The average weight of A, B, C, and D is 67 kg. After including a 5th person E, the average decreases by 2 kg. Then A is replaced by F (whose weight is 4 kg more than E), and the average further becomes 64 kg. Find the weight of A.

Introduction:
Multi-step average problems are handled cleanly by working with totals at each stage. First compute the 5th person’s weight from the first average change, then model the replacement step and solve for the unknown original weight A using the final average.

Given Data / Assumptions:

• Avg(A,B,C,D) = 67 → sum4 = 268.
• Including E: new avg = 65 → sum5 = 325 ⇒ E = 325 − 268 = 57.
• F replaces A; F = E + 4 = 61.
• Final avg with (B,C,D,E,F) = 64 → final sum5' = 320.

Concept / Approach:
Express final sum in terms of A: (B + C + D + E + F) = (268 − A) + 57 + 61. Equate to 320 to solve for A.

Step-by-Step Solution:

Verification / Alternative check:
Check: With A = 66, sum4 = 268; including E = 57 gives 325 (avg 65). Replacing A with F = 61 gives (325 − 66 + 61) = 320 → avg 64. Consistent.

Why Other Options Are Wrong:
58, 62, 75, or 78 break one or both average conditions when back-substituted into the totals.

Common Pitfalls:
Averaging the averages or forgetting to adjust the totals correctly after replacement. Always track sums explicitly.

Final Answer:
66 kg