Proportion with a missing mean: solve for the number x in 50 / x = x / 12 1/2 (treat 12 1/2 as 12.5 and find x exactly).

Introduction / Context:
Proportion problems of the form a/x = x/b are common in aptitude tests. They test recognition that such a relation implies x^2 = a*b, so x is the positive square root of the product (for positive a and b). Here, we handle a mixed number and keep arithmetic clean and exact.

Given Data / Assumptions:

• Equation: 50 / x = x / 12 1/2.
• Interpret 12 1/2 as 12.5.
• All quantities are positive; take the positive root for x.

Concept / Approach:
Cross-multiplying a proportion gives 50 * 12.5 = x * x. This is a direct application of the mean-proportional idea: if a/x = x/b, then x^2 = a*b. After computing the product, take the square root to find x exactly.

Step-by-Step Solution:
Write 12 1/2 as 12.5.Set x^2 = 50 * 12.5.Compute product: 50 * 12.5 = 625.Therefore x = sqrt(625) = 25 (positive value).

Verification / Alternative check:
Substitute x = 25 back: 50/25 = 2 and 25/12.5 = 2. Both sides match, confirming x = 25.

Why Other Options Are Wrong:
25/2 = 12.5 gives 50/12.5 = 4 while 12.5/12.5 = 1, not equal.4 leads to 50/4 = 12.5 but 4/12.5 = 0.32, not equal.50 makes 50/50 = 1 while 50/12.5 = 4, not equal.

Common Pitfalls:
Misreading 12 1/2 as 12/5; forgetting to square-root after forming x^2; or introducing sign errors. Remember that for positive magnitudes, the positive square root is the meaningful solution.

Final Answer:
25