A shopkeeper sold an article for Rs. 455 at a loss (in Rs.). If he sells it for Rs. 490, then he would gain an amount four times the loss, At what price (in Rs.) should he sell the article to gain 25%?
577.50
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:
- Selling Price (SP1) = Rs. 455, resulting in a loss.
- 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
| Item | Value |
|---|---|
| Selling Price 1 (SP1) | Rs. 455 |
| Selling Price 2 (SP2) | Rs. 490 |
| Calculated Cost Price (CP) | Rs. 462 |
| Loss at SP1 | Rs. 7 |
| Gain at SP2 | Rs. 28 |
| Required Gain Percentage | 25% |
| Calculated 25% Gain Amount | Rs. 115.50 |
| Selling Price for 25% Gain (SP3) | Rs. 577.50 |
Revision Table: Key Formulas
| Concept | Formula |
|---|---|
| Loss | \(CP - SP\) |
| Gain | \(SP - CP\) |
| Selling Price (with Gain %) | \(CP \times (1 + \frac{\text{Gain \%}}{100})\) |
| Selling Price (with Loss %) | \(CP \times (1 - \frac{\text{Loss \%}}{100})\) |
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.
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