A trader has 1200 kilograms of rice. He sells part of it at a profit of 5% and the rest at a profit of 11%, so that his overall profit on the entire quantity is 7%. How many kilograms of rice did he sell at 5% profit?

Introduction / Context:
This question combines the idea of weighted average profit with quantities sold at different profit rates. The trader sells one part of the rice stock at a lower profit percentage and the remaining part at a higher profit percentage. The overall profit percentage on the entire stock is known. The aim is to find how much was sold at the lower profit rate. This type of question is common in mixture and alligation topics as well as profit and loss.

Given Data / Assumptions:

• Total quantity of rice = 1200 kilograms.
• Some quantity is sold at 5% profit.
• The rest is sold at 11% profit.
• Overall profit on total 1200 kilograms = 7%.
• We need to find the quantity sold at 5% profit.

Concept / Approach:
The overall profit percentage is a weighted average of the individual profit percentages, with weights equal to the respective quantities sold. If x kilograms are sold at 5% profit and the remaining (1200 - x) kilograms at 11% profit, then the total profit percentage is given by [5x + 11(1200 - x)] / 1200. This must equal 7. Setting up this equation and solving for x yields the quantity sold at 5% profit. This is essentially a linear equation in one variable derived from the weighted average formula.

Step-by-Step Solution:
Step 1: Let x be the quantity sold at 5% profit.Step 2: Then quantity sold at 11% profit is 1200 - x kilograms.Step 3: Total profit (in percentage points times kilograms) = 5x + 11(1200 - x).Step 4: Overall profit percentage is 7, so this total profit divided by 1200 equals 7.Step 5: Write equation: (5x + 11(1200 - x)) / 1200 = 7.Step 6: Multiply both sides by 1200: 5x + 11(1200 - x) = 7 * 1200.Step 7: Expand: 5x + 13200 - 11x = 8400.Step 8: Combine like terms: (5x - 11x) + 13200 = 8400 gives -6x + 13200 = 8400.Step 9: Rearrange: -6x = 8400 - 13200 = -4800.Step 10: So x = (-4800) / (-6) = 800.Step 11: Therefore 800 kilograms of rice were sold at 5% profit.

Verification / Alternative check:
We can quickly verify using the found values. Rice at 5% profit: 800 kilograms. Rice at 11% profit: 1200 - 800 = 400 kilograms. Total profit in percentage points times kilograms = 5 * 800 + 11 * 400 = 4000 + 4400 = 8400. Overall profit percentage = 8400 / 1200 = 7%. This exactly matches the given overall profit percentage, confirming that the derived quantity of 800 kilograms is correct.

Why Other Options Are Wrong:
Option A (900) would give too much weight to the lower profit, reducing overall profit below 7%. Option B (600) would overweight the higher profit, pushing the overall percentage above 7%. Option C (400) would mean very little rice is sold at 5% profit and most at 11%, which again does not match the given 7% overall profit. Only option D, 800 kilograms, satisfies the weighted average condition required by the data.

Common Pitfalls:
Some students mistakenly average 5% and 11% directly to get 8% and then try to adjust, ignoring that different quantities are sold at each rate. Others mix up which quantity corresponds to which profit percentage. Writing the equation clearly with x and 1200 - x, and then carefully expanding and simplifying, is the safest way to avoid such errors. Using the alligation rule is also possible but requires careful handling of ratios and total quantity.

Final Answer:
The trader sells 800 kilograms of rice at a 5% profit.