A and B start a partnership business with initial investments in the ratio 5 : 6. After 6 months, C joins them by investing an amount equal to two-thirds of B's capital. At the end of the year, C's share of the profit is Rs. 21,600. What is the total profit of the business, in rupees, at the end of the year?

Introduction / Context:
This partnership question deals with different joining times and different capital amounts. It focuses on the concept that profit in a partnership is divided in proportion to the product of capital and time, often called money-months or money-time units. Here, A and B start the business together, while C joins later with a specific fraction of B's capital. Based on C's known share of profit, we must work backwards to determine the total profit. Such questions are common in aptitude tests and help reinforce the idea of weighted contributions over time.

Given Data / Assumptions:
- A and B invest in the ratio 5 : 6.- Let B's capital be 6x; then A's capital is 5x.- C joins after 6 months with capital equal to two-thirds of B's capital.- Thus, C's capital is (2/3) * 6x = 4x.- Total business duration is 12 months from the start.- A and B remain invested for 12 months, C for 6 months.- C's share of profit after one year is Rs. 21,600.- We must find the total profit of the business.

Concept / Approach:
The key concept is that profit is shared in the ratio of capital multiplied by the time period for which it is invested. This forms a proportionality ratio. We compute money-time units for each partner: capital * months. These units give us the ratio in which total profit is divided among A, B and C. Once we know the ratio, we can treat C's share as a fraction of the whole and solve for total profit using a simple proportion.

Step-by-Step Solution:
Step 1: Let B's capital be 6x; A's capital is 5x.Step 2: C joins after 6 months with capital (2/3) of B's capital, so C's capital = 4x.Step 3: Time invested:- A: 12 months.- B: 12 months.- C: 6 months (joins after 6 months in a 12 month year).Step 4: Money-time units:- A: 5x * 12 = 60x.- B: 6x * 12 = 72x.- C: 4x * 6 = 24x.Step 5: Ratio of profit shares = 60x : 72x : 24x = 5 : 6 : 2 (dividing by 12x).Step 6: Total parts = 5 + 6 + 2 = 13 parts.Step 7: C gets 2 parts out of 13, and this is given as Rs. 21,600.Step 8: So 2 / 13 of total profit = 21,600.Step 9: Total profit = 21,600 * (13 / 2) = 21,600 * 6.5 = 140,400.

Verification / Alternative check:
Compute 1 part of profit: 140,400 / 13 = 10,800.A's share = 5 * 10,800 = 54,000.B's share = 6 * 10,800 = 64,800.C's share = 2 * 10,800 = 21,600.Sum = 54,000 + 64,800 + 21,600 = 140,400, which matches the calculated total profit, confirming the solution.

Why Other Options Are Wrong:
- 46800: Too small; it ignores the ratio 5 : 6 : 2 and would make C's share unrealistically large as a fraction of total profit.- 56160: Also too small; it does not produce C's share of 21,600 when using the ratio 5 : 6 : 2.- 70200: Exactly half of the correct answer, corresponding to misinterpreting C's share as 4 out of 13 instead of 2 out of 13.

Common Pitfalls:
- Forgetting that C joins after 6 months and incorrectly using 12 months instead of 6 for C's time factor.- Using only the capital ratio 5 : 6 : 4 without incorporating the different time periods.- Miscalculating the ratio by neglecting to reduce 60 : 72 : 24 to 5 : 6 : 2.- Doing percentage-based thinking instead of working directly with proportional parts.

Final Answer:
The total profit of the business at the end of the year is Rs. 1,40,400.