There are four sections A, B, C, and D in a class. Overall average weight is 60 kg. Section-wise averages are: A = 45 kg, B = 50 kg, C = 72 kg, D = 80 kg. Given avg(A & B together) = 48 kg and avg(B & C together) = 60 kg, find the ratio of the number of students in sections A and D.

Introduction:
Weighted averages across multiple groups can be solved by expressing pair-average conditions as equations in group counts. From the pairwise conditions, deduce ratios among A, B, and C. Then use the overall average to relate D and finally compute the required ratio A : D.

Given Data / Assumptions:

• Averages: A = 45, B = 50, C = 72, D = 80, Overall = 60.
• avg(A & B) = 48 → (45a + 50b)/(a + b) = 48.
• avg(B & C) = 60 → (50b + 72c)/(b + c) = 60.
• Let counts be a, b, c, d.

Concept / Approach:
From (45a + 50b) = 48(a + b) ⇒ −3a = −2b ⇒ b = (3/2)a. From (50b + 72c) = 60(b + c) ⇒ 12c = 10b ⇒ c = (5/6)b = (5/4)a. Use the overall equation to relate d to a and solve for d/a, hence A : D.

Step-by-Step Solution:

Verification / Alternative check:
Substitute any convenient a (e.g., a = 4) to get integer counts and confirm the overall average balances at 60 when including D with ratio 3.

Why Other Options Are Wrong:
12:7, 3:2, 8:5, 5:4 contradict at least one of the pair-average conditions or the overall balance when tested.

Common Pitfalls:
Forgetting that averages weight by counts, not by simply averaging the section means; always write equations with a, b, c, d explicitly.

Final Answer:
4 : 3