A shopkeeper sold a pair of headphones for Rs. 5,520 at a gain of 20%. What would have been the gain or loss percent if it had been sold for Rs. 4,370?
Loss 5%
Profit and Loss Percentage Calculation Explained
This problem involves calculating the cost price of an item based on its selling price and gain percentage, and then determining the gain or loss percentage if it were sold at a different price.
Step 1: Calculate the Cost Price (CP)
We are given that the headphones were sold for Rs. 5,520 at a gain of 20%. The selling price (SP) is Rs. 5,520 and the gain percentage is 20%.
The relationship between Selling Price (SP), Cost Price (CP), and Gain Percentage is:
$\text{SP} = \text{CP} \times \left(1 + \frac{\text{Gain \%}}{100}\right)$
We can rearrange this formula to find the Cost Price:
$\text{CP} = \frac{\text{SP}}{1 + \frac{\text{Gain \%}}{100}}$
Substitute the given values:
$\text{CP} = \frac{5520}{1 + \frac{20}{100}}$
$\text{CP} = \frac{5520}{1 + 0.20}$
$\text{CP} = \frac{5520}{1.20}$
Now, calculate the Cost Price:
$\text{CP} = 4600$
So, the cost price of the headphones is Rs. 4,600.
Step 2: Determine Gain or Loss at the New Selling Price
The question asks what would happen if the headphones were sold for Rs. 4,370. This is the new selling price (SP2 = Rs. 4,370).
We compare the new selling price (SP2) with the calculated cost price (CP):
- If SP2 > CP, there is a gain.
- If SP2 < CP, there is a loss.
- If SP2 = CP, there is neither gain nor loss (break-even).
In this case, SP2 = 4370 and CP = 4600.
Since 4370 < 4600, there is a loss.
Step 3: Calculate the Loss Amount
The loss amount is the difference between the Cost Price and the new Selling Price:
Loss = CP - SP2
Loss = 4600 - 4370
Loss = 230
The loss amount is Rs. 230.
Step 4: Calculate the Loss Percentage
The loss percentage is calculated based on the Cost Price:
$\text{Loss \%} = \frac{\text{Loss}}{\text{CP}} \times 100$
Substitute the loss amount and Cost Price:
$\text{Loss \%} = \frac{230}{4600} \times 100$
$\text{Loss \%} = \frac{23}{460} \times 100$
$\text{Loss \%} = \frac{1}{20} \times 100$
$\text{Loss \%} = 5$
The loss percentage is 5%.
Summary of Results
Based on the calculations:
- Cost Price (CP) = Rs. 4,600
- New Selling Price (SP2) = Rs. 4,370
- Comparison: SP2 < CP, indicating a loss.
- Loss Amount = Rs. 230
- Loss Percentage = 5%
Therefore, if the headphones were sold for Rs. 4,370, there would have been a loss of 5%.
Revision Table: Profit and Loss Formulas
| Concept | Formula |
|---|---|
| Gain | Selling Price (SP) - Cost Price (CP) (When SP > CP) |
| Loss | Cost Price (CP) - Selling Price (SP) (When CP > SP) |
| Gain % | $\frac{\text{Gain}}{\text{CP}} \times 100$ |
| Loss % | $\frac{\text{Loss}}{\text{CP}} \times 100$ |
| SP (with Gain %) | $\text{CP} \times \left(1 + \frac{\text{Gain \%}}{100}\right)$ |
| SP (with Loss %) | $\text{CP} \times \left(1 - \frac{\text{Loss \%}}{100}\right)$ |
| CP (from SP and Gain %) | $\frac{\text{SP}}{1 + \frac{\text{Gain \%}}{100}}$ |
| CP (from SP and Loss %) | $\frac{\text{SP}}{1 - \frac{\text{Loss \%}}{100}}$ |
Additional Information: Understanding Profit and Loss
Profit and Loss are fundamental concepts in commercial arithmetic. They help businesses and individuals understand the financial outcome of a transaction.
- Cost Price (CP): This is the price at which an article is purchased. It includes all expenses incurred to get the article ready for sale (like transport, packaging, etc.).
- Selling Price (SP): This is the price at which an article is sold.
- Profit (or Gain): Occurs when the Selling Price is greater than the Cost Price (SP > CP). It represents the positive financial gain from the sale.
- Loss: Occurs when the Cost Price is greater than the Selling Price (CP > SP). It represents the financial deficit from the sale.
- Percentage Calculations: Profit or loss is typically expressed as a percentage of the Cost Price. This allows for easy comparison across different transactions regardless of the absolute value of the price.
Understanding these basic terms and their relationships is crucial for solving various profit and loss problems.
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