A trader has 2,000 kilograms of rice. He sells part of it at a profit of 10% and the remaining part at a profit of 16%, and his overall profit on the entire quantity is 14.2%. How many kilograms of rice does he sell at 10% profit?

Introduction / Context:
This question belongs to the topic of profit and loss combined with weighted averages. A trader sells different parts of the same commodity at different profit percentages but we know only the overall profit on the whole stock. Our goal is to determine how much quantity was sold at one of the given profit rates. Such problems are very useful for competitive exams because they test the understanding of average profit and the ability to set up a correct linear equation based on quantities and percentages.

Given Data / Assumptions:

• Total quantity of rice = 2,000 kg.
• Part of the rice is sold at a profit of 10%.
• The remaining quantity is sold at a profit of 16%.
• Overall profit on the entire 2,000 kg is 14.2%.
• The cost price per kilogram is assumed to be the same for the whole stock.

Concept / Approach:
The overall profit percentage is a weighted average of the individual profit percentages, where the weights are the quantities sold at each rate. If x kilograms are sold at 10% profit and the remaining (2000 − x) kilograms are sold at 16% profit, then the total profit can be expressed in terms of x. By dividing the total profit by the total cost, we can equate this to the given overall profit percentage and solve for x. We do not need the actual cost price per kilogram because it cancels out in the ratio.

Step-by-Step Solution:
Let the cost price per kilogram be Rs. 1 for simplicity (any constant would cancel later).Total cost price for 2,000 kg = 2000 * 1 = Rs. 2,000.Let x kg be sold at 10% profit, so profit on this part = 0.10 * x.The remaining quantity is (2000 − x) kg, sold at 16% profit, so profit on this part = 0.16 * (2000 − x).Total profit = 0.10x + 0.16(2000 − x).Overall profit percentage = (total profit / total cost) * 100 = 14.2%.So, (0.10x + 0.16(2000 − x)) / 2000 = 0.142.Compute: 0.10x + 320 − 0.16x = 0.142 * 2000 = 284.This simplifies to −0.06x + 320 = 284, so −0.06x = −36, hence x = 600.Therefore, 600 kg of rice were sold at 10% profit.

Verification / Alternative check:
If 600 kg are sold at 10%, profit on that part = 0.10 * 600 = 60.Remaining 1,400 kg are sold at 16%, profit on that part = 0.16 * 1400 = 224.Total profit = 60 + 224 = 284.Total cost = 2000 (using Rs. 1 per kg assumption), so overall profit percentage = 284 / 2000 = 0.142 = 14.2%, which matches the given value.

Why Other Options Are Wrong:
If 1,400 kg were taken as the quantity at 10%, the weighted average profit would be much closer to 11.2%, not 14.2%.If 800 kg or 1,000 kg were at 10%, recalculating the overall profit yields percentages different from 14.2%.Only 600 kg satisfies the given overall profit percentage exactly, so it is the correct choice.

Common Pitfalls:
A frequent mistake is to take the simple average of 10% and 16%, which is 13%, ignoring the fact that different quantities are sold at each rate.Another pitfall is to introduce the actual cost price per kilogram and complicate the algebra, even though it cancels out.Students sometimes mis-handle the algebra when distributing the 0.16 multiplier across the bracket.

Final Answer:
The trader sells 600 kg of rice at a profit of 10% to achieve the overall profit of 14.2%.