Calculating Trader's Total Profit Percentage

This problem involves calculating the overall profit percentage when a trader applies both a profit margin on the price and uses a faulty weight measure. We need to consider both factors to find the true profit.

Understanding the Scenario

The trader makes two dishonest gains:

• Selling grains at a profit of 29%. This means they mark up the price.
• Using a weight that is 13% less than stated. This means they give less product than the customer pays for.

We need to find the combined effect of these two actions on the total profit percentage.

Step-by-Step Calculation

Let's assume the trader deals with 1000 grams of grain. We can assign a cost price (CP) to this quantity.

1. Assume Cost Price: Let the cost price (CP) of 1000 grams of grain be $100.

2. Calculate Selling Price (SP) based on Profit Claim: The trader claims a 29% profit. The selling price is calculated on the cost price.

   $\text{SP} = \text{CP} \times (1 + \frac{\text{Profit \%}}{100})$

   $\text{SP} = \$100 \times (1 + \frac{29}{100}) = \$100 \times 1.29 = \$129.$

   This is the price the customer pays, assuming they receive 1000 grams.

3. Account for the Faulty Weight: The trader uses a weight that is 13% less. This means for every 1000 grams the customer *pays for*, the trader actually provides:

   $\text{Actual Weight Supplied} = 1000 \text{ grams} \times (1 - \frac{13}{100})$

   $\text{Actual Weight Supplied} = 1000 \text{ grams} \times (1 - 0.13) = 1000 \text{ grams} \times 0.87 = 870 \text{ grams}.$

4. Determine the Actual Cost Price (CP) for Goods Sold: The trader sold 870 grams. We need to find the cost price for exactly this amount.

   Cost per gram = $\frac{\$100}{1000 \text{ grams}} = \$0.1$ per gram.

   $\text{Actual CP for 870 grams} = 870 \text{ grams} \times \$0.1/\text{gram} = \$87.$

5. Calculate the Actual Profit: The trader received $129 (the selling price for 1000 grams) but only spent $87 (the cost for the 870 grams actually supplied).

   $\text{Actual Profit} = \text{SP} - \text{Actual CP}$

   $\text{Actual Profit} = \$129 - \$87 = \$42.$

6. Calculate the Total Percentage Profit: The profit percentage is calculated based on the actual cost price of the goods sold.

   $\text{Total Profit \%} = \left( \frac{\text{Actual Profit}}{\text{Actual CP}} \right) \times 100$

   $\text{Total Profit \%} = \left( \frac{\$42}{\$87} \right) \times 100$

   $\text{Total Profit \%} \approx 0.4827586 \times 100 \approx 48.27586\%.$

7. Round off the Answer: Rounding the result to the nearest whole number gives 48%.

Alternative Method: Using Combined Profit Formula

We can use a direct formula for combined profit/loss percentage when both profit margin and faulty weight are involved.

Let the claimed profit percentage be $P = 29\%$.

Let the percentage reduction in weight be $L = 13\%$.

The formula for the effective profit percentage is:

$\text{Effective Profit \%} = \left( \frac{1 + P/100}{1 - L/100} - 1 \right) \times 100$

Plugging in the values:

$\text{Effective Profit \%} = \left( \frac{1 + 29/100}{1 - 13/100} - 1 \right) \times 100$

$\text{Effective Profit \%} = \left( \frac{1.29}{0.87} - 1 \right) \times 100$

$\text{Effective Profit \%} = \left( \frac{129}{87} - 1 \right) \times 100$

$\text{Effective Profit \%} = \left( \frac{129 - 87}{87} \right) \times 100$

$\text{Effective Profit \%} = \left( \frac{42}{87} \right) \times 100 \approx 48.27586\%.$

Rounding to the nearest whole number, the total profit percentage is 48%.

Conclusion

By combining the profit margin and the deceptive weight, the trader earns a significant total profit. The calculation shows that after accounting for both factors, the total profit earned by the trader is approximately 48%.