A dishonest dealer sells the goods at `6(1)/(4)%` loss on cost price but uses `12(1)/(2)%` less weight. What is his percentage profit or loss

Answer

To solve the problem step by step, we will first analyze the loss percentage and the weight reduction percentage, and then calculate the overall profit or loss percentage.

### Step 1: Understand the Loss Percentage
The dealer sells the goods at a loss of \(6\frac{1}{4}\%\). This can be converted to a fraction:
\[ 6\frac{1}{4}\% = \frac{25}{4}\% = \frac{25}{400} = \frac{1}{16} \]
This means if the cost price (CP) is 16, the selling price (SP) will be:
\[ SP = CP - \text{Loss} = 16 - 1 = 15 \]

### Step 2: Understand the Weight Reduction Percentage
The dealer uses \(12\frac{1}{2}\%\) less weight. This can also be converted to a fraction:
\[ 12\frac{1}{2}\% = \frac{25}{2}\% = \frac{25}{200} = \frac{1}{8} \]
This means if the dealer is supposed to sell 8 units, he actually sells:
\[ 8 - 1 = 7 \text{ units} \]

### Step 3: Calculate the Actual Cost Price for the Weight Sold
If the dealer is paying for 8 units but only gives 7 units, the effective cost price for the 7 units is:
\[ \text{Effective CP for 7 units} = \frac{CP}{8} \times 7 = \frac{16}{8} \times 7 = 14 \]

### Step 4: Calculate the Selling Price for the 7 Units
The dealer sells these 7 units for 15 (as calculated in Step 1).

### Step 5: Calculate the Profit
Now, we can find the profit made by the dealer:
\[ \text{Profit} = SP - \text{Effective CP} = 15 - 14 = 1 \]

### Step 6: Calculate the Profit Percentage
To find the profit percentage:
\[ \text{Profit Percentage} = \left( \frac{\text{Profit}}{\text{Effective CP}} \right) \times 100 = \left( \frac{1}{14} \right) \times 100 \approx 7.14\% \]

### Conclusion
The dealer makes a profit of approximately \(7.14\%\).

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