Shopkeeper's Total Gain Calculation with Tampered Weights

This problem asks us to calculate the total percentage gain achieved by a shopkeeper who uses tampered weights, cheating by a certain percentage during both the buying and selling processes for fruits.

Understanding the Cheat Mechanism

Cheating while Buying

When the shopkeeper buys fruits, he uses tampered weights to receive more quantity than what he pays for. The cheat is stated as 11%. This means if he pays for 100 units of weight, he actually receives 11% extra weight. The actual weight received is \(100 + 11\%\) of \(100 = 100 + 11 = 111\) units.

Effectively, he gets 111 units of fruit for the price he should have paid for 100 units. This gives him a gain on the quantity of goods acquired relative to his cost.

Cheating while Selling

When the shopkeeper sells fruits, he uses tampered weights to give less quantity than what he charges for. The cheat is again 11%. This means if he charges a customer for 100 units of weight, he actually gives 11% less weight. The actual weight given is \(100 - 11\%\) of \(100 = 100 - 11 = 89\) units.

Effectively, he charges the price of 100 units for only 89 units of fruit. This gives him a gain on the revenue generated relative to the quantity of goods sold.

Calculating the Total Percentage Gain

Let the percentage by which the shopkeeper cheats be \(x\). In this problem, \(x = 11\%\). He cheats by getting \(x\%\) extra while buying and giving \(x\%\) less while selling.

Consider a scenario where the shopkeeper intends to buy a quantity of fruit that would honestly weigh 100 units. He pays the price for 100 units.

• Due to cheating while buying, he receives \(100 + x\) units of actual weight for the price of 100 units. So, the cost incurred is for 100 units, but he possesses \(100 + x\) units of fruit.
• Now, he sells these \(100 + x\) units. When selling, he gives \(100 - x\) actual units but charges the price of 100 units. This means for every \(100 - x\) actual units he sells, he gets the revenue equivalent to 100 units.

The total actual weight he has to sell is \(100 + x\). To find the total revenue, we need to determine how many '100-unit price bundles' he can sell from this quantity. The number of such bundles is the total actual weight divided by the actual weight in one such bundle: \(\frac{100 + x}{100 - x}\).

Each bundle is charged at the price of 100 units. Let the standard price per unit be \(P\). The standard price for 100 units is \(100P\). Assuming the standard buying price equals the standard selling price for simplicity in calculating percentage gain relative to the initial cost paid.

• Cost paid = Price for 100 units = \(100P\).
• Actual quantity received = \(100 + x\) units.
• Quantity charged for while selling \(100+x\) units = \(\frac{100 + x}{100 - x} \times 100\) units.
• Total Revenue = Price for the quantity charged for = \(P \times \left(\frac{100 + x}{100 - x} \times 100\right)\).

Assuming \(P=1\) for simplicity (the percentage gain is independent of the base price):

• Cost = \(100\).
• Revenue = \(\frac{100 + x}{100 - x} \times 100\).

Total Gain = Revenue - Cost = \(\frac{100 + x}{100 - x} \times 100 - 100\)

Total Gain = \(100 \left( \frac{100 + x}{100 - x} - 1 \right) = 100 \left( \frac{100 + x - (100 - x)}{100 - x} \right) = 100 \left( \frac{100 + x - 100 + x}{100 - x} \right)\)

Total Gain = \(100 \left( \frac{2x}{100 - x} \right)\)

Percentage Gain = \(\frac{\text{Total Gain}}{\text{Cost}} \times 100 = \frac{100 \left( \frac{2x}{100 - x} \right)}{100} \times 100 = \frac{2x}{100 - x} \times 100\)

Now, substitute \(x = 11\):

\(\text{Total Gain Percentage} = \frac{2 \times 11}{100 - 11} \times 100\)

\(\text{Total Gain Percentage} = \frac{22}{89} \times 100\)

\(\text{Total Gain Percentage} = \frac{2200}{89}\)

Performing the calculation:

\(\frac{2200}{89} \approx 24.7191\)

Rounding to two decimal places, the total gain percentage is approximately 24.71%.

Final Answer

The calculated total percentage gain for the shopkeeper is approximately 24.71%.

Revision Table: Cheating Weights and Gain

Additional Information: Related Concepts in Profit and Loss

• Dishonest Dealer Problems: Questions involving tampered weights or false measuring tapes fall under this category in competitive exams.
• Profit Calculation Basis: Profit percentage is typically calculated on the cost price. In dishonest dealer problems, identifying the true cost and true revenue is crucial.
• Weight vs. Value: Cheating on weight directly impacts the actual quantity of goods, thereby affecting their true cost or value and the actual revenue generated.
• Successive Percentage Change: While the cheating happens in two steps (buying and selling), the effect is not a simple successive percentage change on the same base. It's a gain achieved on the cost price due to manipulating the quantity of goods.