A trader has 22 quintals of wheat. He sells part of it at a profit of 23 percent and the remaining part at a profit of 33 percent, so that his overall profit on the entire 22 quintals is 27 percent. How many kilograms of wheat are sold at the 33 percent profit?

Introduction / Context:
This question involves a weighted average of profit percentages. A trader sells different portions of a commodity at two different profit rates and the overall profit percentage on the entire quantity is given. The task is to determine how much of the commodity was sold at one of the profit rates. This is a classic application of the concept of weighted averages and linear equations in profit and loss problems.

Given Data / Assumptions:

• Total wheat quantity is 22 quintals.
• Part of the wheat is sold at 23 percent profit.
• The remaining part is sold at 33 percent profit.
• Overall profit on the entire 22 quintals is 27 percent.
• We must find the quantity sold at 33 percent profit, expressed in kilograms.

Concept / Approach:
Since the profit percentages differ on different parts of the quantity but the cost price per quintal is uniform, we can assume a convenient cost price per quintal, for example Rs. 1, to simplify the algebra. Let x quintals be sold at 23 percent profit and the remaining 22 minus x at 33 percent profit. We then compute the total revenue using these profit rates, set it equal to the revenue that would correspond to a uniform 27 percent profit on all 22 quintals, and solve the resulting linear equation for x. Finally we convert the quantity that corresponds to 33 percent profit into kilograms.

Step-by-Step Solution:
Assume cost price per quintal equals 1 rupee for simplicity.Total cost price for 22 quintals equals 22 rupees.Let x quintals be sold at 23 percent profit, so selling price for these is x * 1.23.Remaining quantity equals 22 minus x quintals, sold at 33 percent profit, giving selling price (22 minus x) * 1.33.Total selling price from both parts equals x * 1.23 plus (22 minus x) * 1.33.Overall profit is 27 percent on 22 rupees, so total selling price should also equal 22 * 1.27 equals 27.94 rupees.Set x * 1.23 + (22 minus x) * 1.33 equal to 27.94 and simplify.Expanding gives 1.23x + 29.26 minus 1.33x equals 27.94.This simplifies to minus 0.10x + 29.26 equals 27.94.So minus 0.10x equals minus 1.32, and x equals 13.2 quintals.Therefore, quantity sold at 33 percent profit equals 22 minus 13.2 equals 8.8 quintals.One quintal is 100 kilograms, so 8.8 quintals equals 8.8 * 100 equals 880 kilograms.

Verification / Alternative check:
We can verify quickly. Revenue from 13.2 quintals at 23 percent profit is 13.2 * 1.23 equals 16.236 rupees. Revenue from 8.8 quintals at 33 percent profit is 8.8 * 1.33 equals 11.704 rupees. Total revenue equals 16.236 plus 11.704 equals 27.94 rupees. With total cost price 22 rupees, profit is 5.94 rupees and profit percentage is 5.94 divided by 22 times 100 which is 27 percent. This matches the required overall profit and confirms that 8.8 quintals, or 880 kilograms, were sold at 33 percent profit.

Why Other Options Are Wrong:
1320 kilograms correspond to 13.2 quintals, which is the quantity sold at 23 percent profit, not 33 percent profit. Values 440 kilograms and 1760 kilograms do not satisfy the weighted average equation and would not produce an overall profit of 27 percent. Only 880 kilograms results in a combined selling price that matches the required average profit on the entire stock.

Common Pitfalls:
Some learners try to average the percentages directly using proportions of profit rates alone, without considering the total quantity. Others mix up which part of the quantity corresponds to each profit rate when interpreting the algebraic solution. Using a clear variable definition, writing out the full total selling price expression, and checking units carefully helps avoid these mistakes.

Final Answer:
The trader sells 880 kilograms of wheat at a profit of 33 percent.