Find two positive numbers such that their mean proportional (geometric mean) is 8 and the third proportional to these two numbers is 64.

Introduction / Context:
This question is similar to an earlier one on mean and third proportional, but now a different geometric mean and third proportional are given. We must once again apply the definitions of these special proportions to determine the original pair of numbers. This reinforces conceptual understanding of how proportional relationships translate into algebraic equations.

Given Data / Assumptions:

• The mean proportional (geometric mean) of the two numbers is 8.
• The third proportional to these two numbers is 64.
• Both numbers are positive.
• By definition, if a : b = b : c, then b is the mean proportional between a and c, and c is the third proportional to a and b.

Concept / Approach:
Let the two numbers be a and b. The geometric mean condition gives sqrt(a * b) = 8, so a * b = 8^2 = 64. The third proportional condition tells us that a : b = b : 64, or a / b = b / 64. This yields a second equation. Solving these two equations simultaneously allows us to find unique positive values of a and b that satisfy both conditions.

Step-by-Step Solution:
Let the two numbers be a and b. Given that their geometric mean is 8, so sqrt(a * b) = 8. Square both sides: a * b = 64. (Equation 1) Given that 64 is the third proportional to a and b, so a : b = b : 64. Thus a / b = b / 64, giving b^2 = 64 * a. (Equation 2) From Equation 1, a = 64 / b. Substitute into Equation 2: b^2 = 64 * (64 / b) = (64 * 64) / b. So b^3 = 64 * 64 = 4096. Hence b = cube root of 4096 = 16. Now from a * b = 64, a = 64 / 16 = 4. Therefore the two numbers are 4 and 16.

Verification / Alternative check:
Check the geometric mean: sqrt(4 * 16) = sqrt(64) = 8, which satisfies the first condition. Check the third proportional condition: a : b = 4 : 16 simplifies to 1 : 4. Also, b : 64 = 16 : 64 simplifies to 1 : 4. Hence 4 : 16 = 16 : 64, confirming that 64 is indeed the third proportional to 4 and 16. Both conditions are met, so the solution is consistent.

Why Other Options Are Wrong:
Pairs such as 4 and 8, 8 and 16, 8 and 8 or 2 and 32 either fail to give a geometric mean of 8 or do not yield 64 as the third proportional when tested. Only the pair 4 and 16 satisfies both conditions simultaneously.

Common Pitfalls:
As before, confusion between arithmetic mean and geometric mean can lead to incorrect equations. Another common mistake is misapplying the third proportional definition as a : b = 64 : b instead of a : b = b : 64. Ensuring that definitions are clearly remembered and carefully translating them into algebraic form prevents such errors.

Final Answer:
The required two numbers are 4 and 16.