Calculating Shopkeeper's Total Gain with Tampered Weights

This problem involves a shopkeeper who uses tampered weights to cheat during both the buying and selling processes. We need to calculate the total percentage gain the shopkeeper makes due to this dishonesty.

Understanding the Cheating Mechanism

When a shopkeeper cheats using tampered weights, it means they manipulate the quantity of goods involved relative to the price paid or charged. Let's analyze the cheating in buying and selling separately, assuming the cheating is by 9%.

1. Cheating while Buying: The shopkeeper pays for a certain weight but receives more than that weight. Cheating by 9% while buying means that for the price of 100 units of weight, the shopkeeper actually receives 100 + 9 = 109 units of weight. Essentially, they get 109 units for the cost of 100 units.
2. Cheating while Selling: The shopkeeper charges for a certain weight but delivers less than that weight. Cheating by 9% while selling means that for the price of 100 units of weight, the shopkeeper only delivers 100 - 9 = 91 units of weight. Essentially, they sell 91 units for the price of 100 units.

Calculating the Total Gain Percentage

Let's assume the true price of 1 unit of fruit is Rs. 1. We will track the cost and selling price for a specific quantity of fruit.

Consider the purchase where the shopkeeper pays for 100 units.

• Cost Price (CP): While buying, the shopkeeper pays the price of 100 units. If the true price of 1 unit is Re 1, the cost incurred is Rs. 100.

• Quantity Received: Due to cheating by 9% while buying, the shopkeeper receives 100 + 9 = 109 units of fruit for this cost.

Now, the shopkeeper sells the 109 units of fruit they received.

• Selling Price (SP): While selling, the shopkeeper charges the price of 100 units for every 91 units delivered. This means the selling price per unit is \(\frac{\text{Price of 100 units}}{\text{Quantity delivered (91 units)}} = \frac{\text{Rs. 100}}{91 \text{ units}}\). The selling price per unit is Rs. \(\frac{100}{91}\).

• The shopkeeper sells 109 units. The total selling price for 109 units will be the quantity multiplied by the selling price per unit: \(\text{SP} = 109 \times \text{Rs. } \frac{100}{91} = \text{Rs. } \frac{10900}{91}\).

Now we can calculate the total gain:

• Total Gain = Selling Price - Cost Price

  Total Gain = Rs. \(\frac{10900}{91}\) - Rs. 100

  Total Gain = Rs. \(\frac{10900 - 100 \times 91}{91}\)

  Total Gain = Rs. \(\frac{10900 - 9100}{91}\)

  Total Gain = Rs. \(\frac{1800}{91}\)

The gain percentage is calculated on the cost price:

• Gain Percentage = \(\left( \frac{\text{Total Gain}}{\text{Cost Price}} \right) \times 100\)

  Gain Percentage = \(\left( \frac{1800/91}{100} \right) \times 100\)

  Gain Percentage = \(\left( \frac{1800}{91 \times 100} \right) \times 100\)

  Gain Percentage = \(\left( \frac{1800}{9100} \right) \times 100\)

  Gain Percentage = \(\frac{18}{91} \times 100\)

Calculating the value:

\(\frac{18}{91} \times 100 \approx 0.197802 \times 100 \approx 19.78\)

The total gain in percentage is approximately 19.78%.

Verification with Options

Let's compare our calculated gain percentage with the given options:

Our calculated value of approximately 19.78% matches option C.

Conclusion

By cheating 9% during both buying (getting more quantity) and selling (giving less quantity), the shopkeeper makes a total gain of 19.78%.

Revision Table: Key Concepts

Additional Information: Percentage Gain with Faulty Weights

Problems involving faulty weights are common in competitive exams. The key is to correctly interpret how the cheating affects the cost and selling price relative to the actual quantity of goods.

• When a shopkeeper uses a weight that is \(x\)% less than the standard weight while selling (e.g., uses 910g instead of 1000g), the gain percentage on the transaction is given by \(\frac{x}{100-x} \times 100\). In our selling case, \(x=9\), so gain from selling is \(\frac{9}{100-9} \times 100 = \frac{9}{91} \times 100\).
• When a shopkeeper uses a weight that is \(x\)% less than the standard weight while buying (meaning they get more weight than marked, e.g., uses a 910g marked weight to measure something they should get 1000g of, thus getting more), or more commonly interpreted as getting \(x\)% extra quantity for the price, the gain is added to the quantity bought.
• In cases where cheating occurs at both buying and selling ends using weight manipulation, consider the quantity received for a fixed cost and the revenue generated from selling that quantity. The combined effect leads to a higher total gain than simply adding the individual percentage gains calculated on different bases. The method used in the solution, tracking the cost and revenue for the actual quantity of goods that passes through the shopkeeper's hands, is a reliable way to solve such problems.