Question

A dishonest shopkeeper professes to sell his goods at cost price but uses a false weight of 920 grams for a 1 kg weight. What is his overall profit percentage?

• A. 8%
• B. 8.69%
• C. 7.41%
• D. 9.2%

Hint:

For dishonest dealing with false weights, remember the trick:
Profit % = $\frac{\text{Error}}{\text{Weight Used} - \text{Error}} \times 100$.
Error = $1000 - 920 = 80$ grams.
Weight Used = 920 grams.
Profit % = $\frac{80}{920} \times 100 = 8.69\%$.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem describes a dishonest shopkeeper who claims to sell goods at the cost price (CP) but uses a false weight, effectively selling less quantity than he charges for. We need to calculate his actual profit percentage.

Step 2: Key Formula or Approach:
For such problems, it's easiest to assume a cost price per unit weight (e.g., per gram).
Let the cost price of 1 gram of goods be Re. 1.
1. Calculate the cost price of the quantity the shopkeeper actually sells.
2. Calculate the selling price for the quantity the shopkeeper claims to sell.
3. Calculate the profit.
4. Calculate the profit percentage based on the actual cost price.
Alternatively, a direct formula can be used:
\[ \text{Profit \%} = \left( \frac{\text{Error}}{\text{True Value} - \text{Error}} \right) \times 100 \]
Here, Error = (True Weight - False Weight) and True Value = True Weight (or 1000 grams).

Step 3: Detailed Explanation:
1. True Weight and False Weight:
True weight (what he should give) = 1 kg = 1000 grams.
False weight (what he actually gives) = 920 grams.
2. Assumed Cost Price:
Let the cost price (CP) of 1 gram of goods be Re. 1.
3. Shopkeeper's Actual Cost (for quantity sold):
The shopkeeper actually sells 920 grams of goods.
So, his actual cost price for this quantity is Rs. $920 \times 1 = \text{Rs. } 920$.
4. Shopkeeper's Selling Price (for quantity claimed):
He professes to sell at the cost price of 1 kg (1000 grams).
So, he charges the customer for 1000 grams at Re. 1/gram, which is Rs. $1000 \times 1 = \text{Rs. } 1000$.
5. Calculate Profit:
Profit = Selling Price - Actual Cost Price
Profit = Rs. $1000 - \text{Rs. } 920 = \text{Rs. } 80$.
6. Calculate Profit Percentage:
Profit % = $\left( \frac{\text{Profit}}{\text{Actual Cost Price}} \right) \times 100$
Profit % = $\left( \frac{80}{920} \right) \times 100$
Profit % = $\left( \frac{8}{92} \right) \times 100 = \left( \frac{2}{23} \right) \times 100$
Profit % = $\frac{200}{23} \approx 8.6956... \%$
Rounding to two decimal places, Profit % $\approx$ 8.69%.

Step 4: Final Answer:
His overall profit percentage is 8.69%.

Answer 2

A different way to look at this is to imagine the shopkeeper's actual physical stock being handed out in false "kilograms," and to see how many extra false-kilograms he can squeeze out of his real stock, checking this against every option.

1. 8%: This figure would come from taking the shortfall of 80 g directly over the true weight of 1000 g, i.e. \( \frac{80}{1000}\times 100 = 8\% \), but the profit must be measured against what the shopkeeper actually spent (on 920 g), not against the full 1000 g, so this understates the real profit.
2. 8.69%: Suppose the shopkeeper has 1000 g of true stock on hand, bought at Re. 1/g. Using his false 920 g weight as one unit of "1 kg," the number of such false units he can dispense from his 1000 g of real stock is \( \frac{1000}{920} \approx 1.08696 \). Since he charges the cost-price rate of Rs. 1000 for every one of these false "kg" units, his total revenue from the 1000 g stock is \( 1.08696 \times 1000 \approx 1086.96 \). His cost was Rs. 1000, so profit \( \approx 86.96 \), and profit percentage \( = \frac{86.96}{1000}\times 100 \approx 8.69\% \). This matches.
3. 7.41%: This would result from dividing the profit of Rs. 80 by an incorrect base such as Rs. 1080 (i.e. \( \frac{80}{1080}\times100 \approx 7.41\% \)), which mixes up the selling price with the wrong reference amount instead of the true cost basis of Rs. 920.
4. 9.2%: This is simply the raw shortfall percentage \( \frac{1000-920}{1000}\times 100 = 8\% \) computed differently, or a similar surface-level percentage of the weights themselves, and does not represent the shopkeeper's true profit margin on what he actually spent.

Working from the actual quantity of stock and how many false units it yields confirms that the true profit percentage is 8.69%.

Therefore, the correct answer is 8.69%.