A trader has 9 quintals of wheat. He sells part of it at a profit of 10% and the rest at a profit of 20%, resulting in an overall profit of 14%. How many kilograms of wheat are sold at 20% profit?

Introduction / Context:
This problem involves a mixture of two different profit percentages applied to different portions of the same commodity. It tests your ability to use weighted averages or alligation concepts on profit percentages. The total quantity and the overall profit percentage are given, and you must work backwards to find how much quantity was sold at the higher profit rate.

Given Data / Assumptions:

• Total wheat = 9 quintals.
• 1 quintal = 100 kg, so total quantity = 900 kg.
• Part is sold at 10% profit.
• Remaining part is sold at 20% profit.
• Overall profit percentage on the entire 9 quintals is 14%.
• We need the quantity (in kg) sold at 20% profit.

Concept / Approach:
We use a weighted average of profit percentages:

• Let x quintals be sold at 20% profit.
• Then (9 - x) quintals are sold at 10% profit.
• Overall profit percentage = (Total profit) / (Total cost) * 100.
• Since total cost is proportional to total quantity, we can equate the weighted average of percentages directly.

Thus, we set up an equation based on the average of 10% and 20% giving 14% over the whole 9 quintals.

Step-by-Step Solution:
Step 1: Let x quintals be sold at 20% profit. Step 2: Then 9 - x quintals are sold at 10% profit. Step 3: Total profit (in percentage terms) on 9 quintals is given as 14%. Step 4: Weighted average equation: (10 * (9 - x) + 20 * x) / 9 = 14. Step 5: Simplify numerator: 10 * (9 - x) + 20 * x = 90 - 10x + 20x = 90 + 10x. Step 6: So (90 + 10x) / 9 = 14. Step 7: Multiply both sides by 9: 90 + 10x = 126. Step 8: 10x = 126 - 90 = 36, so x = 36 / 10 = 3.6 quintals. Step 9: Convert to kilograms: 3.6 quintals * 100 kg per quintal = 360 kg.

Verification / Alternative check:
Assume cost price per kg is Rs 1 to simplify:

• Total cost = 900 kg * 1 = Rs 900.
• Quantity at 20% profit = 360 kg, so profit there = 0.20 * 360 = Rs 72.
• Quantity at 10% profit = 900 - 360 = 540 kg, so profit there = 0.10 * 540 = Rs 54.
• Total profit = 72 + 54 = Rs 126.
• Overall profit percentage = 126 / 900 * 100 = 14%.

This matches the given overall profit percentage, confirming that the quantity at 20% profit is indeed 360 kg.

Why Other Options Are Wrong:
540 kg: If this much were sold at 20% profit, the weighted average profit would be higher than 14%.
180 kg: This would make the average profit less than 14% because most wheat would earn only 10%.
720 kg: This would heavily favor the 20% side, leading to a much larger overall profit percentage.
Only 360 kg satisfies the overall 14% profit condition.

Common Pitfalls:
Students sometimes confuse the units, forgetting to convert quintals to kilograms or vice versa. Others incorrectly average the percentages as (10 + 20) / 2 = 15% without considering quantity distribution, which is wrong here because different quantities are sold at different rates. It is crucial to use the weighted average formula based on quantities, not just average the percentages blindly.

Final Answer:
The trader sells 360 kg of wheat at a 20% profit.