A trader has 960 kg of rice in stock. He sells a part of this rice at a profit of 20% and sells the remaining rice at a profit of 8%, so that his overall profit on the entire 960 kg is exactly 12%. How many kilograms of rice does he sell at the lower profit rate of 8%?

Introduction / Context:
This is a classic mixture-type profit and loss problem in which a trader sells one part of his stock at a higher profit percentage and the remaining part at a lower profit percentage. The average or overall profit percentage for the entire quantity is given, and we must determine how much quantity was sold at one of the specific profit rates. Such questions test understanding of weighted averages and profit percentage calculations, which are very important for aptitude exams involving commercial arithmetic and business mathematics.

Given Data / Assumptions:
- Total quantity of rice = 960 kg. - Part of the rice is sold at a profit of 20%. - The rest of the rice is sold at a profit of 8%. - Overall profit on the entire 960 kg is 12%. - We must find the quantity sold at 8% profit.

Concept / Approach:
The core idea is that overall profit percentage is a weighted average of the individual profit percentages, weighted by the quantities sold at each rate. Let x be the quantity sold at 20% profit; then 960 - x is sold at 8% profit. By assuming the cost price per kilogram is the same for all rice, we can either work with an assumed cost price (like Rs. 1 per kg) or cancel it out. We then equate the weighted profit to the given overall profit and solve for x. The remaining quantity is simply 960 - x, which gives the amount sold at the lower profit rate.

Step-by-Step Solution:
Assume cost price per kg = Re. 1 (this assumption simplifies calculations without affecting percentages). Total cost price of 960 kg = 960 * 1 = Rs. 960. Let x kg be sold at 20% profit; so its profit = 0.20 * x. Then the remaining (960 - x) kg is sold at 8% profit; profit on this part = 0.08 * (960 - x). Total profit = 0.20x + 0.08(960 - x). Overall profit is 12%, so total profit = 0.12 * 960. Set up the equation: 0.20x + 0.08(960 - x) = 0.12 * 960. Compute the right side: 0.12 * 960 = 115.2. Left side: 0.20x + 76.8 - 0.08x = 0.12x + 76.8. So, 0.12x + 76.8 = 115.2. Hence, 0.12x = 115.2 - 76.8 = 38.4. x = 38.4 / 0.12 = 320. Quantity sold at 20% profit = 320 kg; quantity at 8% profit = 960 - 320 = 640 kg.

Verification / Alternative check:
Profit from 320 kg at 20% = 0.20 * 320 = 64. Profit from 640 kg at 8% = 0.08 * 640 = 51.2. Total profit = 64 + 51.2 = 115.2. Overall profit percentage = 115.2 / 960 * 100 = 12%, which matches the given overall profit, confirming the answer.

Why Other Options Are Wrong:
- 460 kg, 560 kg, 540 kg, and 600 kg do not satisfy the weighted average equation for 12% overall profit. - For each of these quantities, if you compute the combined profit, it will not equal 12% of the total cost price. - Only 640 kg at 8% and 320 kg at 20% give the correct overall profit percentage.

Common Pitfalls:
Students often misinterpret the overall profit as a simple average of 20% and 8%, which would give 14%, not 12%. Another mistake is to forget that the quantities sold at each profit rate are different, so the average must be weighted. Some learners also attempt to work with rupee amounts directly without simplifying via an assumed cost price, which can lead to arithmetic errors.

Final Answer:
The trader sells 640 kg of rice at 8% profit.