Understanding Profit and Loss Calculations

This problem involves calculating the cost price of an article based on a known selling price and loss percentage, and then determining the new selling price required to achieve a specific profit percentage.

Key Concepts:

• Cost Price (CP): The original price at which the article was bought.
• Selling Price (SP): The price at which the article is sold.
• Profit: When SP > CP. Profit % is calculated on CP.
• Loss: When SP < CP. Loss % is calculated on CP.

Formulas Used:

• When there is a loss: $\text{SP} = \text{CP} \times \left( \frac{100 - \text{Loss}\%}{100} \right)$
• When there is a profit: $\text{SP} = \text{CP} \times \left( \frac{100 + \text{Profit}\%}{100} \right)$

Calculating the Cost Price (CP)

We are given that the article was sold for Rs. 355 with a loss of 29%.

Here, SP = Rs. 355 and Loss % = 29%.

Using the loss formula, we can write:

$\text{SP} = \text{CP} \times \left( \frac{100 - \text{Loss}\%}{100} \right)$

$355 = \text{CP} \times \left( \frac{100 - 29}{100} \right)$

$355 = \text{CP} \times \left( \frac{71}{100} \right)$

To find CP, we rearrange the equation:

$\text{CP} = 355 \times \left( \frac{100}{71} \right)$

Now, we calculate the value of CP:

$\text{CP} = \frac{35500}{71}$

$\text{CP} = 500$

So, the cost price of the article is Rs. 500.

Calculating the New Selling Price for 31% Profit

Now, we want to sell the article to gain a profit of 31%.

The cost price is CP = Rs. 500 and the desired Profit % = 31%.

Using the profit formula, let the new selling price be SP2:

$\text{SP2} = \text{CP} \times \left( \frac{100 + \text{Profit}\%}{100} \right)$

$\text{SP2} = 500 \times \left( \frac{100 + 31}{100} \right)$

$\text{SP2} = 500 \times \left( \frac{131}{100} \right)$

Now, we calculate the value of SP2:

$\text{SP2} = \frac{500 \times 131}{100}$

$\text{SP2} = 5 \times 131$

$\text{SP2} = 655$

Thus, the article should be sold for Rs. 655 to gain a 31% profit.

Step-by-Step Solution Summary

1. Identify the given Selling Price (SP1) and Loss Percentage.
2. Use the formula $\text{SP} = \text{CP} \times \left( \frac{100 - \text{Loss}\%}{100} \right)$ to calculate the Cost Price (CP).
3. Identify the desired Profit Percentage.
4. Use the formula $\text{SP} = \text{CP} \times \left( \frac{100 + \text{Profit}\%}{100} \right)$ with the calculated CP and desired Profit Percentage to find the new Selling Price (SP2).
5. Perform the calculations to get the final price.

Final Answer

The article should be sold for Rs. 655 to gain 31% of profit.

Revision Table: Profit and Loss Concepts

Additional Information: Percentage Basics

Percentages are often used in profit and loss calculations. A percentage represents a fraction of 100. For example, 29% means 29 out of 100, which can be written as $\frac{29}{100}$ or 0.29.

• To find a percentage of a number, convert the percentage to a decimal or fraction and multiply. For instance, 71% of CP is $0.71 \times \text{CP}$ or $\frac{71}{100} \times \text{CP}$.
• If something decreases by a percentage (like loss), the remaining value is (100 - percentage)% of the original.
• If something increases by a percentage (like profit), the new value is (100 + percentage)% of the original.

Understanding these percentage basics is crucial for solving profit and loss problems efficiently.