The arithmetic mean of 2 numbers is 34 and their geometric mean is 16. One of the numbers will be a. 4 b. 16 c. 18 d. 12?

Question

The arithmetic mean of 2 numbers is 34 and their geometric mean is 16. One of the numbers will be a. 4 b. 16 c. 18 d. 12?

2 Answers

Answer Question

Answer 1

This is equivalent to saying the product of the two numbers is 16^2 = 256 and the sum of the 2 numbers is 2*34 = 68. The numbers are the solutions to the quadratic
  x(68-x) = 256
This has solutions
  x = 34 ± √(1156-256)
  x = 34 ± 30

One of the numbers is 4; the other number is 64.

Answer 2

Answers C and D can be eliminated because they both contain factors of 3. If the geometric mean is 16, one of the numbers cannot be 16 unless the other one is, too. In that case, the mean would be 16, so answer B can be eliminated.
Answer A is the correct one.

The product of the two numbers must be 16^2 = 256. Thus each of the two numbers must be a power of 2. Their total must be 68 = 2*34.
The numbers 64 = 2^6 and 4 = 2^2 are the two numbers.