Two-number system with sum–difference relation: For positive numbers P and Q, the sum equals 2.5 times their difference, and their product is 84. Find P + Q.

Introduction / Context:
This problem links sum, difference, and product of two positive numbers. Expressing the difference in terms of the sum and using the identity for product in terms of sum and difference leads to a single solvable equation in one variable (the sum).

Given Data / Assumptions:

• P and Q are positive.
• Sum S = P + Q, difference D = |P − Q|.
• Condition: S = 2.5 * D, and product P*Q = 84.

Concept / Approach:
Use the relation for two numbers: P*Q = (S^2 − D^2)/4. Replace D by S/2.5 = (2/5)S to get a single equation in S. Solve for S, then choose the matching option.

Step-by-Step Solution:
Given S = 2.5D ⇒ D = S/2.5 = (2/5)S.Use product identity: P*Q = (S^2 − D^2)/4.Substitute D^2 = (4/25)S^2 ⇒ P*Q = (S^2 − (4/25)S^2)/4 = ((21/25)S^2)/4 = (21/100)S^2.Given P*Q = 84 ⇒ (21/100)S^2 = 84 ⇒ S^2 = 84 * (100/21) = 400.Thus S = 20 (positive case).

Verification / Alternative check:
If S = 20 and D = (2/5)*20 = 8, then numbers are (S ± D)/2 = (20 ± 8)/2 = 14 and 6. Their product is 84, confirming consistency.

Why Other Options Are Wrong:
26, 24, and 22 do not satisfy the derived relation (21/100)S^2 = 84; only S = 20 works.

Common Pitfalls:
Mixing S and D; using S = 2.5/D instead of S = 2.5*D; forgetting the product identity.

Final Answer:
20