By selling a watch for Rs. 2,000, a shopkeeper loses 20%. How much would he gain or lose by selling it for Rs. 3,000?
20% gain
Understanding the Watch Selling Problem
This problem involves calculating profit and loss percentages related to the selling price and cost price of a watch. We are given the selling price at which a loss occurred and need to find the gain or loss percentage if the watch is sold at a different price.
To solve this, we first need to determine the original cost price (CP) of the watch. Once we know the CP, we can compare it to the new selling price (SP) of Rs. 3,000 to find the gain or loss and then calculate the percentage.
Calculating the Cost Price (CP) of the Watch
We know the first selling price (SP1) and the loss percentage at that price:
- Selling Price (SP1) = \(\text{Rs. } 2,000\)
- Loss Percentage = \(20\%\)
The formula relating Selling Price, Cost Price, and Loss Percentage is:
\(\text{SP} = \text{CP} \times \left(1 - \frac{\text{Loss } \%}{100}\right)\)
We can rearrange this formula to find the Cost Price:
\(\text{CP} = \frac{\text{SP}}{1 - \frac{\text{Loss } \%}{100}}\)
Now, substitute the given values:
\(\text{CP} = \frac{2000}{1 - \frac{20}{100}}\)
\(\text{CP} = \frac{2000}{1 - 0.20}\)
\(\text{CP} = \frac{2000}{0.80}\)
To calculate this, we can write 0.80 as 80/100 or 4/5:
\(\text{CP} = \frac{2000}{80/100} = 2000 \times \frac{100}{80} = 2000 \times \frac{10}{8} = 2000 \times \frac{5}{4}\)
\(\text{CP} = 500 \times 5\)
\(\text{CP} = 2500\)
So, the original Cost Price (CP) of the watch was Rs. 2,500.
Finding Gain or Loss with New Selling Price (SP2)
Now we are given a new selling price (SP2):
- New Selling Price (SP2) = \(\text{Rs. } 3,000\)
- Cost Price (CP) = \(\text{Rs. } 2,500\)
To determine if there is a gain or loss, we compare the New SP with the CP:
\(\text{SP2} = 3000\)
\(\text{CP} = 2500\)
Since \(\text{SP2} > \text{CP}\) (\(3000 > 2500\)), there is a gain.
The amount of gain is calculated as:
\(\text{Gain} = \text{New SP} - \text{CP}\)
\(\text{Gain} = 3000 - 2500\)
\(\text{Gain} = 500\)
The gain amount is Rs. 500.
Calculating the Gain Percentage
The gain percentage is calculated on the Cost Price (CP). The formula for Gain Percentage is:
\(\text{Gain } \% = \frac{\text{Gain}}{\text{CP}} \times 100\)
Substitute the gain amount and the CP:
\(\text{Gain } \% = \frac{500}{2500} \times 100\)
\(\text{Gain } \% = \frac{1}{5} \times 100\)
\(\text{Gain } \% = 20\)
So, the shopkeeper would gain 20% by selling the watch for Rs. 3,000.
Summary of Calculation Steps
| Step | Description | Calculation | Result |
|---|---|---|---|
| 1 | Find CP using SP1 and Loss % | \(\text{CP} = \frac{2000}{1 - 0.20}\) | Rs. 2500 |
| 2 | Compare CP with New SP (SP2) | New SP (\(3000\)) vs CP (\(2500\)) | Gain (since SP2 > CP) |
| 3 | Calculate Gain Amount | \(\text{Gain} = 3000 - 2500\) | Rs. 500 |
| 4 | Calculate Gain % on CP | \(\text{Gain } \% = \frac{500}{2500} \times 100\) | 20% |
Revision Table: Profit and Loss Formulas
| Concept | Formula |
|---|---|
| Profit | \(\text{Profit} = \text{SP} - \text{CP}\) (when SP > CP) |
| Loss | \(\text{Loss} = \text{CP} - \text{SP}\) (when CP > SP) |
| Profit Percentage | \(\text{Profit } \% = \frac{\text{Profit}}{\text{CP}} \times 100\) |
| Loss Percentage | \(\text{Loss } \% = \frac{\text{Loss}}{\text{CP}} \times 100\) |
| SP with Profit | \(\text{SP} = \text{CP} \times \left(1 + \frac{\text{Profit } \%}{100}\right)\) |
| SP with Loss | \(\text{SP} = \text{CP} \times \left(1 - \frac{\text{Loss } \%}{100}\right)\) |
| CP from SP (Profit) | \(\text{CP} = \frac{\text{SP}}{1 + \frac{\text{Profit } \%}{100}}\) |
| CP from SP (Loss) | \(\text{CP} = \frac{\text{SP}}{1 - \frac{\text{Loss } \%}{100}}\) |
Additional Information: Key Terms in Profit and Loss
- Cost Price (CP): The price at which an article is purchased. This is the base value for calculating profit or loss percentages.
- Selling Price (SP): The price at which an article is sold.
- Profit: Occurs when the Selling Price is greater than the Cost Price (SP > CP).
- Loss: Occurs when the Cost Price is greater than the Selling Price (CP > SP).
- Profit Percentage: The profit expressed as a percentage of the Cost Price.
- Loss Percentage: The loss expressed as a percentage of the Cost Price.
Understanding these basic terms and formulas is crucial for solving profit and loss problems quickly and accurately in exams.
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