A shopkeeper gains 20% in place of 16% loss if the selling price of an article is increased by Rs. 324. The cost price of the article is:
Finding the Cost Price with Profit and Loss Changes
This problem involves understanding how changes in selling price affect profit and loss percentages relative to the cost price. We are given two scenarios: one where there is a loss and one where there is a gain, with a specific increase in the selling price causing this change.
Understanding Profit and Loss Percentages
Profit or loss percentage is always calculated based on the cost price (CP) of the article unless stated otherwise.
- If there is a profit, the selling price (SP) is CP + Profit. Profit percentage \( = \left(\frac{\text{Profit}}{\text{CP}}\right) \times 100\% \).
- If there is a loss, the selling price (SP) is CP - Loss. Loss percentage \( = \left(\frac{\text{Loss}}{\text{CP}}\right) \times 100\% \).
Setting up the Problem Equations
Let the Cost Price of the article be \( \text{CP} \).
Scenario 1: 16% Loss
When there is a 16% loss, the selling price (\( \text{SP}_1 \)) is 16% less than the cost price. \( \text{SP}_1 = \text{CP} - 16\% \text{ of } \text{CP} \) \( \text{SP}_1 = \text{CP} - \frac{16}{100} \times \text{CP} \) \( \text{SP}_1 = \text{CP} - 0.16 \text{ CP} \) \( \text{SP}_1 = (1 - 0.16) \text{ CP} \) \( \text{SP}_1 = 0.84 \text{ CP} \)
Scenario 2: 20% Gain
When there is a 20% gain (profit), the new selling price (\( \text{SP}_2 \)) is 20% more than the cost price. \( \text{SP}_2 = \text{CP} + 20\% \text{ of } \text{CP} \) \( \text{SP}_2 = \text{CP} + \frac{20}{100} \times \text{CP} \) \( \text{SP}_2 = \text{CP} + 0.20 \text{ CP} \) \( \text{SP}_2 = (1 + 0.20) \text{ CP} \) \( \text{SP}_2 = 1.20 \text{ CP} \)
Calculating the Increase in Selling Price
The problem states that the selling price is increased by Rs. 324. This increase is the difference between the new selling price (\( \text{SP}_2 \)) and the original selling price (\( \text{SP}_1 \)).
Increase in SP \( = \text{SP}_2 - \text{SP}_1 \)
We are given that the Increase in SP = Rs. 324.
So, \( \text{SP}_2 - \text{SP}_1 = 324 \)
Substitute the expressions for \( \text{SP}_1 \) and \( \text{SP}_2 \) in terms of CP:
\( 1.20 \text{ CP} - 0.84 \text{ CP} = 324 \)
Solving for the Cost Price (CP)
Combine the terms involving CP:
\( (1.20 - 0.84) \text{ CP} = 324 \)
\( 0.36 \text{ CP} = 324 \)
Now, isolate CP by dividing 324 by 0.36:
\( \text{CP} = \frac{324}{0.36} \)
To make the division easier, we can write 0.36 as \( \frac{36}{100} \):
\( \text{CP} = \frac{324}{\frac{36}{100}} \)
\( \text{CP} = 324 \times \frac{100}{36} \)
Now, we can simplify the expression. We know that \( 324 \div 36 = 9 \).
\( \text{CP} = 9 \times 100 \)
\( \text{CP} = 900 \)
So, the cost price of the article is Rs. 900.
Verification
Let's quickly verify our answer. If CP = Rs. 900:
- 16% loss: Loss \( = 16\% \text{ of } 900 = 0.16 \times 900 = 144 \). \( \text{SP}_1 = 900 - 144 = 756 \).
- 20% gain: Profit \( = 20\% \text{ of } 900 = 0.20 \times 900 = 180 \). \( \text{SP}_2 = 900 + 180 = 1080 \).
The difference between the selling prices is \( \text{SP}_2 - \text{SP}_1 = 1080 - 756 = 324 \).
This matches the given increase of Rs. 324, confirming our calculated cost price is correct.
The cost price of the article is Rs. 900.
| Scenario | Percentage | Calculation Relative to CP | Selling Price (SP) in terms of CP |
|---|---|---|---|
| Loss | 16% | CP - 16% of CP | \(0.84 \times \text{CP}\) |
| Gain | 20% | CP + 20% of CP | \(1.20 \times \text{CP}\) |
Revision Table: Profit and Loss Terms
| Term | Definition | How it Relates to Cost Price (CP) and Selling Price (SP) |
|---|---|---|
| Cost Price (CP) | The price at which an article is bought. | Basis for calculating profit or loss percentage. |
| Selling Price (SP) | The price at which an article is sold. | \( \text{SP} = \text{CP} + \text{Profit} \) (if profit) \( \text{SP} = \text{CP} - \text{Loss} \) (if loss) |
| Profit | When SP > CP. | Profit \( = \text{SP} - \text{CP} \) |
| Loss | When SP < CP. | Loss \( = \text{CP} - \text{SP} \) |
| Profit % | \( (\text{Profit} / \text{CP}) \times 100 \) | Indicates profit per Rs. 100 of CP. |
| Loss % | \( (\text{Loss} / \text{CP}) \times 100 \) | Indicates loss per Rs. 100 of CP. |
Additional Information: Percentage Point Change
In problems like this, the change from a 16% loss to a 20% gain represents a total percentage change relative to the cost price. A 16% loss means the SP is 16% below CP. A 20% gain means the SP is 20% above CP.
The total difference as a percentage of CP is the distance from -16% to +20% on a number line centered at CP (0%).
Total percentage difference \( = 20\% - (-16\%) = 20\% + 16\% = 36\% \)
This 36% of the cost price corresponds to the Rs. 324 increase in the selling price.
So, \( 36\% \text{ of CP} = 324 \)
\( \frac{36}{100} \times \text{CP} = 324 \)
\( \text{CP} = 324 \times \frac{100}{36} \)
\( \text{CP} = 9 \times 100 \)
\( \text{CP} = 900 \)
This method provides a quicker way to solve such problems once the concept is understood.
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