A, B, and C enter into partnership. A invests an amount a from the start. After 6 months, B invests 2a. After 8 months, C invests 3a. If the annual profit is ₹ 54,000, what is C's share?

Introduction / Context:
Here, each partner's capital-time product happens to be equal, leading to equal profit shares. This occurs because later partners invest proportionally larger capitals for proportionally shorter times.

Given Data / Assumptions:

• A: a for 12 months ⇒ 12a.
• B: 2a for (12 − 6) = 6 months ⇒ 12a.
• C: 3a for (12 − 8) = 4 months ⇒ 12a.
• Total profit = ₹ 54,000.

Concept / Approach:
Since all capital-time weights are 12a, profits are equal among A, B, and C. Divide the total profit by 3 to find each share.

Step-by-Step Solution:
Weights: A = 12a; B = 12a; C = 12a. Ratio = 1 : 1 : 1. Each share = 54,000 / 3 = ₹ 18,000.

Verification / Alternative check:
Any base a cancels because all three weights equal 12a. Thus equal sharing is robust.

Why Other Options Are Wrong:
Any amount other than ₹ 18,000 would violate the equal-weight conclusion.

Common Pitfalls:
Forgetting that partners who join later may have higher capitals to balance the shorter duration, resulting in equal weights.

Final Answer:
₹ 18,000