Calculating Selling Price for Profit Gain

This problem involves calculating the cost price of an article when there is a loss and then determining the new selling price required to achieve a desired profit percentage. We will use the concepts of Cost Price (CP), Selling Price (SP), Loss Percentage, and Profit Percentage.

Understanding Key Terms in Profit and Loss

• Cost Price (CP): The price at which an article is purchased.
• Selling Price (SP): The price at which an article is sold.
• Loss: Occurs when SP < CP. Loss % = \(\frac{\text{Loss}}{\text{CP}} \times 100\)
• Profit (Gain): Occurs when SP > CP. Profit % = \(\frac{\text{Profit}}{\text{CP}} \times 100\)

Step-by-Step Solution to Find the New Selling Price

Step 1: Calculate the Cost Price (CP)

We are given that the shopkeeper sells the article for Rs. 360 and incurs a 10% loss. The formula relating SP, CP, and Loss % is:

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

Substituting the given values:

\(360 = \text{CP} \times \left(\frac{100 - 10}{100}\right)\)

\(360 = \text{CP} \times \left(\frac{90}{100}\right)\)

To find CP, we rearrange the formula:

\(\text{CP} = \frac{360 \times 100}{90}\)

\(\text{CP} = \frac{36000}{90}\)

\(\text{CP} = 400\)

So, the Cost Price of the article is Rs. 400.

Step 2: Calculate the New Selling Price for a 30% Gain

Now, the shopkeeper wants to gain 30% on the article. We know the CP is Rs. 400. The formula relating New SP, CP, and Gain % is:

\(\text{New SP} = \text{CP} \times \left(\frac{100 + \text{Gain \%}}{100}\right)\)

Substituting the values (CP = Rs. 400, Gain % = 30):

\(\text{New SP} = 400 \times \left(\frac{100 + 30}{100}\right)\)

\(\text{New SP} = 400 \times \left(\frac{130}{100}\right)\)

\(\text{New SP} = \frac{400 \times 130}{100}\)

\(\text{New SP} = 4 \times 130\)

\(\text{New SP} = 520\)

Therefore, to gain 30%, the selling price of the article should be Rs. 520.

Summary of Calculations

Based on the calculations, the selling price should be Rs. 520 to achieve a 30% gain.

Revision Table: Profit and Loss Formulas

Additional Information: Calculating Percentage Changes

Percentage change is a way to express how much a quantity changes relative to its original value. In profit and loss, the base for calculating profit or loss percentage is always the Cost Price (CP).

If a value increases by P%, the new value is Original Value \(\times (1 + \frac{P}{100})\).

If a value decreases by L%, the new value is Original Value \(\times (1 - \frac{L}{100})\).

In our problem:

• When the shopkeeper lost 10%, the Selling Price (Rs. 360) was the Cost Price decreased by 10%: \(360 = \text{CP} \times (1 - \frac{10}{100}) = \text{CP} \times 0.90\).
• When the shopkeeper wants to gain 30%, the New Selling Price will be the Cost Price increased by 30%: \(\text{New SP} = \text{CP} \times (1 + \frac{30}{100}) = \text{CP} \times 1.30\).

Using these factors:

From \(360 = \text{CP} \times 0.90\), we get \(\text{CP} = \frac{360}{0.90} = 400\).

Then, \(\text{New SP} = 400 \times 1.30 = 520\).

This method gives the same result and is a quick way to handle percentage increases and decreases in profit and loss calculations.