A trader has 630 kg of rice. He sells part of it at a profit of 15% and the remaining part at a profit of 8%, so that his overall profit on the entire quantity is 12%. How many kilograms of rice does he sell at 8% profit?

Introduction / Context:
This question is a classic example of a weighted average profit problem. A trader sells parts of the same commodity at different profit percentages, and we are given the overall profit on the total quantity. Such questions test understanding of profit percentage, total cost, total profit, and how to set up and solve linear equations for mixtures of profit rates.

Given Data / Assumptions:

• Total quantity of rice = 630 kg.
• Part of the rice is sold at 15% profit.
• The rest is sold at 8% profit.
• Overall profit on the entire 630 kg is 12%.
• We assume the cost price per kilogram is uniform.
• We need to find the quantity sold at 8% profit.

Concept / Approach:
We can treat this as a weighted average of profit percentages. If the cost price per kilogram is C, then:

• Profit on the portion sold at 15% is 0.15 C per kg.
• Profit on the portion sold at 8% is 0.08 C per kg.
• Overall profit is 0.12 C per kg for all 630 kg.

We introduce a variable for the quantity sold at 15% profit, then express the weighted average of profit amounts and equate it to the total profit at 12%.

Step-by-Step Solution:
Let x be the quantity of rice (in kg) sold at 15% profit. Then the remaining quantity sold at 8% profit is (630 - x) kg. Assume cost price per kg is C (same for all). Total profit from x kg at 15% = 0.15 * C * x. Total profit from (630 - x) kg at 8% = 0.08 * C * (630 - x). Overall profit at 12% on 630 kg = 0.12 * C * 630. Set up the equation: 0.15Cx + 0.08C(630 - x) = 0.12C * 630. Divide both sides by C: 0.15x + 0.08(630 - x) = 0.12 * 630. 0.15x + 50.4 - 0.08x = 75.6. 0.07x = 25.2, so x = 25.2 / 0.07 = 360. Quantity at 8% profit = 630 - 360 = 270 kg.

Verification / Alternative check:
Check using total profit directly. Suppose C = 1 per kg (for simplicity). Then:

• 360 kg at 15% gives profit 360 * 0.15 = 54.
• 270 kg at 8% gives profit 270 * 0.08 = 21.6.
• Total profit = 54 + 21.6 = 75.6.
• Total cost = 630 * 1 = 630.
• Overall profit percent = (75.6 / 630) * 100 = 12%.

Everything matches the given condition, so the answer is correct.

Why Other Options Are Wrong:
If 300 kg were at 8%, the weighted average profit would change and would not equal 12%. Similarly, 280 kg, 260 kg, or 240 kg lead to different average profit percentages once total profit is recomputed, and none match the required 12% overall. Only 270 kg balances the weighted average correctly.

Common Pitfalls:
A common error is to average 15% and 8% directly to get 11.5% instead of forming an equation based on quantities. Another mistake is confusing which part corresponds to which percentage or making arithmetic errors when solving the equation. Always define the variable clearly, write the total profit expression carefully, and simplify step by step.

Final Answer:
The trader sells 270 kg of rice at 8% profit.