Understanding the Profit Scenario

This problem involves a trader who uses two methods to increase profit: a faulty weighing balance and marking up the price. We need to calculate the overall net profit percentage achieved by the trader.

• The trader's weighing balance shows 1300 g when the actual weight is 1000 g (1 kg). This means for every 1300 g shown, he is only giving 1000 g of the product.
• Additionally, the trader marks up the cost price by 15%. This means he increases the price he sells at by 15% over his initial cost for the quantity he claims to sell.

Step-by-Step Profit Calculation

Let's assume a base value to make the calculation straightforward. We will assume the trader's cost price for 1000 grams (1 kg actual weight) is Rs. 100.

Step 1: Determine the Actual Quantity Given vs. Quantity Charged

The faulty balance shows 1300 g for 1000 g. When the trader sells something showing 1300 g on the scale, he is actually giving only 1000 g.

• Actual quantity given = 1000 g
• Quantity shown/charged for = 1300 g

This is equivalent to saying that for every 1 g shown, the actual weight is $1000/1300 = 10/13$ grams.

Step 2: Calculate the Marked Price

The trader marks up his cost price by 15%. Let's assume the cost price of 1000 g (actual) is Rs. 100.

The trader bases his marked price on the weight shown on the scale. He pretends his cost for 1000g shown is Rs 100. Then he marks it up.

• Assumed Cost Price (for 1000 g shown) = Rs. 100
• Mark-up percentage = 15%
• Marked Price (MP) for 1000 g shown = Cost Price + 15% of Cost Price
• MP = $100 + \left(\frac{15}{100} \times 100\right) = 100 + 15 = \text{Rs. } 115$

So, the marked price for 1000 g shown on the scale is Rs. 115.

Step 3: Determine the Selling Price for the Actual Quantity Sold

The trader sells 1000 g (actual weight) but charges the price for 1300 g *shown* on his scale. The price per gram shown is based on the marked price for 1000 g shown.

• Price per gram (based on marked price for shown weight) = $\frac{\text{Marked Price for 1000 g shown}}{\text{1000 g}} = \frac{115}{1000} = \text{Rs. } 0.115 \text{ per gram shown}$
• The trader sells 1000 g actual weight, but charges the price of 1300 g shown.
• Selling Price (SP) for the 1000 g actual weight = Price of 1300 g shown
• SP = $1300 \times (\text{Price per gram shown}) = 1300 \times 0.115 = \text{Rs. } 149.5$

The actual cost price for the 1000 g he sold is Rs. 100 (our initial assumption).

Step 4: Calculate the Net Profit

Net Profit = Selling Price (for actual quantity sold) - Actual Cost Price (of actual quantity sold)

• Net Profit = Rs. 149.5 - Rs. 100 = Rs. 49.5

Step 5: Calculate the Net Profit Percentage

Net Profit Percentage = $\left(\frac{\text{Net Profit}}{\text{Actual Cost Price}}\right) \times 100$

• Net Profit Percentage = $\left(\frac{49.5}{100}\right) \times 100 = 49.5\%$

The net profit percentage is 49.5%.

Summary Table of Profit Calculation

Revision Table: Key Concepts in Profit and Loss

Additional Information: Dealing with Faulty Weights

Problems involving faulty weighing balances are common in profit and loss. The key idea is to understand the relationship between the weight the trader claims to sell and the actual weight they provide.

If a trader uses a weight 'W_shown' instead of the actual weight 'W_actual' (where W_shown > W_actual), they are making a profit from the weighing itself. The gain from the faulty weight can be calculated as:

Gain % from faulty weight = $\left(\frac{\text{W}_{\text{shown}} - \text{W}_{\text{actual}}}{\text{W}_{\text{actual}}}\right) \times 100$

In this problem, the trader claims to sell 1300g for 1000g actual. The gain from weight alone, if there was no mark-up, would be:

Gain % = $\left(\frac{1300 - 1000}{1000}\right) \times 100 = \left(\frac{300}{1000}\right) \times 100 = 30\%$

However, the trader also applies a mark-up. When there is both a gain from faulty weight and a mark-up on price, these effects combine. A common way to think about it is that the trader gets the goods at a certain cost for the actual quantity but sells them at a price based on the charged quantity and the marked-up rate.

Let Actual CP per unit weight be $c$. Trader sells W_actual units, but charges price for W_shown units. Trader marks up price by M%. Selling Price per unit charged is $c \times (1 + M/100)$. Total SP for W_actual is $W_{\text{shown}} \times c \times (1 + M/100)$. Actual CP for W_actual is $W_{\text{actual}} \times c$. Profit % = $\left(\frac{W_{\text{shown}} \times c \times (1 + M/100) - W_{\text{actual}} \times c}{W_{\text{actual}} \times c}\right) \times 100$ Profit % = $\left(\frac{W_{\text{shown}} \times (1 + M/100) - W_{\text{actual}}}{W_{\text{actual}}}\right) \times 100$ Profit % = $\left(\frac{W_{\text{shown}}}{W_{\text{actual}}} \times (1 + M/100) - 1\right) \times 100$

Using values from the question:

W_shown = 1300, W_actual = 1000, M = 15

Profit % = $\left(\frac{1300}{1000} \times (1 + 15/100) - 1\right) \times 100$

Profit % = $\left(1.3 \times (1.15) - 1\right) \times 100$

Profit % = $(1.495 - 1) \times 100$

Profit % = $0.495 \times 100 = 49.5\%$

This formula confirms the step-by-step calculation.