Aman starts a business by investing Rs. 70,000. After 6 months, Rakhi joins with Rs. 1,05,000. After another 6 months, Sagar joins them with Rs. 1,40,000. If the profit is calculated 3 years after Aman starts the business, in what ratio will it be shared among Aman, Rakhi and Sagar?

Introduction / Context:
This partnership problem involves three partners who join the business at different times with different capital amounts. The business runs for 3 years from the start of Aman's investment. We must compute the effective capital * time for each partner and then derive the ratio in which the profit will be distributed among Aman, Rakhi and Sagar.

Given Data / Assumptions:

• Aman invests Rs. 70,000 at the start and remains invested for the full 3 years (36 months).
• After 6 months, Rakhi joins with Rs. 1,05,000 and remains for the remaining 30 months.
• After another 6 months (that is, 12 months from the start), Sagar joins with Rs. 1,40,000 and remains for the remaining 24 months.
• Profit is calculated at the end of 3 years and shared in proportion to effective investments.
• Effective investment = capital * time.

Concept / Approach:
When partners join at different times, the simple ratio of capital is not enough. Instead, we must calculate capital * time for each partner in rupee months. This gives the relative contribution of each partner to the business over the whole period. The ratio of these effective investments is then used to divide the final profit, independent of the exact profit amount.

Step-by-Step Solution:
Step 1: Effective investment of Aman: Aman invests 70,000 for 36 months. Effective investment = 70,000 * 36 = 25,20,000 rupee months. Step 2: Effective investment of Rakhi: Rakhi joins after 6 months, so she invests for 30 months. Effective investment = 1,05,000 * 30 = 31,50,000 rupee months. Step 3: Effective investment of Sagar: Sagar joins after 12 months, so he invests for 24 months. Effective investment = 1,40,000 * 24 = 33,60,000 rupee months. Step 4: Write their effective investments as a ratio: Aman : Rakhi : Sagar = 25,20,000 : 31,50,000 : 33,60,000. Step 5: Divide all by 2,10,000 to simplify: 25,20,000 / 2,10,000 = 12, 31,50,000 / 2,10,000 = 15, 33,60,000 / 2,10,000 = 16. So the ratio becomes 12 : 15 : 16. Step 6: Therefore, the profit will be shared among Aman, Rakhi and Sagar in the ratio 12 : 15 : 16.

Verification / Alternative check:
We can confirm the logic by observing that Sagar has the largest effective investment, followed by Rakhi and then Aman. This aligns with the final ratio, where Sagar has the largest share (16 parts), Rakhi the next (15 parts) and Aman the lowest (12 parts). Any actual profit figure, for example Rs. 43,000, would be split proportionally according to these numbers, and the proportionality would remain correct regardless of the exact amount.

Why Other Options Are Wrong:
Ratios such as 11 : 13 : 15, 11 : 13 : 17 or 12 : 17 : 18 do not correspond to the actual effective investments. For instance, a ratio with much larger share for Sagar compared to Rakhi would not be justified because Rakhi's effective investment is only slightly smaller than Sagar's. Therefore these options are inconsistent with the capital * time contributions.

Common Pitfalls:
A common error is to ignore the different entry times and simply compare the capital amounts 70,000, 1,05,000 and 1,40,000. Another mistake is to treat each year equally without using exact months, which can lead to incorrect time durations for each partner. Always determine how many months each capital amount stays in the business and then multiply capital by time before forming ratios.

Final Answer:
The profit will be shared in the ratio 12 : 15 : 16 among Aman, Rakhi and Sagar.