Understanding Profit and Loss Concepts

This problem involves the fundamental concepts of profit and loss in business. When an article is sold for less than its cost price, there is a loss. When it is sold for more than its cost price, there is a gain or profit.

• Cost Price (CP): The price at which an article is bought.
• Selling Price (SP): The price at which an article is sold.
• Loss: Occurs when SP < CP. Loss = CP - SP
• Gain (Profit): Occurs when SP > CP. Gain = SP - CP

Setting up the Equations for Profit and Loss

We are given two scenarios for selling the article:

1. Selling Price (SP1) = Rs. 455, resulting in a loss.
2. Selling Price (SP2) = Rs. 490, resulting in a gain.

Let the Cost Price of the article be Rs. C. Let the amount of loss in the first scenario be L Rs., and the amount of gain in the second scenario be G Rs.

From the definitions above:

• In the first scenario (Loss): \(L = C - 455\)
• In the second scenario (Gain): \(G = 490 - C\)

The problem states that the gain would be four times the loss. We can write this relationship as:

\(G = 4 \times L\)

Calculating the Cost Price (CP)

Now, we can substitute the expressions for L and G into the equation \(G = 4 \times L\):

\(490 - C = 4 \times (C - 455)\)

Let's solve this equation for C:

\(490 - C = 4C - 4 \times 455\)

\(490 - C = 4C - 1820\)

Now, group the terms with C on one side and the constant terms on the other side:

\(490 + 1820 = 4C + C\)

\(2310 = 5C\)

Divide by 5 to find the value of C:

\(C = \frac{2310}{5}\)

\(C = 462\)

So, the Cost Price of the article is Rs. 462.

Verifying the Loss and Gain Amounts

Let's check if the cost price we found satisfies the conditions:

• Loss at SP1 (Rs. 455): Loss = CP - SP1 = \(462 - 455 = 7\) Rs.
• Gain at SP2 (Rs. 490): Gain = SP2 - CP = \(490 - 462 = 28\) Rs.

Is the gain four times the loss? \(4 \times 7 = 28\). Yes, the gain (28) is four times the loss (7). This confirms that our calculated Cost Price of Rs. 462 is correct.

Calculating Selling Price for a 25% Gain

The final part of the question asks for the selling price needed to gain 25%. The gain percentage is always calculated on the Cost Price.

Required Gain = 25% of CP

Required Gain Amount = \(25\%\) of Rs. 462

Required Gain Amount = \(\frac{25}{100} \times 462\)

Required Gain Amount = \(0.25 \times 462\)

Required Gain Amount = \(115.50\) Rs.

To find the Selling Price (SP3) for this gain, we add the gain amount to the Cost Price:

Selling Price (SP3) = CP + Required Gain Amount

Selling Price (SP3) = \(462 + 115.50\)

Selling Price (SP3) = \(577.50\)

Therefore, the shopkeeper should sell the article for Rs. 577.50 to gain 25%.

Summary of Calculations

Revision Table: Key Formulas

Additional Information: Profit and Loss Percentage

Profit or loss is usually expressed as a percentage of the cost price. The formulas are:

• Profit Percentage = \(\frac{\text{Gain}}{\text{CP}} \times 100\%\)
• Loss Percentage = \(\frac{\text{Loss}}{\text{CP}} \times 100\%\)

In this problem, we first used the absolute values of loss and gain to find the cost price, and then used the cost price to calculate the selling price for a specific percentage gain.

Understanding the relationship between CP, SP, absolute loss/gain, and percentage loss/gain is crucial for solving such problems.