A, B and C start a company with initial capital investments in the ratio 2 : 3 : 4. After 6 months, A adds more capital so that from then on his total investment equals B's initial investment. If B's annual share of profit is Rs. 3000, what is the total profit of the company for the year?

Introduction / Context:
This partnership problem involves a change in capital contribution by one partner during the year. A, B and C initially invest in a certain ratio, but after 6 months A increases his capital. To find the total profit, we must carefully account for different capital amounts over different time periods and then use the known profit share of B to determine the total profit.

Given Data / Assumptions:

• Initial capital ratio of A : B : C is 2 : 3 : 4.
• Business runs for 1 year, that is 12 months.
• For the first 6 months, A invests according to the initial ratio.
• After 6 months, A increases his capital so that his capital becomes equal to B's initial capital.
• B and C keep their initial capitals for the full 12 months.
• B's share of annual profit is Rs. 3000.
• Profits are shared in proportion to capital multiplied by time.

Concept / Approach:
Let the common capital factor be k. Initially, A invests 2k, B invests 3k and C invests 4k. After 6 months, A increases his capital to 3k, equal to B's original investment. We must compute the capital time units for each partner over the 12 month period. The profit ratio will then be proportional to these capital time products. Once the ratio is formed, we use B's known share to find the total profit by treating the ratio as parts of the total.

Step-by-Step Solution:
Step 1: Let initial capitals be A = 2k, B = 3k and C = 4k. Step 2: For the first 6 months, A invests 2k. For the next 6 months, A invests 3k after adding more capital. Step 3: Therefore, A's capital time units are (2k * 6) + (3k * 6) = 12k + 18k = 30k. Step 4: B invests 3k for all 12 months, so B's capital time units are 3k * 12 = 36k. Step 5: C invests 4k for all 12 months, so C's capital time units are 4k * 12 = 48k. Step 6: Thus the ratio of capital time units for A : B : C is 30k : 36k : 48k. Step 7: Divide by 6k to simplify the ratio to 5 : 6 : 8. Step 8: Total parts in the ratio are 5 + 6 + 8 = 19. Step 9: B's share corresponds to 6 parts and is given as Rs. 3000, so one part is 3000 / 6 = Rs. 500. Step 10: Therefore, total profit equals 19 * 500 = Rs. 9500.

Verification / Alternative check:
We can compute each partner's share from the total profit to verify. A's share is 5 * 500 = Rs. 2500. B's share is 6 * 500 = Rs. 3000, which matches the given information. C's share is 8 * 500 = Rs. 4000. Adding these, 2500 + 3000 + 4000 = Rs. 9500, which confirms that our calculation for the total profit is correct.

Why Other Options Are Wrong:
If the total profit were Rs. 10600, 7500, 8900 or 8400, then B's share as 6 parts out of 19 would not equal Rs. 3000. For example, with a total profit of Rs. 8400, one part would be 8400 / 19, which is not 500, and B's share would not be Rs. 3000. Only a total profit of Rs. 9500 produces a per part value of Rs. 500 and a B share of Rs. 3000 consistent with the ratio 5 : 6 : 8.

Common Pitfalls:
One frequent mistake is to ignore the change in A's capital after 6 months and simply use the initial ratio 2 : 3 : 4 for profit sharing. Another error is to assume that A's capital becomes equal to C's instead of B's. It is also easy to forget to add the two separate contributions of A over different time intervals when computing capital time units. A stepwise calculation that clearly separates the first 6 months and the final 6 months helps to avoid these errors.

Final Answer:
The total profit of the company for the year is Rs. 9500, which corresponds to option A.