A trader declares that he will sell his commodity at $10\%$ profit over his cost price. However, he alters his weighing balance to sell $10\%$ less to each customer. What is his true profit margin in percent?

The trader claims $10\%$ profit but cheats by giving $10\%$ less weight.

Let the cost price of $1$ kg commodity $= ₹100$

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Declared selling price with $10\%$ profit:

Declared selling price $= ₹100 + 10\% \text{ of } ₹100$

$= ₹100 + ₹10$

$= ₹110$ per kg

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What actually happens:

• Customer pays $₹110$ (thinking they buy $1$ kg)
• Trader gives only $0.9$ kg (due to rigged scale)
• Customer gets $0.9$ kg for $₹110$

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If $0.9$ kg costs $₹110$, then $1$ kg costs:

Actual selling price per kg $= \dfrac{₹110}{0.9}$

$= ₹122.22$

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Cost price per kg $= ₹100$

Actual selling price per kg $= ₹122.22$

True profit $= ₹122.22 - ₹100$

$= ₹22.22$

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True profit percentage $= \dfrac{\text{True profit}}{\text{Cost price}} \times 100$

$= \dfrac{₹22.22}{₹100} \times 100$

$= 22.22\%$

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Alternative approach:

True profit percentage $= \left[\dfrac{1 + \text{declared profit percentage}}{1 - \text{weight reduction percentage}} - 1\right] \times 100$

$= \left[\dfrac{1.10}{0.90} - 1\right] \times 100$

$= [1.2222 - 1] \times 100$

$= 22.22\%$

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The trader's true profit margin is $22.22\%$