A shopkeeper defrauds both the dealer and the customer by measuring weights incorrectly. When he is purchasing items from the dealer, he takes 20% more than the indicated weight and when he is selling them to the customer, he gives 20% less than the indicated weight. If the price that the shopkeeper charges his customer is the same as what the dealer charges the shopkeeper, then what profit percentage does the shopkeeper make?

Options

A. A)`40%`
B. B)`44%`
C. C)`50%`
D. D)`60%`

Answer

To solve the problem step-by-step, we need to analyze the transactions made by the shopkeeper with both the dealer and the customer.

### Step 1: Understand the shopkeeper's transaction with the dealer
- When the shopkeeper purchases items from the dealer, he takes 20% more than the indicated weight.
- If the indicated weight is 100 units, the shopkeeper actually takes:
\[ \text{Actual weight taken} = 100 + 20\% \text{ of } 100 = 100 + 20 = 120 \text{ units} \]
- Therefore, if the cost price (CP) for the indicated weight (100 units) is assumed to be 100 rupees, the cost price for the actual weight (120 units) is also 100 rupees.

### Step 2: Calculate the effective cost price per unit
- The effective cost price per unit for the shopkeeper when he takes 120 units is:
\[ \text{Effective CP per unit} = \frac{100 \text{ rupees}}{120 \text{ units}} = \frac{5}{6} \text{ rupees/unit} \]

### Step 3: Understand the shopkeeper's transaction with the customer
- When selling to the customer, the shopkeeper gives 20% less than the indicated weight.
- If the indicated weight is 100 units, the shopkeeper actually gives:
\[ \text{Actual weight given} = 100 - 20\% \text{ of } 100 = 100 - 20 = 80 \text{ units} \]
- The selling price (SP) charged to the customer for the indicated weight (100 units) is the same as the cost price from the dealer, which is 100 rupees.

### Step 4: Calculate the effective selling price per unit
- The effective selling price per unit for the shopkeeper when he gives 80 units is:
\[ \text{Effective SP per unit} = \frac{100 \text{ rupees}}{80 \text{ units}} = \frac{5}{4} \text{ rupees/unit} \]

### Step 5: Calculate the profit per unit
- Now, we can calculate the profit per unit:
\[ \text{Profit per unit} = \text{Effective SP per unit} - \text{Effective CP per unit} = \frac{5}{4} - \frac{5}{6} \]
- To subtract these fractions, we need a common denominator, which is 12:
\[ \frac{5}{4} = \frac{15}{12}, \quad \frac{5}{6} = \frac{10}{12} \]
- Therefore:
\[ \text{Profit per unit} = \frac{15}{12} - \frac{10}{12} = \frac{5}{12} \text{ rupees} \]

### Step 6: Calculate the profit percentage
- The profit percentage is calculated as:
\[ \text{Profit Percentage} = \left(\frac{\text{Profit per unit}}{\text{Effective CP per unit}}\right) \times 100 \]
- Substituting the values:
\[ \text{Profit Percentage} = \left(\frac{\frac{5}{12}}{\frac{5}{6}}\right) \times 100 = \left(\frac{5}{12} \times \frac{6}{5}\right) \times 100 = \left(\frac{6}{12}\right) \times 100 = 50\% \]

### Final Answer
The profit percentage that the shopkeeper makes is **50%**.

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