The average of A and B is 20. If C replaces A, the average becomes 19; if C replaces B, the average becomes 21. What are the ages of A, B, and C?

Introduction:
This system uses pairwise averages that translate directly into pairwise sums. Solving the resulting simultaneous linear equations yields unique values for A, B, and C.

Given Data / Assumptions:

• (A + B)/2 = 20 ⇒ A + B = 40.
• (C + B)/2 = 19 ⇒ B + C = 38.
• (A + C)/2 = 21 ⇒ A + C = 42.

Concept / Approach:
Use elimination by subtracting equations to find differences, then combine with a known sum. Specifically, subtract (B + C) from (A + C) to get A − B, then combine with A + B.

Step-by-Step Solution:

Verification / Alternative check:
Check: (22 + 18)/2 = 20; (20 + 18)/2 = 19; (22 + 20)/2 = 21. All conditions satisfied.

Why Other Options Are Wrong:
The other permutations break at least one of the three pairwise average constraints on substitution.

Common Pitfalls:
Mixing which person is replaced in each scenario or averaging the given averages instead of forming equations on sums.

Final Answer:
22, 18, 20