A and B begin a business with investments of Rs. 3000 and Rs. 4000 respectively. After 8 months, A withdraws Rs. 1000 from his capital and B increases his capital by Rs. 1000. At the end of one year, their total profit is Rs. 630. What is A's share of the profit?

Introduction / Context:
This question involves a partnership where both partners change their investments during the year. The concept tested is the calculation of effective capital time contributions for each partner and then using that to divide a known total profit between them.

Given Data / Assumptions:

• A initially invests Rs. 3000.
• B initially invests Rs. 4000.
• After 8 months, A withdraws Rs. 1000, so his capital becomes Rs. 2000.
• After 8 months, B adds Rs. 1000, so his capital becomes Rs. 5000.
• Total business duration is 12 months.
• Total profit at the end of the year is Rs. 630.
• We must find A's share in this profit.

Concept / Approach:
When capital changes during the partnership, we split the time into segments where the capital remains constant and calculate capital time products for each segment. We then add these products for each partner to get their effective investments. The ratio of these effective investments gives the ratio of profit shares, which is applied to the total profit to find individual amounts.

Step-by-Step Solution:
Step 1: For A, from month 0 to month 8, capital is Rs. 3000. Step 2: Capital time for this period is 3000 * 8 = 24,000. Step 3: From month 8 to month 12, A's capital is Rs. 2000. Step 4: Capital time for this period is 2000 * 4 = 8000. Step 5: Total effective capital time for A is 24,000 + 8000 = 32,000. Step 6: For B, from month 0 to month 8, capital is Rs. 4000. Step 7: Capital time for this period is 4000 * 8 = 32,000. Step 8: From month 8 to month 12, B's capital is Rs. 5000. Step 9: Capital time for this period is 5000 * 4 = 20,000. Step 10: Total effective capital time for B is 32,000 + 20,000 = 52,000. Step 11: Ratio of A : B in capital time is 32,000 : 52,000 = 8 : 13. Step 12: Sum of ratio parts = 8 + 13 = 21. Step 13: A's share of profit = (8 / 21) * 630 = 240.

Verification / Alternative check:
If A gets Rs. 240, then B gets the remaining profit 630 - 240 = Rs. 390. Check the ratio: 240 : 390 simplifies by dividing by 30 to 8 : 13, which matches the capital time ratio 32,000 : 52,000. Therefore the distribution of profit is consistent with the investments and time periods, confirming that A's share is correct.

Why Other Options Are Wrong:
Rs. 75, Rs. 125 and Rs. 354 do not maintain the required 8 : 13 ratio when compared to B's share. Using any of these values leads either to a different ratio or to a total different from Rs. 630, so they cannot represent A's correct share of profit.

Common Pitfalls:
It is easy to forget to split the year into two parts and instead use only the original capitals for the whole 12 months. Another frequent error is to change the capital amounts but not adjust the months correctly when forming the capital time products. Always handle each time segment separately and then combine them carefully for each partner.

Final Answer:
A's share of the profit is Rs. 240.