A and B start a business by investing Rs. 12,000 and Rs. 14,400 respectively. At the end of the year, B receives 12% of the total profit as management fees, and the remaining profit is shared between A and B in the ratio of their investments. If the total annual profit is Rs. 7,500, what is the difference between their profits?

Introduction / Context:
This partnership problem introduces the idea of a management fee taken by one partner before the remaining profit is divided in proportion to capital. A and B invest different amounts, and B, as the managing partner, receives a fixed percentage of the total profit. The question asks for the difference between the final profits of A and B after these steps are followed.

Given Data / Assumptions:

• A invests Rs. 12,000.
• B invests Rs. 14,400.
• Total annual profit is Rs. 7,500.
• B receives 12% of the total profit as management fees.
• The remaining profit is shared between A and B in the ratio of their capital investments.
• We must find the difference between the profits received by A and B.

Concept / Approach:
First, compute B's management fee as a fixed percentage of the total profit. Subtract this from the total profit to obtain the remaining amount, which is then shared in proportion to capital. We find the capital ratio, calculate each partner's share from the remaining profit and finally add B's management fee to B's share. The difference between A's share and B's total share gives the required answer.

Step-by-Step Solution:
Step 1: Total profit is Rs. 7500. Step 2: B receives 12% of Rs. 7500 as management fees. So management fee = 12/100 * 7500 = 0.12 * 7500 = Rs. 900. Step 3: Remaining profit for distribution based on capital is 7500 - 900 = Rs. 6600. Step 4: Capital ratio of A : B is 12000 : 14400. Divide both by 1200 to get 10 : 12, which simplifies further to 5 : 6. Step 5: Total ratio parts are 5 + 6 = 11. Step 6: Value of one part of the remaining profit is 6600 / 11 = Rs. 600. Step 7: A's share from the remaining profit is 5 * 600 = Rs. 3000. Step 8: B's share from the remaining profit is 6 * 600 = Rs. 3600. Step 9: B also receives the management fee of Rs. 900, so B's total profit share is 3600 + 900 = Rs. 4500. Step 10: The difference between their profits is B's share minus A's share = 4500 - 3000 = Rs. 1500.

Verification / Alternative check:
Confirm that the total profit is correctly accounted for. A receives Rs. 3000. B receives Rs. 3600 from the remaining profit plus Rs. 900 as a management fee, totalling Rs. 4500. Sum of the two partners' profits is 3000 + 4500 = Rs. 7500, which matches the given total profit. This verifies that the distribution has been done correctly and the calculated difference is accurate.

Why Other Options Are Wrong:
Differences such as Rs. 1250, Rs. 600, Rs. 900 or Rs. 750 would imply different allocations of the 6600 and a different management fee or capital ratio. When checked against the correct capital ratio of 5 : 6 and the fixed management fee of 12% of Rs. 7500, none of these alternative differences are consistent. Only Rs. 1500 arises from the valid computations described above.

Common Pitfalls:
One frequent mistake is to calculate the 12% management fee on only part of the profit instead of the total profit. Another error is to forget to add the management fee to B's share when computing the difference. Some learners also miscalculate the capital ratio by not simplifying it correctly. Carefully separate the calculation into steps and always check that the final sums match the total profit.

Final Answer:
The difference between the profits of A and B is Rs. 1500, which corresponds to option D.