Third proportional computation: Calculate the third proportional to 15 and 30 (i.e., find x such that 15 : 30 = 30 : x).

Introduction / Context:
The third proportional to a and b is the number x satisfying a : b = b : x. This is a direct application of cross-multiplication and proportional reasoning.

Given Data / Assumptions:

• a = 15.
• b = 30.
• Find x such that 15 : 30 = 30 : x.

Concept / Approach:
For a : b = b : x, we have a * x = b * b ⇒ x = b^2 / a. Substitute the values and simplify to obtain an integer result.

Step-by-Step Solution:
x = b^2 / a = 30^2 / 15.30^2 = 900; 900 / 15 = 60.

Verification / Alternative check:
Check ratio equality: 15 : 30 = 1 : 2 and 30 : 60 = 1 : 2, so the proportion holds.

Why Other Options Are Wrong:

• 55 and 65 do not satisfy 30 : x = 1 : 2.
• 15 repeats the first term and breaks the proportion.

Common Pitfalls:

• Mistaking “third proportional” for “mean proportional.”
• Arithmetic slip in squaring b or dividing by a.

Final Answer:
60