AM and GM (easy)

Algebra Level 3

The A.M of two numbers is smaller than the larger of the two numbers by 24 and the G.M of the same numbers exceeds the smaller number by 12.

Find those two numbers.

Enter your answer as the sum of cubes of these numbers.


The answer is 157680.

Sai Ram by Sai Ram , 19, India

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2 solutions

Sai Ram
Jun 29, 2015

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Toby M
Aug 2, 2020

From equation (2) in Sai Ram's post, a b = b 2 + 24 b + 144 ab = b^2+24b+144 . But a = b + 48 a = b + 48 from equation (1), so b ( b + 48 ) = b 2 + 24 b + 144 b(b+48) = b^2+24b+144 . This implies that b 2 + 48 b = b 2 + 24 b + 144 b^2+48b = b^2+24b+144 or 24 b = 144 ⇒ b = 6 24b = 144 \Rightarrow b = 6 . From here a = 54 a = 54 and you can verify that both equations (1) and (2) hold true.

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