Four friends with overlapping averages and a sum condition: A, B, C, and D are four friends. Average age of A and C is 35 years (so A + C = 70). Average age of B and D is 40 years (so B + D = 80). The average age of B, C and D is 40 years (so B + C + D = 120). Also, A + D = B + C. Find the ages (A, B, C, D) in years.

Introduction / Context:
Multiple overlapping average conditions combined with a balancing sum (A + D = B + C) form a compact linear system. The key is to translate each statement into equations and solve systematically.

Given Data / Assumptions:

• A + C = 70.
• B + D = 80.
• B + C + D = 120.
• A + D = B + C.

Concept / Approach:
Use B + C + D = 120 and A + D = B + C to eliminate variables stepwise. Express A in terms of D, then find C and B, ensuring all constraints hold.

Step-by-Step Solution:
1) From A + D = B + C and B + C + D = 120 ⇒ (A + D) + D = 120 ⇒ A + 2D = 120 ⇒ A = 120 − 2D.2) From A + C = 70 ⇒ (120 − 2D) + C = 70 ⇒ C = 2D − 50.3) From B + D = 80 ⇒ B = 80 − D.4) Check B + C + D = (80 − D) + (2D − 50) + D = 30 + 2D; set equal to 120 ⇒ 2D = 90 ⇒ D = 45.5) Then A = 120 − 90 = 30; C = 2×45 − 50 = 40; B = 80 − 45 = 35.

Verification / Alternative check:
A + C = 30 + 40 = 70; B + D = 35 + 45 = 80; B + C + D = 35 + 40 + 45 = 120; A + D = 30 + 45 = 75 = B + C (35 + 40). All conditions satisfied.

Why Other Options Are Wrong:

• Other tuples violate at least one of the sums or the balancing condition A + D = B + C.

Common Pitfalls:
Mixing averages with totals incorrectly or skipping the A + D = B + C constraint leads to inconsistent sets.

Final Answer:
30, 35, 40, 45