A, B and C start a business where their initial capital investments are in the ratio 2 : 3 : 4. At the end of 6 months, A increases his capital so that his total capital becomes equal to C's initial capital. If B's annual share of profit is Rs. 3,000, what is the total profit earned by the business?

Introduction / Context:
In this partnership question, three partners A, B and C invest in a ratio but one partner increases his capital after 6 months. Because the capital of one partner changes midway, we must compute effective investments taking time into account. We are told B's annual profit share and must determine the total profit of the business.

Given Data / Assumptions:

• Initial capital ratio of A : B : C = 2 : 3 : 4.
• Business duration is 1 year (12 months).
• For the first 6 months, all partners invest in the original ratio 2 : 3 : 4.
• At the end of 6 months, A increases his capital so that his new capital equals C's initial capital.
• Thus, during the last 6 months A's capital becomes equal to 4 parts, the same as C's initial capital.
• B keeps his capital unchanged throughout the year.
• B's share of the annual profit is Rs. 3,000.
• Profit is shared in proportion to capital * time.

Concept / Approach:
We treat the year in two halves of 6 months each. For each half, we record the capital ratio of A, B and C and compute the effective investment by multiplying by 6 months. After summing each partner's effective investment, we obtain the overall ratio for profit sharing. Using B's share, we can then scale the ratio to find the total profit.

Step-by-Step Solution:
Step 1: Let the unit capital for the ratio be k. Initially: A = 2k, B = 3k, C = 4k. Step 2: For the first 6 months, effective investments are: A = 2k * 6 = 12k, B = 3k * 6 = 18k, C = 4k * 6 = 24k. Step 3: After 6 months, A increases his capital so that it becomes equal to C's initial 4k. So for the last 6 months: A = 4k, B = 3k, C = 4k. Step 4: Effective investments for the second 6 months are: A = 4k * 6 = 24k, B = 3k * 6 = 18k, C = 4k * 6 = 24k. Step 5: Total effective investments across the year: A = 12k + 24k = 36k, B = 18k + 18k = 36k, C = 24k + 24k = 48k. Step 6: Divide by 12k to simplify the ratio: A : B : C = 3 : 3 : 4. Step 7: Total number of parts = 3 + 3 + 4 = 10. B's share is 3 parts which correspond to Rs. 3,000. Step 8: Value of 1 part = 3,000 / 3 = Rs. 1,000. Step 9: Total profit = 10 * 1,000 = Rs. 10,000.

Verification / Alternative check:
We can verify by computing A's and C's shares. A gets 3 parts = Rs. 3,000 and C gets 4 parts = Rs. 4,000. Adding A's, B's and C's shares gives 3,000 + 3,000 + 4,000 = 10,000, which matches the total profit we calculated. The ratio of profits 3 : 3 : 4 is consistent with the ratio of effective investments 36k : 36k : 48k.

Why Other Options Are Wrong:
If the total profit were Rs. 8,640, Rs. 9,850 or Rs. 11,220, B's share calculated as 3/10 of that amount would not equal Rs. 3,000. For example, 3/10 of 8,640 is 2,592, not 3,000. Therefore those options do not satisfy the condition that B's annual share is exactly Rs. 3,000.

Common Pitfalls:
A common error is to use only the initial ratio 2 : 3 : 4 for the whole year, ignoring the increase in A's capital after 6 months. Another mistake is to add capital ratios directly across periods instead of computing capital * time. Always break the timeline into segments whenever there is a change in capital and then sum effective investments to build the final ratio.

Final Answer:
The total profit earned by the business is Rs. 10,000.