Understanding the Dishonest Merchant Problem

This question involves a common type of profit and loss problem where a dishonest merchant manipulates both the selling price and the weight used. The merchant claims to sell at a loss to appear fair, but recovers this (and more) by using a false weight.

Analyzing the Merchant's Claim and Action

Let's break down the information given:

• Claimed Action: Sells goods at a 12.5% loss on the cost price (CP).
• Actual Action: Uses 28 g weight instead of 36 g.

This means that for every 36 grams of goods the customer intends to buy and pays for, they actually receive only 28 grams. The merchant effectively sells 28 grams but charges a price based on 36 grams, calculated with a claimed loss.

Calculating the Actual Profit or Loss Percentage

To find the actual profit or loss, we need to compare the cost price of the goods actually delivered with the selling price received for those goods.

Let's assume the cost price per gram of the goods is \(\text{Re } 1\).

If the cost price per gram is \(\text{Re } 1\), then:

• The cost price of 36 g is \(36 \times 1 = \text{Re } 36\).
• The cost price of 28 g (the amount actually sold) is \(28 \times 1 = \text{Re } 28\).

The merchant claims to sell at a 12.5% loss on the cost price of 36 g. The selling price (SP) is calculated based on the cost of 36 g:

Claimed SP for 36 g = CP of 36 g \(\times (1 - \text{Loss Percentage})\)

Claimed SP for 36 g = \(36 \times (1 - 0.125)\)

Claimed SP for 36 g = \(36 \times 0.875\)

Claimed SP for 36 g = \(31.5\)

So, the merchant charges \(\text{Re } 31.5\) for what is claimed to be 36 g.

However, the merchant only provides 28 g of goods for this price. The actual cost price for the merchant for the goods delivered is the cost of 28 g, which is \(\text{Re } 28\).

Now we can compare the actual cost price and the actual selling price for the quantity of goods delivered:

• Actual Cost Price (for 28 g) = \(\text{Re } 28\)
• Actual Selling Price (received for 28 g) = \(\text{Re } 31.5\)

Since the actual selling price is greater than the actual cost price, there is a profit.

Actual Profit = Actual Selling Price - Actual Cost Price

Actual Profit = \(31.5 - 28 = \text{Re } 3.5\)

Now, we calculate the profit percentage on the actual cost price:

Profit Percentage = \(\frac{\text{Actual Profit}}{\text{Actual Cost Price}} \times 100\%\)

Profit Percentage = \(\frac{3.5}{28} \times 100\%\)

Profit Percentage = \(\frac{3.5 \times 10}{28 \times 10} \times 100\%\)

Profit Percentage = \(\frac{35}{280} \times 100\%\)

Profit Percentage = \(\frac{1}{8} \times 100\%\)

Profit Percentage = \(12.5\%\)

Since the result is positive, it represents a gain (profit).

Summary of Calculation

The merchant makes a profit of 12.5% despite claiming a loss.

Revision Table: Profit and Loss Concepts

Additional Information: Dishonest Weighing

Problems involving dishonest merchants using false weights are common. The key is to compare the cost price of the actual quantity sold with the selling price received for that quantity (which is based on the declared or false weight).

A common shortcut formula for a merchant who sells at cost price but uses a false weight (less than the true weight) is:

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

In this problem, the merchant also claims a loss. So, the method used above (comparing actual CP vs. actual SP) is more general and reliable when both factors (false weight and claimed profit/loss) are involved.

Let the cost price per unit be \(C\). The merchant sells quantity \(Q_{actual}\) but charges for quantity \(Q_{false}\). Let the claimed loss percentage be \(L\%\).

• Cost of quantity actually sold (\(Q_{actual}\)) = \(C \times Q_{actual}\)
• Selling Price received (charged for \(Q_{false}\) at \(L\%\) loss on its cost) = \((C \times Q_{false}) \times (1 - L/100)\)

Actual Profit/Loss = Selling Price - Cost of actual quantity

Actual Profit/Loss = \((C \times Q_{false}) \times (1 - L/100) - (C \times Q_{actual})\)

Actual Profit/Loss % = \(\frac{(C \times Q_{false}) \times (1 - L/100) - (C \times Q_{actual})}{C \times Q_{actual}} \times 100\%\)

Actual Profit/Loss % = \(\left(\frac{Q_{false}}{Q_{actual}} \times (1 - L/100) - 1\right) \times 100\%\)

In our case, \(Q_{false} = 36\), \(Q_{actual} = 28\), \(L = 12.5\). \(1 - L/100 = 1 - 12.5/100 = 1 - 0.125 = 0.875\).

Actual Profit/Loss % = \(\left(\frac{36}{28} \times 0.875 - 1\right) \times 100\%\)

Actual Profit/Loss % = \(\left(\frac{9}{7} \times \frac{7}{8} - 1\right) \times 100\%\)

Actual Profit/Loss % = \(\left(\frac{9}{8} - 1\right) \times 100\%\)

Actual Profit/Loss % = \(\left(\frac{9-8}{8}\right) \times 100\%\)

Actual Profit/Loss % = \(\frac{1}{8} \times 100\%\)

Actual Profit/Loss % = \(12.5\%\)

Since the result is positive, it is a profit or gain of 12.5%.