Fourth proportional: Find the fourth proportional to 4, 16, and 7 (i.e., determine x such that 4 : 16 = 7 : x).

Introduction / Context: The fourth proportional to three numbers a, b, c is a number x that satisfies a : b = c : x. Such problems are direct applications of the property of equal ratios and cross multiplication.

Given Data / Assumptions:

• a = 4, b = 16, c = 7.
• Find x such that 4 : 16 = 7 : x.

Concept / Approach: Use a : b = c : x → a/b = c/x. Cross multiply to solve for x: a*x = b*c → x = (b*c)/a.

Step-by-Step Solution: Set up proportion: 4 : 16 = 7 : x. Cross multiplication: 4x = 16 * 7. Compute 16 * 7 = 112. Thus x = 112 / 4 = 28.

Verification / Alternative check: Check ratios as decimals: 4/16 = 0.25. With x = 28, 7/28 = 0.25. Ratios match, confirming x = 28.

Why Other Options Are Wrong:

• 29, 22, 25, 32 do not satisfy 7 : x = 4 : 16.
• Only 28 yields identical ratios on both sides.

Common Pitfalls:

• Inverting the ratio (using 16 : 4) and getting 7 : x wrong.
• Avoid arithmetic mistakes when multiplying 16 and 7.

Final Answer: 28