Ram and Raj start a business with investments of Rs. 2,500 and Rs. 2,250 respectively. After 2 months, Ram withdraws Rs. 1,250 from his capital. After 1 more month, Raj withdraws Rs. 750 from his capital, and at that time Rakesh joins them with Rs. 3,500. At the end of 6 months from the start, the total profit is Rs. 2,540. What is the share of each member in the profit?

Introduction / Context:
This is a multi stage partnership question where partners change their capital contributions at different times and a new partner joins later. To determine the correct profit share for each person, we must calculate the capital time (money months) for each segment of the 6 month period and then form a ratio based on those capital time totals.

Given Data / Assumptions:

• Ram invests Rs. 2,500 initially.
• Raj invests Rs. 2,250 initially.
• After 2 months, Ram reduces his investment by Rs. 1,250, so his new investment becomes Rs. 1,250.
• After 3 months from the start (that is, 1 month later), Raj withdraws Rs. 750, leaving Rs. 1,500 invested.
• At the same time (end of 3rd month), Rakesh joins with Rs. 3,500.
• The business runs for a total of 6 months.
• Total profit at the end is Rs. 2,540.

Concept / Approach:
When capital changes during the business period, we cannot use a single static ratio of capitals. Instead, we divide the total time into segments where capital remains constant for each partner, compute capital time for every partner in each segment and then sum these to obtain their overall contribution. The profit is shared in the ratio of these overall capital time values.

Step-by-Step Solution:
Step 1: From month 0 to month 2 (2 months), Ram has Rs. 2,500 and Raj has Rs. 2,250. Rakesh has not yet joined.Step 2: Ram's capital time for this segment = 2,500 * 2 = 5,000. Raj's capital time = 2,250 * 2 = 4,500.Step 3: From month 2 to month 3 (1 month), Ram reduces to Rs. 1,250, while Raj remains at Rs. 2,250. Rakesh has not joined yet.Step 4: In this segment, Ram's capital time = 1,250 * 1 = 1,250; Raj's capital time = 2,250 * 1 = 2,250.Step 5: From month 3 to month 6 (3 months), Ram has Rs. 1,250, Raj has Rs. 1,500, and Rakesh has Rs. 3,500.Step 6: In this segment, Ram's capital time = 1,250 * 3 = 3,750; Raj's capital time = 1,500 * 3 = 4,500; Rakesh's capital time = 3,500 * 3 = 10,500.Step 7: Total capital time for Ram = 5,000 + 1,250 + 3,750 = 10,000.Step 8: Total capital time for Raj = 4,500 + 2,250 + 4,500 = 11,250.Step 9: Total capital time for Rakesh = 10,500.Step 10: Ratio Ram : Raj : Rakesh = 10,000 : 11,250 : 10,500.Step 11: Divide all by 250 to simplify: 40 : 45 : 42.Step 12: Total ratio parts = 40 + 45 + 42 = 127.Step 13: Each part of profit = 2,540 / 127 = Rs. 20.Step 14: Ram's share = 40 * 20 = Rs. 800; Raj's share = 45 * 20 = Rs. 900; Rakesh's share = 42 * 20 = Rs. 840.

Verification / Alternative check:
Add the three computed shares: 800 + 900 + 840 = Rs. 2,540, which matches the given total profit. This confirms that the capital time ratio and the final distribution of profit are consistent and correctly calculated.

Why Other Options Are Wrong:
The distributions in the other options do not respect the capital time ratio 40 : 45 : 42 when converted back to parts of 20 rupees. For example, any option that changes Rakesh's share to 940 or swaps Ram and Raj's amounts will no longer add to Rs. 2,540 in the correct ratio. 'None of these' is incorrect because option with Ram - Rs. 800, Raj - Rs. 900, Rakesh - Rs. 840 perfectly matches the ratio derived from the capital time analysis.

Common Pitfalls:
Candidates often forget to segment the timeline correctly and may assume fixed capitals throughout the six months. Another frequent mistake is to ignore the effect of late entry by Rakesh or to average capitals instead of using capital time. Always break the problem into time segments and compute money months for each partner in each segment.

Final Answer:
The correct distribution of profit is Ram - Rs. 800, Raj - Rs. 900, Rakesh - Rs. 840.