A grocer has 1600 kg of wheat. He sells part of it at a profit of 20% and the remaining part at a profit of 12%, so that his overall profit on the entire quantity is 17%. How many kilograms of wheat does he sell at 20% profit?

Introduction / Context:
This question is another weighted average profit problem, similar to earlier rice and mango examples. The grocer sells some wheat at one profit rate and the rest at a different profit rate, but we know the overall profit percentage on the complete stock. Such questions assess a student's ability to set up and solve linear equations involving percentage profit and total quantities.

Given Data / Assumptions:

• Total wheat = 1600 kg.
• Part sold at 20% profit.
• Remaining part sold at 12% profit.
• Overall profit on the entire 1600 kg is 17%.
• Cost price per kilogram is uniform.
• We must find the quantity sold at 20% profit.

Concept / Approach:
Let cost price per kg be C. If x kg is sold at 20% profit:

• Profit from x kg at 20% = 0.2Cx.
• Remaining quantity = 1600 - x kg, with profit 12%.
• Profit from (1600 - x) kg at 12% = 0.12C(1600 - x).
• Total profit = 0.17 * 1600C.

Set up the equation for total profit using these profits, divide out C, and solve for x.

Step-by-Step Solution:
Let x be the quantity of wheat in kg sold at 20% profit. Then 1600 - x kg is sold at 12% profit. Assume cost price per kg is C. Total profit from x kg at 20% = 0.2Cx. Total profit from 1600 - x kg at 12% = 0.12C(1600 - x). Overall profit at 17% on all 1600 kg = 0.17 * 1600C. Equation: 0.2Cx + 0.12C(1600 - x) = 0.17 * 1600C. Divide both sides by C: 0.2x + 0.12(1600 - x) = 0.17 * 1600. 0.2x + 192 - 0.12x = 272. 0.08x = 80, hence x = 80 / 0.08 = 1000. So 1000 kg of wheat is sold at 20% profit.

Verification / Alternative check:
Verify with assumed cost price C = 1 per kg:

• 1000 kg at 20% gives profit 1000 * 0.2 = 200.
• 600 kg at 12% gives profit 600 * 0.12 = 72.
• Total profit = 200 + 72 = 272.
• Total cost = 1600 * 1 = 1600.
• Overall profit percent = (272 / 1600) * 100 = 17%.

This confirms that the calculated distribution is correct.

Why Other Options Are Wrong:
If 600 kg were at 20% profit, the weighted profit would be much lower than 17%. Values like 800 kg or 1200 kg also lead to different overall profit percentages when computed carefully. Only 1000 kg at 20% and 600 kg at 12% produce exactly 17% overall profit.

Common Pitfalls:
A common mistake is to average 20% and 12% directly without considering quantities. Another error is to set up the profit equation incorrectly, mixing up which term corresponds to which quantity. Always represent the unknown quantity as a variable and write the total profit as the sum of profits from each part before equating it to the overall profit.

Final Answer:
The grocer sells 1000 kg of wheat at 20% profit.