S, T and U started a business by investing Rs. 4,200, Rs. 6,300 and Rs. 8,400 respectively. S withdrew from the business after 3 months, T withdrew after 6 months, and U remained invested for the full one year. At the end of the year, the total profit is Rs. 6,000. What is U's share in the profit, in rupees?

Introduction / Context:
This question examines profit sharing when partners invest for different lengths of time. Although each person invests a certain capital amount, they do not keep their money in the business for the same duration. Therefore, the share of profit must be based on both capital and time. This approach is crucial in partnership problems and is frequently tested in banking, SSC and other competitive exams.

Given Data / Assumptions:
- Initial investments are S = Rs. 4,200, T = Rs. 6,300, U = Rs. 8,400.- S withdraws after 3 months, so S is invested for 3 months.- T withdraws after 6 months, so T is invested for 6 months.- U remains invested for the full 12 months (1 year).- Total profit at the end of the year is Rs. 6,000.- We need to find U's share of the profit.

Concept / Approach:
The central concept is that profit is proportional to capital multiplied by time of investment. These products are often referred to as money-time units, such as rupee-months. We calculate these units for each partner, form a ratio, and then divide the total profit according to that ratio. Finally, U's fraction of the total is applied to Rs. 6,000 to find U's share.

Step-by-Step Solution:
Step 1: Determine the duration of investment for each partner.- S: 3 months.- T: 6 months.- U: 12 months.Step 2: Compute money-time units (capital * time).- S: 4,200 * 3 = 12,600.- T: 6,300 * 6 = 37,800.- U: 8,400 * 12 = 100,800.Step 3: Form the ratio of profit shares = 12,600 : 37,800 : 100,800.Step 4: Simplify the ratio by dividing all three terms by 4,200.- S: 12,600 / 4,200 = 3.- T: 37,800 / 4,200 = 9.- U: 100,800 / 4,200 = 24.So the ratio is S : T : U = 3 : 9 : 24.Step 5: Sum of ratio parts = 3 + 9 + 24 = 36.Step 6: U's share fraction = 24 / 36 = 2 / 3.Step 7: Total profit = Rs. 6,000. So U's share = (2 / 3) * 6,000 = Rs. 4,000.

Verification / Alternative check:
Compute each partner's share directly.Each part of the ratio corresponds to 6,000 / 36 = Rs. 166.67 approximately.S's share = 3 * 166.67 ≈ 500.T's share = 9 * 166.67 ≈ 1,500.U's share = 24 * 166.67 ≈ 4,000.The approximate sum is 500 + 1,500 + 4,000 = 6,000, consistent with total profit. Exact fractional calculations give exactly Rs. 4,000 for U.

Why Other Options Are Wrong:
- 4200: This exceeds the total profit share that U can reasonably get given the ratio and total of 6,000.- 3200: This would correspond to a smaller fraction than 2/3 of the total profit and does not match the money-time ratio.- 3600: This is 60% of the profit, but according to the ratio, U must receive 2/3, that is about 66.67% of the profit.

Common Pitfalls:
- Ignoring time and using only the capital amounts 4,200 : 6,300 : 8,400 which would be incorrect.- Miscalculating the duration, for example assuming S and T also stay for 12 months.- Arithmetic mistakes when simplifying the ratio or multiplying by the total profit.- Forgetting to express U's share as a fraction of the total ratio before applying it to the profit.

Final Answer:
U's share of the profit is Rs. 4,000.