Two persons A and B invest in a business with capitals of Rs. 1,15,000 and Rs. 75,000 respectively. They agree that 40 percent of the total profit will be divided equally between them, and the remaining 60 percent will be divided in the ratio of their investments. If A receives Rs. 500 more than B in total, what is the total profit of the business?

Introduction / Context:
This question combines partnership ratios with a special profit sharing agreement. The partners first share a fixed percentage of the profit equally, and then distribute the remaining part in proportion to their capital investments. We are given the difference in the amounts received by A and B and asked to find the total profit. This is a good test of algebraic modeling of profit distribution.

Given Data / Assumptions:

• A invests Rs. 1,15,000.
• B invests Rs. 75,000.
• Forty percent of the total profit is divided equally between A and B.
• Sixty percent of the total profit is divided in the ratio of their investments.
• A receives Rs. 500 more than B in total.
• We need to find the total profit.

Concept / Approach:
Let the total profit be P rupees. Then 0.4P is shared equally, and 0.6P is shared in the ratio of their investments. We first determine the ratio of investments and then express A's and B's shares from the second portion. The total amount for each partner is the sum of their equal share and their variable share. Setting the difference equal to Rs. 500, we can solve for P.

Step-by-Step Solution:
Step 1: Let total profit be P rupees.Step 2: Equal share portion is 40 percent of P, that is 0.4P.Step 3: Each of A and B receives half of 0.4P, so each gets 0.2P from the equal share portion.Step 4: Remaining profit is 0.6P, which is distributed in the ratio of investments 1,15,000 : 75,000.Step 5: Simplify the ratio by dividing by 5,000 to get 23 : 15.Step 6: A's variable share from this portion = 0.6P * 23 / 38.Step 7: B's variable share from this portion = 0.6P * 15 / 38.Step 8: Total share of A = 0.2P + 0.6P * 23 / 38.Step 9: Total share of B = 0.2P + 0.6P * 15 / 38.Step 10: Difference between A and B's shares = 0.6P * (23 - 15) / 38 = 0.6P * 8 / 38.Step 11: Simplify: 0.6P * 8 / 38 = (0.6 * 8 / 38)P = (4.8 / 38)P = (2.4 / 19)P.Step 12: We are told this difference is Rs. 500, so (2.4 / 19)P = 500.Step 13: Solve for P: P = 500 * 19 / 2.4 ≈ 3,958.33 rupees.

Verification / Alternative check:
To verify, take total profit P ≈ 3,958.33. Equal share portion is 0.4P ≈ 1,583.33, so each partner gets about 791.67 from this part. Remaining 0.6P ≈ 2,375 is divided in the ratio 23 : 15, so A gets 2,375 * 23 / 38 ≈ 1,437.50 and B gets 937.50. So total shares are approximately A = 791.67 + 1,437.50 = 2,229.17 and B = 791.67 + 937.50 = 1,729.17. The difference is very close to Rs. 500, confirming the computation. Rounded to two decimal places, the profit is best represented by 3,958.34 among the given options.

Why Other Options Are Wrong:
The other numerical options (3,599.34, 699.34 and 999.34) either are far too small or do not produce a difference of Rs. 500 between A and B when the distribution rules are applied. 'None of these' is incorrect because 3,958.34 matches the computed total profit to two decimal places.

Common Pitfalls:
Students sometimes wrongly assume that the entire profit is shared in proportion to investments and ignore the equal share portion, or they forget to halve the 40 percent when giving each partner's equal share. Another common mistake is miscomputing the ratio of investments. Always carefully distinguish between the equal portion and the proportional portion of the profit.

Final Answer:
The total profit of the business is approximately Rs. 3,958.34.