Rajeev starts a business with Rs. 10,000. After 2 months, Deepu joins with an investment that is 20% more than Rajeev's. After another 2 months, Shakti joins with an investment that is 40% less than Deepu's. If the total profit at the end of the year is equal to twice the difference between Rajeev's investment and ten times Shakti's investment, what is Rajeev's share of the profit?

Introduction / Context:
This is a challenging partnership question with three partners joining at different times and with investments related by percentage changes. Additionally, the total profit is defined indirectly using a relationship involving the initial investments. We must first work out the individual investments and their effective capital * time contributions, then translate the given condition about total profit into a numerical value, and finally compute Rajeev's share of this profit.

Given Data / Assumptions:

• Rajeev invests Rs. 10,000 at the beginning of the year.
• After 2 months, Deepu joins with 20% more than Rajeev, so Deepu invests Rs. 12,000.
• After another 2 months, Shakti joins with 40% less than Deepu, so Shakti invests Rs. 7,200.
• The business runs for a total of 12 months.
• Rajeev invests for 12 months, Deepu for 10 months and Shakti for 8 months.
• Total profit P is equal to twice the difference between Rajeev's investment and ten times Shakti's investment.
• Profit is shared in proportion to capital * time.

Concept / Approach:
We first compute each partner's capital * time product to get the profit sharing ratio. Then we use the condition for total profit to find P. Finally we apply the ratio to P to calculate Rajeev's share. The key ideas are careful percentage calculation for investments, correct time periods and algebraic use of the profit condition P = 2 * (10 * Shakti investment minus Rajeev investment).

Step-by-Step Solution:
Step 1: Rajeev's capital = 10,000 for 12 months, so effective investment = 10,000 * 12 = 1,20,000 rupee months. Step 2: Deepu invests 20% more than Rajeev, so Deepu's capital = 10,000 * 1.2 = 12,000. Deepu invests for 12 - 2 = 10 months, so effective investment = 12,000 * 10 = 1,20,000 rupee months. Step 3: Shakti invests 40% less than Deepu, so Shakti's capital = 12,000 * 0.6 = 7,200. Shakti invests for 12 - 4 = 8 months, so effective investment = 7,200 * 8 = 57,600 rupee months. Step 4: The effective investment ratio is 1,20,000 : 1,20,000 : 57,600. Divide all by 2,400 to get 50 : 50 : 24. Step 5: Total ratio parts = 50 + 50 + 24 = 124. Step 6: According to the question, total profit P = twice the difference between Rajeev's investment and ten times Shakti's investment. Ten times Shakti's investment = 10 * 7,200 = 72,000. Difference = 72,000 - 10,000 = 62,000. So P = 2 * 62,000 = Rs. 1,24,000. Step 7: Rajeev's share of profit = P * (Rajeev's ratio part / total parts) = 1,24,000 * (50 / 124) = Rs. 50,000.

Verification / Alternative check:
We can also compute Deepu's and Shakti's shares. Deepu's share = 1,24,000 * (50 / 124) = 50,000 and Shakti's share = 1,24,000 * (24 / 124) = 24,000. Adding all three shares gives 50,000 + 50,000 + 24,000 = 1,24,000 which exactly matches the total profit. Rajeev's share is therefore consistent with both the ratio and the given profit condition.

Why Other Options Are Wrong:
If Rajeev's profit were Rs. 48,000, Rs. 38,000 or Rs. 40,000, the resulting shares for Deepu and Shakti using the fixed ratio 50 : 50 : 24 would not sum to a total profit that equals 2 times the difference between 10,000 and ten times 7,200. These options break either the ratio condition or the total profit condition, so they cannot be correct.

Common Pitfalls:
Common mistakes include misinterpreting the phrase "20% more" or "40% less", forgetting the exact times for which each partner invests and misreading the profit condition. Some learners also incorrectly compute P as twice the difference between Rajeev's and Shakti's investments, without multiplying Shakti's investment by ten. Careful reading and systematic algebra are essential here.

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