A starts a business with a capital of Rs 1800. After 4 months B joins with Rs 3600, and 6 months after the business started C joins with Rs 1200. If the total profit at the end of the year is Rs 1760, what is the share of A in the profit?

Introduction / Context:
This problem involves a partnership where different partners join the business at different times and with different capitals. In such questions, profit is shared according to capital multiplied by time in months, often called money months. The aim is to compute the effective contribution of each partner and then find the share of profit belonging to partner A, given the total profit at the end of one year.

Given Data / Assumptions:
A invests Rs 1800 from the start of the business. B joins after 4 months with Rs 3600. C joins after 6 months from the start with Rs 1200. Total duration of the business is 12 months. Total profit at the end of the year is Rs 1760.

Concept / Approach:
In partnerships with varying joining times, each partner share is proportional to capital * time. We calculate money months for each partner, form a ratio, and then divide the total profit according to this ratio. The steps are to compute A, B and C contributions over their respective time spans, simplify the ratio and finally compute A share of the total profit of Rs 1760.

Step-by-Step Solution:
Step 1: A invests Rs 1800 for the entire 12 months. Step 2: Money months of A = 1800 * 12 = 21600. Step 3: B joins after 4 months, so B is in business for 12 - 4 = 8 months. Step 4: Money months of B = 3600 * 8 = 28800. Step 5: C joins after 6 months, so C is in business for 12 - 6 = 6 months. Step 6: Money months of C = 1200 * 6 = 7200. Step 7: Ratio of contributions A : B : C = 21600 : 28800 : 7200. Step 8: Divide all by 7200 to simplify, giving 3 : 4 : 1. Step 9: Total parts = 3 + 4 + 1 = 8 parts. Step 10: Each part of profit = 1760 / 8 = Rs 220. Step 11: Share of A = 3 parts = 3 * 220 = Rs 660.

Verification / Alternative check:
We can also find shares of B and C. Share of B = 4 * 220 = Rs 880, share of C = 1 * 220 = Rs 220. The total of all shares is 660 + 880 + 220 = Rs 1760, which matches the given total profit. This confirms that the ratio of 3 : 4 : 1 based on money months has been applied correctly and that A share of Rs 660 is accurate.

Why Other Options Are Wrong:
Rs 540 corresponds to 540 / 220 = 2.45 parts, which does not align with an integer ratio. Rs 840 would represent 3.82 parts and cannot be justified by any correct ratio of contributions. Rs 720 would be 3.27 parts, which again is inconsistent. Only Rs 660 lines up perfectly with 3 parts of 8 total parts in the profit sharing ratio derived from the actual capital and time contributions.

Common Pitfalls:
Learners sometimes ignore the months and divide profit only according to capital, which is wrong when partners join at different times. Another mistake is to miscount months, for example taking B contribution as 4 months instead of 8. Forgetting to simplify the ratio or using inconsistent units, such as mixing months and years, can also lead to incorrect answers. Using money months consistently and carefully counting the duration solves this type of partnership question reliably.

Final Answer:
The share of A in the profit at the end of the year is Rs 660.