Understanding the Profit and Loss Problem

This question involves calculating the selling price of an item under different profit and loss scenarios. We are given information about a sale at a loss, and how the selling price changes to achieve a profit. We need to find the cost price first, and then calculate the selling price required for a specific profit percentage.

Key Concepts in Profit and Loss

• Cost Price (CP): The original price at which an item is bought.
• Selling Price (SP): The price at which an item is sold.
• Profit: When SP > CP. Profit = SP - CP.
• Loss: When SP < CP. Loss = CP - SP.
• Profit Percentage: \( \frac{\text{Profit}}{\text{CP}} \times 100\% \)
• Loss Percentage: \( \frac{\text{Loss}}{\text{CP}} \times 100\% \)

Step-by-Step Solution for Calculating Selling Price

Let's denote the Cost Price (CP) of the item by \( C \). The selling price is calculated based on this cost price and the profit or loss percentage.

Scenario 1: Item Sold at a Loss of 13%

When the item is sold at a loss of 13%, the selling price (SP1) is 13% less than the cost price.

SP1 = \( C - 13\% \text{ of } C \)

SP1 = \( C - 0.13C \)

SP1 = \( 0.87C \)

Scenario 2: Item Sold for Rs. 49 More, Resulting in 22% Profit

If the item had been sold for Rs. 49 more than SP1, the new selling price (SP2) would have resulted in a profit of 22%. The selling price with a 22% profit is 22% more than the cost price.

SP2 = \( C + 22\% \text{ of } C \)

SP2 = \( C + 0.22C \)

SP2 = \( 1.22C \)

According to the problem, SP2 is Rs. 49 more than SP1.

SP2 = SP1 + 49

Substituting the expressions for SP1 and SP2 in terms of \( C \):

\( 1.22C = 0.87C + 49 \)

Calculating the Cost Price (CP)

Now, we can solve this equation to find the value of \( C \).

\( 1.22C - 0.87C = 49 \)

\( (1.22 - 0.87)C = 49 \)

\( 0.35C = 49 \)

\( C = \frac{49}{0.35} \)

To simplify the division, we can write 0.35 as \( \frac{35}{100} \):

\( C = \frac{49}{\frac{35}{100}} = \frac{49 \times 100}{35} \)

Now, simplify the fraction:

\( C = \frac{49}{35} \times 100 = \frac{7 \times 7}{5 \times 7} \times 100 = \frac{7}{5} \times 100 \)

\( C = 7 \times \frac{100}{5} = 7 \times 20 \)

\( C = 140 \)

So, the Cost Price of the item is Rs. 140.

Calculating Selling Price for 30% Profit

We need to find the selling price required to gain a profit of 30%. This means the selling price (SP3) should be 30% more than the cost price.

SP3 = \( C + 30\% \text{ of } C \)

SP3 = \( C + 0.30C \)

SP3 = \( 1.30C \)

Substitute the calculated value of \( C = 140 \):

SP3 = \( 1.30 \times 140 \)

SP3 = \( 1.3 \times 140 \)

SP3 = \( 182 \)

Therefore, the selling price of the item should be Rs. 182 to gain a profit of 30%.

Summary of Calculations

The selling price required to achieve a 30% profit is Rs. 182.

Revision Table: Profit and Loss Basics

Additional Information on Profit and Loss Calculations

Understanding the relationship between CP, SP, and percentage profit/loss is fundamental. We can express SP directly using percentages:

• If there is a profit of \( P\% \), then \( \text{SP} = \text{CP} \times (1 + \frac{P}{100}) \).
• If there is a loss of \( L\% \), then \( \text{SP} = \text{CP} \times (1 - \frac{L}{100}) \).

In this problem, the difference in selling prices corresponds to the combined percentage change from -13% to +22%, which is \( 22\% - (-13\%) = 35\% \). This 35% of the cost price is equal to the Rs. 49 difference in selling price. So, \( 35\% \text{ of } CP = 49 \), which leads to \( 0.35 \times CP = 49 \), and \( CP = \frac{49}{0.35} = 140 \).

Once the cost price is known, calculating the selling price for any desired profit percentage is straightforward using the formula SP = \( \text{CP} \times (1 + \frac{\text{Desired Profit}\%}{100}) \). For a 30% profit, SP = \( 140 \times (1 + \frac{30}{100}) = 140 \times 1.30 = 182 \).