Understanding the Dishonest Merchant Problem

This problem deals with a common type of profit and loss question involving a dishonest merchant who uses false weights. The merchant claims to sell goods at the cost price (CP), meaning they should ideally make no profit. However, by using a weight less than the standard weight (900g instead of 1000g for 1kg), they effectively sell a smaller quantity of goods while charging for a larger quantity. The profit comes from the difference in the actual quantity sold and the quantity they charge for.

Calculating the Gain with Faulty Weight

Let's break down how to calculate the gain percentage in this scenario. We need to determine the cost of the goods actually sold and the price at which they are sold.

• The merchant pretends to sell 1 kg, which is equal to 1000 g.
• They claim to sell at CP. Let's assume the Cost Price (CP) of 1 g of goods is Re 1.
• So, the CP of 1000 g would be 1000 × Re 1 = Rs 1000.
• Since they claim to sell at CP, the Selling Price (SP) they charge the customer for the transaction (which is supposed to be for 1000g) is Rs 1000.

However, the merchant does not actually give 1000 g of goods. They use a faulty weight of 900 g.

• The actual quantity of goods sold is 900 g.
• The Cost Price (CP) of these 900 g (at our assumed rate of Re 1 per g) is 900 × Re 1 = Rs 900.
• So, the merchant spends Rs 900 to acquire the goods they actually sell.
• They sell these 900 g for the price of what 1000 g should cost, which is Rs 1000 (their claimed SP).

Now we can calculate the profit made by the merchant on this transaction:

• Actual Cost Price of goods sold = Rs 900
• Selling Price of goods sold = Rs 1000
• Profit = Selling Price - Cost Price
• Profit = Rs 1000 - Rs 900 = Rs 100

The profit of Rs 100 is made on the actual cost of the goods sold, which is Rs 900. To find the gain percentage, we use the formula:

Gain % = $\backslash left(\; \backslash frac\{\backslash text\{Profit\}\}\{\backslash text\{Actual\; CP\; of\; goods\; sold\}\}\; \backslash right)\; \backslash times\; 100$

Substituting the values:

Gain % = $\backslash left(\; \backslash frac\{100\}\{900\}\; \backslash right)\; \backslash times\; 100$

Gain % = $\backslash left(\; \backslash frac\{1\}\{9\}\; \backslash right)\; \backslash times\; 100$

Gain % = $\backslash frac\{100\}\{9\}$ %

Now, we convert the fraction to a decimal or mixed number:

$\backslash frac\{100\}\{9\}\; =\; 11\; \backslash frac\{1\}\{9\}$

As a decimal, $\backslash frac\{1\}\{9\}\; \backslash approx\; 0.1111...$

So, the gain percentage is approximately 11.11%.

Summary of Calculation

The profit percentage for a dishonest merchant using a faulty weight can be calculated using the formula based on the error in weight:

Gain % = $\backslash left(\; \backslash frac\{\backslash text\{Error\; in\; weight\}\}\{\backslash text\{Weight\; used\}\}\; \backslash right)\; \backslash times\; 100$

In this case:

• Standard Weight (what should be given) = 1 kg = 1000 g
• Weight Used (what is actually given) = 900 g
• Error in weight = Standard Weight - Weight Used = 1000 g - 900 g = 100 g

Gain % = $\backslash left(\; \backslash frac\{100\; \backslash text\{\; g\}\}\{900\; \backslash text\{\; g\}\}\; \backslash right)\; \backslash times\; 100$

Gain % = $\backslash left(\; \backslash frac\{1\}\{9\}\; \backslash right)\; \backslash times\; 100$

Gain % = $\backslash frac\{100\}\{9\}$ % $\backslash approx\; 11.11$%

Comparing this result with the given options, 11.11% is the correct gain percentage.

Revision Table: Key Concepts

Additional Information: Dishonest Dealing Variations

Dishonest merchant problems can have variations. Understanding the core concept helps solve them:

• Selling at a loss on CP but using false weights: The merchant claims a loss percentage on CP, but the gain from using a false weight might offset this loss or still result in an overall gain. You calculate the effective SP based on the claimed loss, then find the actual CP of the goods sold using the false weight, and finally calculate the net profit/loss.
• Marking up price and using false weights: The merchant increases the price above CP and also uses a false weight. The total gain is a combination of the profit from the markup and the profit from the false weight.
• Error in weighing both while buying and selling: The merchant might use a false weight to buy more than paid for (while buying) and use another false weight to sell less than charged for (while selling), compounding the profit.

In all cases, the key is to determine the actual cost price of the goods that change hands and the actual selling price received for that exact quantity of goods.