A sweet seller sells three fifths of his sweets at a profit of 10% and the remaining sweets at a loss of 5%. If his overall profit on the entire transaction is Rs. 1500, what is the total cost price of the sweets?

Introduction / Context:
This is a classic mixed transaction problem in profit and loss. A trader sells part of the goods at a gain and the remainder at a loss, yet overall he makes a net profit. The question asks us to compute the total cost price from the known overall profit and the given profit and loss percentages on the respective parts. Understanding weighted profit and loss is key here.

Given Data / Assumptions:

• Fraction of sweets sold at profit = 3/5 of total quantity.
• Profit on this part = 10%.
• Remaining 2/5 of sweets are sold at a loss of 5%.
• Overall profit on all sweets = Rs. 1500.
• We must find the total cost price (CP) of all sweets.
• All sweets are identical, so cost is proportional to quantity.

Concept / Approach:
Let the total cost price be x rupees. Then the cost of the first part is (3/5)x and the cost of the remaining part is (2/5)x. We calculate the selling price of each part by applying the respective profit or loss percentage. Adding these two selling prices gives the total selling price. The overall profit equals total selling price minus total cost price, which is given as Rs. 1500. This leads to an equation in x that we can solve to obtain the total cost price.

Step-by-Step Solution:
Step 1: Let total cost price be x rupees. Step 2: Cost of first part (3/5 of sweets) = (3/5)x. Step 3: Selling price of this part with 10% profit = (3/5)x * 1.10 = (3.3/5)x. Step 4: Cost of remaining part (2/5 of sweets) = (2/5)x. Step 5: Selling price of this part with 5% loss = (2/5)x * 0.95 = (1.9/5)x. Step 6: Total selling price = (3.3/5)x + (1.9/5)x = (5.2/5)x = 1.04x. Step 7: Overall profit = total SP - total CP = 1.04x - x = 0.04x. Step 8: Given overall profit = Rs. 1500, so 0.04x = 1500. Step 9: Therefore, x = 1500 / 0.04 = 1500 * 25 = Rs. 37,500.

Verification / Alternative check:
If total CP is Rs. 37,500, then CP of first part = (3/5) * 37,500 = 22,500. SP1 at 10% profit = 22,500 * 1.10 = 24,750. CP of second part = (2/5) * 37,500 = 15,000. SP2 at 5% loss = 15,000 * 0.95 = 14,250. Total SP = 24,750 + 14,250 = 39,000. Overall profit = 39,000 - 37,500 = Rs. 1,500, matching the given profit exactly.

Why Other Options Are Wrong:
Any other value for total cost such as Rs. 36,400, Rs. 37,415, Rs. 36,500, or Rs. 35,000 would lead to a different overall profit when the same percentages are applied. None of those values yield exactly Rs. 1,500 as the net gain, so they cannot be correct total cost prices under the given conditions.

Common Pitfalls:
Students sometimes wrongly average the profit and loss percentages as simple percentages without considering the fractions 3/5 and 2/5. Another mistake is to assume 10% and 5% apply to the same cost base or to forget that total profit is the difference between total selling price and total cost price. Always form equations based on money values, not just the percentages alone.

Final Answer:
The total cost price of all the sweets is Rs. 37,500.