Understanding the Vendor's Claim and Cheating

The problem involves a vendor who claims to sell wheat at a specific loss percentage but uses incorrect weights to increase their actual profit.

Calculating Actual Profit Percentage

We need to find the vendor's real profit percentage considering both the claimed loss and the weight discrepancy.

Step 1: Define Variables and Base Cost

Let the cost price (CP) of 1 unit of weight be $1$. Assume the vendor claims to sell $W$ units of weight.

• The cost price for $W$ units is assumed to be $W \times 1 = W$.
• The vendor claims a 50% loss. This means the Selling Price (SP) is 50% of the cost price.
• Claimed SP for $W$ units = $W \times (1 - 0.50) = 0.5W$.

Step 2: Account for Cheating in Weights

The vendor uses weights that are 65% less than the stated weight ($W$).

• Actual weight used by the vendor = $W \times (1 - 0.65) = W \times 0.35 = 0.35W$.
• The actual cost incurred by the vendor is for this reduced weight: Actual CP = $0.35W \times 1 = 0.35W$.

Step 3: Determine Actual Revenue and Profit

The vendor receives the selling price based on the *claimed* weight ($W$), not the actual weight supplied.

• Actual Revenue = Claimed SP = $0.5W$.
• Actual Profit = Actual Revenue - Actual CP
• Actual Profit = $0.5W - 0.35W = 0.15W$.

Step 4: Calculate the Actual Profit Percentage

The profit percentage is calculated based on the *actual cost* incurred by the vendor.

• Actual Profit Percentage = $\frac{\text{Actual Profit}}{\text{Actual CP}} \times 100$
• Actual Profit Percentage = $\frac{0.15W}{0.35W} \times 100$
• Actual Profit Percentage = $\frac{0.15}{0.35} \times 100 = \frac{15}{35} \times 100 = \frac{3}{7} \times 100$
• Actual Profit Percentage $\approx 0.42857 \times 100 = 42.857\%$

Step 5: Round the Result

Rounding the calculated profit percentage to two decimal places:

• Actual Profit Percentage $\approx 42.86\%$