Calculating Actual Profit Percentage with False Weights

This problem involves a shopkeeper who uses a faulty weight and also sells goods at a stated profit margin. We need to determine the true profit percentage the shopkeeper makes by combining these two factors.

Here's how we can break down the problem:

• The shopkeeper intends to sell 1 kg (1000 gm) but actually sells only 940 gm. This is a deception related to the quantity.
• The shopkeeper claims to sell at a 4% profit on the price he expects for 1 kg. This is the stated, or nominal, profit.
• The actual profit is based on the difference between the actual cost of the goods sold and the price received for that quantity.

Step-by-Step Profit Calculation

Let's assume a base price to make the calculation clear. Suppose the shopkeeper buys goods at a cost price (CP) of Rs. 1 per gram.

1. Calculate the Actual Cost of Goods Sold:
   The shopkeeper sells 940 gm of goods.
   The actual cost price for 940 gm is \( 940 \text{ gm} \times \text{Rs. } 1/\text{gm} = \text{Rs. } 940 \).
   So, the actual CP for the quantity sold is Rs. 940.

2. Calculate the Selling Price Charged:
   The shopkeeper claims to sell based on the weight of 1 kg (1000 gm).
   The cost price for 1 kg would be \( 1000 \text{ gm} \times \text{Rs. } 1/\text{gm} = \text{Rs. } 1000 \).
   He sells at a nominal profit of 4% on this price.
   Nominal Selling Price (SP) for 1 kg = CP of 1 kg + 4% of CP of 1 kg
   Nominal SP for 1 kg = \( \text{Rs. } 1000 + \frac{4}{100} \times \text{Rs. } 1000 \)
   Nominal SP for 1 kg = \( \text{Rs. } 1000 + \text{Rs. } 40 = \text{Rs. } 1040 \).
   This Rs. 1040 is the price he charges the customer for the 940 gm he actually provides.
   So, the actual SP for the quantity sold is Rs. 1040.

3. Calculate the Actual Profit:
   Actual Profit = Actual Selling Price - Actual Cost Price
   Actual Profit = Rs. 1040 - Rs. 940 = Rs. 100.

4. Calculate the Actual Profit Percentage:
   The actual profit percentage is calculated on the actual cost price of the goods sold.
   Actual Profit Percentage \( = \frac{\text{Actual Profit}}{\text{Actual Cost Price}} \times 100\% \)
   Actual Profit Percentage \( = \frac{\text{Rs. } 100}{\text{Rs. } 940} \times 100\% \)
   Actual Profit Percentage \( = \frac{100}{940} \times 100\% \)
   Actual Profit Percentage \( = \frac{1000}{94}\% \)
   Actual Profit Percentage \( = \frac{500}{47}\% \)

5. Convert the Fraction to Decimal and Round Off:
   Now, we calculate the decimal value of \( \frac{500}{47} \):
   \( \frac{500}{47} \approx 10.63829... \)
   Rounding this to two decimal places gives 10.64%.

So, despite claiming a 4% profit, the shopkeeper makes an actual profit of approximately 10.64% due to using a false weight.

Revision Table: Key Terms

Understanding the key terms helps in solving problems involving profit and loss, especially with false weights.

• Cost Price (CP): The price at which an article is purchased.
• Selling Price (SP): The price at which an article is sold.
• Nominal Profit: The profit calculated based on the declared or intended weight and price, before considering any deceit like false weights.
• Actual Profit: The true profit calculated based on the actual quantity of goods sold and their actual cost.
• False Weight: Using a weight less than the declared weight while selling goods, increasing the actual profit margin.

Additional Information on False Weight Problems

Problems involving false weights are common in profit and loss calculations. The core idea is that the shopkeeper is selling less quantity than what is being paid for. The profit comes from two sources:

• The declared profit margin on the nominal weight.
• The extra amount gained by selling a smaller quantity at the price of a larger one.

The crucial step is to correctly identify the actual cost of the goods that are physically handed over to the customer and the price received for them. The actual profit percentage is always calculated on the actual cost price of the quantity sold.

A quick formula for profit percentage in case of selling at cost price but using false weight:

\( \text{Profit % } = \frac{\text{(True Weight - False Weight)}}{\text{False Weight}} \times 100\% \)

In our problem, the shopkeeper also adds a 4% profit. Let's say CP of 1gm is Rs. 1. He sells 940gm but charges for 1000gm at 4% profit on cost of 1000gm. Actual CP of 940gm = Rs. 940. SP = 1000 * 1.04 = Rs. 1040. Profit = 1040 - 940 = Rs. 100. Profit % = \( \frac{100}{940} \times 100 = \frac{1000}{94} = \frac{500}{47} \approx 10.64\% \). This confirms the method used above which accounts for both false weight and nominal profit.