Sekar starts a business in 1999 by investing Rs. 25,000. In 2000 he adds another Rs. 10,000 and Rajeev joins with Rs. 35,000. In 2001 Sekar adds another Rs. 10,000 and Jatin joins with Rs. 35,000. If the profit at the end of 3 years from the start is Rs. 1,50,000, what is Rajeev's share of this profit?

Introduction / Context:
This is a multi stage partnership problem where partners join at different times and existing partners change their capital during the life of the business. The profit must be divided according to each partner's capital multiplied by the time his capital remains invested. The goal is to compute the share of Rajeev in the total profit of Rs. 1,50,000 earned over three years.

Given Data / Assumptions:

• Sekar invests Rs. 25,000 at the start of 1999.
• In 2000, Sekar adds Rs. 10,000 more, so his investment becomes Rs. 35,000.
• In 2000, Rajeev joins with Rs. 35,000.
• In 2001, Sekar adds another Rs. 10,000, so his investment becomes Rs. 45,000.
• In 2001, Jatin joins with Rs. 35,000.
• Each year is taken as 12 months and the changes happen at the beginning of each year.
• Total profit at the end of 3 years is Rs. 1,50,000.
• Profit is distributed in proportion to capital * time for each partner.

Concept / Approach:
We break the three year period into yearly segments. In each segment, we note the capital of each partner and multiply it by the duration of that segment to get the effective investment in rupee months. After summing the effective investments for all partners across all segments, we obtain the ratio in which the total profit must be shared. Finally, we calculate Rajeev's part from this ratio.

Step-by-Step Solution:
Step 1: Year 1 (1999): Only Sekar invests Rs. 25,000 for 12 months. Effective investment = 25,000 * 12 = 3,00,000 rupee months. Step 2: Year 2 (2000): Sekar now has Rs. 35,000 and Rajeev invests Rs. 35,000. Each is invested for 12 months, so Sekar contributes 35,000 * 12 = 4,20,000 and Rajeev contributes 35,000 * 12 = 4,20,000 rupee months. Step 3: Year 3 (2001): Sekar invests Rs. 45,000, Rajeev keeps Rs. 35,000 and Jatin joins with Rs. 35,000. Each again invests for 12 months. So Sekar contributes 45,000 * 12 = 5,40,000, Rajeev contributes 35,000 * 12 = 4,20,000 and Jatin contributes 35,000 * 12 = 4,20,000. Step 4: Total effective investments: Sekar = 3,00,000 + 4,20,000 + 5,40,000 = 12,60,000, Rajeev = 4,20,000 + 4,20,000 = 8,40,000, Jatin = 4,20,000. Step 5: Divide these by 4,20,000 to get the ratio: Sekar : Rajeev : Jatin = 3 : 2 : 1. Step 6: Total number of ratio parts = 3 + 2 + 1 = 6. Each part of profit = 1,50,000 / 6 = Rs. 25,000. Step 7: Rajeev gets 2 parts, so Rajeev's share = 2 * 25,000 = Rs. 50,000.

Verification / Alternative check:
We can verify by computing the other partners' profits. Sekar gets 3 parts = 3 * 25,000 = 75,000 and Jatin gets 1 part = 25,000. Adding all shares, 75,000 + 50,000 + 25,000 = 1,50,000, which matches the total profit available, confirming the correctness of the ratio and Rajeev's share.

Why Other Options Are Wrong:
Options Rs. 45,000, Rs. 70,000 and Rs. 75,000 do not fall in the correct 3 : 2 : 1 ratio when fitted into the total profit of Rs. 1,50,000. For example, if Rajeev received Rs. 45,000, the remaining profit and the shares of Sekar and Jatin would not match the required proportional relationship defined by the effective investments.

Common Pitfalls:
Learners often ignore the fact that capital changes over time or assume that every partner invests for the full period. Another common mistake is to sum only capital amounts without multiplying by the time of investment. Always convert every partner's contribution into capital * time units before forming ratios.

Final Answer:
Rajeev's share of the profit is Rs. 50,000.