Calculating Shopkeeper Profit with False Weights

This problem involves a shopkeeper who uses a fraudulent weight. The shopkeeper claims to sell goods at the cost price, but makes a profit by giving less quantity than what is paid for. This is a common type of problem involving dishonest dealings and percentage calculations.

Understanding the False Weight Concept

When a shopkeeper uses a weight of 460 grams instead of the promised 500 grams, they are effectively selling 460 grams but charging the price of 500 grams. The profit comes from this difference in weight.

• The customer pays for 500 grams.
• The customer receives only 460 grams.

The actual item sold is 460 grams, and the profit is the value of the weight difference.

Calculating the Profit in Weight

The difference between the weight the customer pays for and the weight they receive is the profit in terms of weight.

Difference in weight = Advertised weight - Actual weight given

Difference in weight = \(500 \text{ grams} - 460 \text{ grams} = 40 \text{ grams}\)

So, for every 460 grams sold, the shopkeeper makes a profit equivalent to the cost of 40 grams.

Calculating the Profit Percentage

The profit percentage is calculated based on the actual quantity sold, which is 460 grams in this case, not the advertised 500 grams. The formula for profit percentage is:

Profit Percentage = \(\left( \frac{\text{Profit}}{\text{Actual Quantity Sold}} \right) \times 100\%\)

Here, the profit is equivalent to 40 grams, and the actual quantity sold is 460 grams.

Profit Percentage = \(\left( \frac{40 \text{ grams}}{460 \text{ grams}} \right) \times 100\%\)

Now, we simplify the fraction \(\frac{40}{460}\):

\(\frac{40}{460} = \frac{4}{46} = \frac{2}{23}\)

So, the profit percentage is:

Profit Percentage = \(\left( \frac{2}{23} \right) \times 100\%\)

Profit Percentage = \(\frac{200}{23}\%\)

Converting to a Mixed Fraction

To express \(\frac{200}{23}\%\) as a mixed fraction, we divide 200 by 23:

• \(200 \div 23\)
• \(23 \times 8 = 184\)
• Remainder = \(200 - 184 = 16\)

So, \(\frac{200}{23}\) can be written as \(8\) with a remainder of \(16\), or \(8 \frac{16}{23}\).

Therefore, the profit percentage is \(8 \frac{16}{23}\%\).

Summary of Calculation

The shopkeeper makes a profit equivalent to 40 grams for every 460 grams sold. The profit percentage is calculated on the actual weight given.

Profit Percentage = \(\frac{\text{Weight difference}}{\text{Weight given}} \times 100\%\)

Profit Percentage = \(\frac{500 - 460}{460} \times 100\%\)

Profit Percentage = \(\frac{40}{460} \times 100\%\)

Profit Percentage = \(\frac{2}{23} \times 100\%\)

Profit Percentage = \(\frac{200}{23}\%\)

Profit Percentage = \(8 \frac{16}{23}\%\)

Revision Table: Key Concepts in False Weight Problems

Additional Information: False Weight Scenarios

False weight problems are a classic example of profit and loss calculations where the profit is hidden. Here are a few points to remember:

• If the shopkeeper sells at cost price using a false weight, the profit percentage is always calculated on the actual weight delivered.
• If the shopkeeper also marks up the price, the calculation becomes a bit more complex, combining the profit from the price markup and the profit from the false weight.
• Always identify the 'cost' amount (what the shopkeeper effectively spends or the actual quantity given) and the 'selling' amount (what the shopkeeper charges for or the advertised quantity).
• In this problem, the shopkeeper 'sells' 460g but gets paid for 500g. The 'cost' is the value of 460g, and the 'selling price' is the value of 500g. The profit is the value of 40g. Profit % = (Profit / Cost) * 100 = (Value of 40g / Value of 460g) * 100 = (40/460) * 100.