Mean proportional (geometric mean): Find the mean proportional between 9 and 64 (i.e., find x such that 9 : x = x : 64).

Introduction / Context:
The mean proportional (geometric mean) x between a and b satisfies a : x = x : b, or x^2 = a * b. This is a standard concept in ratio and proportion and geometric mean computations.

Given Data / Assumptions:

• a = 9.
• b = 64.
• Find x such that x^2 = a * b.

Concept / Approach:
Compute the product a * b and then take the positive square root, since a and b are positive.

Step-by-Step Solution:
x^2 = 9 * 64 = 576.x = sqrt(576) = 24.

Verification / Alternative check:
Check proportionality: 9 : 24 = 3 : 8 and 24 : 64 = 3 : 8, confirming correctness.

Why Other Options Are Wrong:

• 25, 27, 35 are not the square root of 576 and do not keep equal ratios with 9 and 64.

Common Pitfalls:

• Confusing third proportional with mean proportional.
• Arithmetic mistakes in multiplying or in the square root.

Final Answer:
24