On selling a painting at Rs. 1,498 , the gain is 25% more than the loss incurred on selling it at Rs.1,300. In order to gain 25%, the selling price will be:
Rs. 1,735
Understanding Gain and Loss in Pricing
This problem involves calculating the cost price of a painting using information about two different selling prices and the relationship between the resulting gain and loss. Once the cost price is known, we can determine the selling price required to achieve a specific profit percentage.
Defining Key Terms: Cost Price, Selling Price, Gain, 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.
- Gain (Profit): Occurs when Selling Price > Cost Price. Gain = SP - CP.
- Loss: Occurs when Cost Price > Selling Price. Loss = CP - SP.
Setting up the Problem Equations
We are given two selling scenarios:
- Selling Price 1 (SP1) = Rs. 1,498, resulting in a gain.
- Selling Price 2 (SP2) = Rs. 1,300, resulting in a loss.
Let CP be the Cost Price of the painting.
The gain when selling at Rs. 1,498 is:
\( \text{Gain} = \text{SP1} - \text{CP} = 1498 - \text{CP} \)
The loss when selling at Rs. 1,300 is:
\( \text{Loss} = \text{CP} - \text{SP2} = \text{CP} - 1300 \)
The problem states that the gain is 25% more than the loss. This can be written as:
\( \text{Gain} = \text{Loss} + 25\% \text{ of Loss} \)
\( \text{Gain} = \text{Loss} + 0.25 \times \text{Loss} \)
\( \text{Gain} = 1.25 \times \text{Loss} \)
Calculating the Cost Price (CP)
Now, substitute the expressions for Gain and Loss into the equation:
\( 1498 - \text{CP} = 1.25 \times (\text{CP} - 1300) \)
Distribute 1.25 on the right side:
\( 1498 - \text{CP} = 1.25 \times \text{CP} - 1.25 \times 1300 \)
\( 1498 - \text{CP} = 1.25 \times \text{CP} - 1625 \)
Now, we need to isolate CP. Let's move the CP terms to one side and the constant terms to the other side:
Add CP to both sides:
\( 1498 = 1.25 \times \text{CP} + \text{CP} - 1625 \)
\( 1498 = 2.25 \times \text{CP} - 1625 \)
Add 1625 to both sides:
\( 1498 + 1625 = 2.25 \times \text{CP} \)
\( 3123 = 2.25 \times \text{CP} \)
To find CP, divide 3123 by 2.25:
\( \text{CP} = \frac{3123}{2.25} \)
\( \text{CP} = 1388 \)
So, the Cost Price of the painting is Rs. 1,388.
Calculating the Selling Price for 25% Gain
The question asks for the selling price required to gain 25%. The desired gain is 25% of the Cost Price.
Desired Gain = 25% of CP
\( \text{Desired Gain} = 0.25 \times 1388 \)
\( \text{Desired Gain} = 347 \)
The Selling Price (SP3) for a 25% gain is the Cost Price plus the Desired Gain:
\( \text{SP3} = \text{CP} + \text{Desired Gain} \)
\( \text{SP3} = 1388 + 347 \)
\( \text{SP3} = 1735 \)
Alternatively, a 25% gain means the selling price is 125% of the cost price:
\( \text{SP3} = \text{CP} \times (1 + 0.25) \)
\( \text{SP3} = 1388 \times 1.25 \)
\( \text{SP3} = 1735 \)
Therefore, the selling price must be Rs. 1,735 in order to gain 25%.
Summary of Calculations
| Item | Value | Calculation / Reason |
|---|---|---|
| Selling Price 1 (SP1) | Rs. 1498 | Given (Gain) |
| Selling Price 2 (SP2) | Rs. 1300 | Given (Loss) |
| Relationship | Gain = 1.25 Loss | Given (Gain is 25% more than Loss) |
| Cost Price (CP) | Rs. 1388 | Calculated from Gain = 1.25 Loss equation |
| Desired Gain Percentage | 25% | Given |
| Selling Price for 25% Gain (SP3) | Rs. 1735 | \( \text{CP} \times 1.25 \) |
Revision Table: Profit and Loss Concepts
| Concept | Formula | Condition |
|---|---|---|
| Gain (Profit) | SP - CP | SP > CP |
| Loss | CP - SP | CP > SP |
| Gain % | \(\left(\frac{\text{Gain}}{\text{CP}}\right) \times 100\) | SP > CP |
| Loss % | \(\left(\frac{\text{Loss}}{\text{CP}}\right) \times 100\) | CP > SP |
| Selling Price (with Gain) | \(\text{CP} \times \left(\frac{100 + \text{Gain}\%}{100}\right)\) | Known CP and Gain % |
| Selling Price (with Loss) | \(\text{CP} \times \left(\frac{100 - \text{Loss}\%}{100}\right)\) | Known CP and Loss % |
Additional Information on Solving Profit and Loss Problems
Profit and loss problems often require setting up algebraic equations based on the given relationships between cost price, selling price, gain, and loss. It's crucial to correctly express percentages as decimals or fractions when using them in equations. Remember that gain and loss percentages are typically calculated with respect to the cost price unless otherwise specified.
In this problem, the key was translating "gain is 25% more than the loss" into the equation \( \text{Gain} = 1.25 \times \text{Loss} \). This allowed us to find the unknown cost price, which is the base for calculating any required selling price for a target profit percentage.
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