What is the average age of the six family members A, B, C, D, E, F? I. Average age of D and E is 14 years. II. Average age of A, B, C and F is 50 years. III. Average age of A, B, D and E is 40 years.

Introduction / Context:
We need the overall average for six people using given subset averages. Data-sufficiency asks which combination of statements uniquely determines that average.

Given Data / Assumptions:

• I: (D + E)/2 = 14 ⇒ D + E = 28.
• II: (A + B + C + F)/4 = 50 ⇒ A + B + C + F = 200.
• III: (A + B + D + E)/4 = 40 ⇒ A + B + D + E = 160.

Concept / Approach:
Express the grand total S = A + B + C + D + E + F in terms of the partial sums and check which set of statements suffices to find S.

Step-by-Step Solution:
From I: D + E = 28.From III: A + B + D + E = 160 ⇒ A + B = 160 − 28 = 132.From II: A + B + C + F = 200 ⇒ C + F = 200 − (A + B) = 68.Total S = (A + B) + (C + F) + (D + E) = 132 + 68 + 28 = 228 ⇒ Average = 228/6 = 38.

Why Other Options Are Wrong:
Any two statements leave an undetermined pair-sum; all three together fix S uniquely.

Common Pitfalls:
Trying to average the averages; you must convert to sums first.

Final Answer:
I, II and III.