq.(2) into Eq.(1):
b ( b + 4 8 ) = b + 1 2 b ( b + 4 8 ) = ( b + 1 2 ) 2 b 2 + 4 8 b = b 2 + 2 4 b + 1 4 4 2 4 b = 1 4 4 b = 6 a = 6 + 4 8 = 5 4 a + b = 5 4 + 6 = 6 0