Mean proportional in a continued proportion If 8 : y = y : 18 (i.e., 8/y = y/18), find the value of y.

Introduction / Context:
In a continued proportion a : b = b : c, the middle term b is the mean proportional, satisfying b^2 = a * c. This question directly applies that identity to solve for y.

Given Data / Assumptions:

• 8 : y = y : 18.

Concept / Approach:
Set up the equality 8 / y = y / 18, cross-multiply, and solve the resulting quadratic y^2 = 8 * 18. Take the positive root since ratios of magnitudes are nonnegative here.

Step-by-Step Solution:
8 / y = y / 18 ⇒ y^2 = 8 * 18 = 144.y = √144 = 12 (taking the positive value).

Verification / Alternative check:
Check: 8 : 12 = 2 : 3 and 12 : 18 = 2 : 3; ratios match, confirming y = 12.

Why Other Options Are Wrong:
144 and 72 are products or doubles, not the mean proportional; 26 is arbitrary; 36 would imply 8:36 ≠ 36:18.

Common Pitfalls:
Taking the product 8*18 as the answer or forgetting to take the square root to find the mean proportional.

Final Answer:
12