Understanding Dishonest Shopkeeper Problems and Profit

Dishonest shopkeeper problems are common in profit and loss calculations. They involve a seller who uses faulty weights or measures to cheat customers, thereby making extra profit. In this specific problem, a shopkeeper uses a faulty weight instead of the standard 1 kg weight. This means they sell less than 1 kg of goods but charge the price of 1 kg. The profit comes from the value of the weight that was withheld.

Calculating Profit Percentage with Faulty Weight

Let's break down how the profit is calculated in this scenario. The shopkeeper's "cost price" is effectively the value of the actual weight of goods they give to the customer. The "selling price" is the value they receive, which is equivalent to the price of the weight they claim to sell (1 kg in this case).

• Actual weight sold = \( W \) grams (This is the faulty weight).
• Claimed weight (what the customer pays for) = 1 kg = 1000 grams.
• Profit is made because the shopkeeper gives \( W \) grams but gets paid for 1000 grams.
• The profit percentage is given as 25%. This profit is based on the cost, which is the value of the \( W \) grams actually sold.

We can set up a relationship between the cost price (CP) and selling price (SP) in terms of weight.

Let the cost per gram be \( c \).

• Cost Price (CP) for the shopkeeper = Cost of actual weight sold = \( W \times c \)
• Selling Price (SP) for the shopkeeper = Price of claimed weight = Cost of 1000 grams = \( 1000 \times c \)
• Profit = SP - CP = \( (1000 \times c) - (W \times c) = (1000 - W) \times c \)
• Profit Percentage = \( \frac{\text{Profit}}{\text{CP}} \times 100 \)
• Given Profit Percentage = 25%

So, we have the equation:

\( 25 = \frac{(1000 - W) \times c}{W \times c} \times 100 \)

Notice that \( c \) cancels out, which means the actual cost per gram doesn't matter, only the weights do.

\( 25 = \frac{1000 - W}{W} \times 100 \)

Step-by-Step Calculation to Find the Faulty Weight

Let's solve the equation for \( W \), the faulty weight.

1. Divide both sides by 100:

   \( \frac{25}{100} = \frac{1000 - W}{W} \)

   \( 0.25 = \frac{1000 - W}{W} \)

2. Multiply both sides by \( W \):

   \( 0.25 \times W = 1000 - W \)

3. Add \( W \) to both sides:

   \( 0.25W + W = 1000 \)

   \( 1.25W = 1000 \)

4. Divide both sides by 1.25:

   \( W = \frac{1000}{1.25} \)

   \( W = \frac{1000}{\frac{5}{4}} \)

   \( W = 1000 \times \frac{4}{5} \)

   \( W = \frac{4000}{5} \)

   \( W = 800 \)

The faulty weight used by the dishonest shopkeeper is 800 grams. This means the shopkeeper gives 800 grams of product but charges the price of 1000 grams, making a 25% profit on the actual quantity sold (800 grams).

Let's quickly check the profit:

• Cost (value of 800g) = \( 800c \)
• Selling Price (value of 1000g) = \( 1000c \)
• Profit = \( 1000c - 800c = 200c \)
• Profit Percentage = \( \frac{\text{Profit}}{\text{CP}} \times 100 = \frac{200c}{800c} \times 100 = \frac{1}{4} \times 100 = 25\% \)

This matches the given profit percentage.

Comparing with Options

Let's look at the given options for the faulty weight:

• 250 grams
• 800 grams
• 640 grams
• 980 grams

Our calculated faulty weight is 800 grams, which matches one of the options.

The table confirms that using a faulty weight of 800 grams results in a 25% profit.

Revision Table: Key Concepts in Dishonest Dealing

Additional Information on Dishonest Weight Profit

Problems involving dishonest weights can be approached using a simple formula derived from the cost/selling price logic.

If a shopkeeper sells goods using a faulty weight instead of a true weight, and the error is the difference between the true weight and the faulty weight, the profit percentage is given by:

\( \text{Profit}\% = \frac{\text{Error}}{\text{Faulty Weight}} \times 100 \)

Where:

• Error = True Weight - Faulty Weight
• Faulty Weight = True Weight - Error

In our problem:

• True Weight = 1000 grams
• Let Faulty Weight = \( W \) grams
• Error = \( 1000 - W \) grams
• Profit Percentage = 25%

Using the formula:

\( 25 = \frac{1000 - W}{W} \times 100 \)

This is the same equation we solved earlier, leading to \( W = 800 \) grams.

This formula is a shortcut based on the understanding that the profit (Error) is a percentage of the cost (Faulty Weight).