Remi earns a profit of 20% on selling an article at a certain price If she sells the articles for Rs. 8 more, she will gain 30% What is the original cost price of 16 such articles?
Rs. 1,280
Understanding Profit and Cost Price
The question asks for the original cost price of 16 articles based on profit percentages at different selling prices. We are given two scenarios involving profit on selling the article and the difference in selling price between these scenarios.
Setting up the Problem
Let's denote the original cost price of one article as \(CP\).
We are given two profit scenarios:
- Scenario 1: Remi earns a profit of 20% on selling an article at a certain price.
- Scenario 2: If she sells the article for Rs. 8 more, she gains 30%.
Calculating Selling Prices
The selling price (\(SP\)) is calculated as \(SP = CP + \text{Profit}\). Profit is a percentage of the cost price.
In Scenario 1, the profit is 20% of the cost price. So, the selling price (\(SP_1\)) is:
\[ SP_1 = CP + 20\% \text{ of } CP \]
\[ SP_1 = CP + \frac{20}{100} \times CP \]
\[ SP_1 = CP + 0.20 \times CP \]
\[ SP_1 = 1.20 \times CP \]
In Scenario 2, the profit is 30% of the cost price. So, the selling price (\(SP_2\)) is:
\[ SP_2 = CP + 30\% \text{ of } CP \]
\[ SP_2 = CP + \frac{30}{100} \times CP \]
\[ SP_2 = CP + 0.30 \times CP \]
\[ SP_2 = 1.30 \times CP \]
Finding the Cost Price of One Article
We are told that \(SP_2\) is Rs. 8 more than \(SP_1\). We can write this relationship as an equation:
\[ SP_2 = SP_1 + 8 \]
Now, substitute the expressions for \(SP_1\) and \(SP_2\) in terms of \(CP\):
\[ 1.30 \times CP = 1.20 \times CP + 8 \]
To find the value of \(CP\), subtract \(1.20 \times CP\) from both sides of the equation:
\[ 1.30 \times CP - 1.20 \times CP = 8 \]
\[ (1.30 - 1.20) \times CP = 8 \]
\[ 0.10 \times CP = 8 \]
To isolate \(CP\), divide both sides by 0.10:
\[ CP = \frac{8}{0.10} \]
\[ CP = \frac{8}{\frac{10}{100}} = 8 \times \frac{100}{10} \]
\[ CP = 8 \times 10 \]
\[ CP = 80 \]
So, the original cost price of one article is Rs. 80.
Calculating the Cost Price of 16 Articles
The question asks for the original cost price of 16 such articles. To find this, multiply the cost price of one article by 16:
\[ \text{Cost price of 16 articles} = 16 \times CP \]
\[ \text{Cost price of 16 articles} = 16 \times 80 \]
\[ \text{Cost price of 16 articles} = 1280 \]
The original cost price of 16 such articles is Rs. 1,280.
Summary of Calculation Steps
- Define the cost price of one article as \(CP\).
- Write expressions for the selling price in the 20% profit scenario (\(SP_1 = 1.20 \times CP\)).
- Write expressions for the selling price in the 30% profit scenario (\(SP_2 = 1.30 \times CP\)).
- Use the information that \(SP_2 = SP_1 + 8\) to set up the equation \(1.30 \times CP = 1.20 \times CP + 8\).
- Solve the equation to find the value of \(CP\), which is Rs. 80.
- Multiply the cost price of one article by 16 to find the cost price of 16 articles: \(16 \times 80 = 1280\).
| Item | Calculation | Result |
|---|---|---|
| Cost Price (1 article) | \(CP\) | Rs. 80 |
| Selling Price (20% profit) | \(1.20 \times CP\) | \(1.20 \times 80 = 96\) |
| Selling Price (30% profit) | \(1.30 \times CP\) | \(1.30 \times 80 = 104\) |
| Difference in Selling Prices | \(104 - 96\) | 8 |
| Cost Price (16 articles) | \(16 \times 80\) | 1280 |
The difference in selling prices (Rs. 8) corresponds exactly to the difference in profit percentages (30% - 20% = 10%). Therefore, 10% of the Cost Price must equal Rs. 8.
\[ 10\% \text{ of } CP = 8 \]
\[ \frac{10}{100} \times CP = 8 \]
\[ 0.10 \times CP = 8 \]
\[ CP = \frac{8}{0.10} = 80 \]
This confirms our calculation for the cost price of one article. Then, the cost price of 16 articles is \(16 \times 80 = 1280\).
Revision Table: Profit and Cost Price
| Concept | Formula | Description |
|---|---|---|
| Cost Price (CP) | - | The original price at which an item is bought. |
| Selling Price (SP) | - | The price at which an item is sold. |
| Profit | \(SP - CP\) (when \(SP > CP\)) | The gain earned from selling an item. |
| Profit Percentage | \( \frac{\text{Profit}}{CP} \times 100\% \) | Profit expressed as a percentage of the cost price. |
| Selling Price with Profit | \( CP \times (1 + \frac{\text{Profit \%}}{100}) \) | Calculates SP when CP and profit percentage are known. |
Additional Information on Profit and Loss Concepts
- Loss: Occurs when the selling price is less than the cost price (\(SP < CP\)). Loss = \(CP - SP\).
- Loss Percentage: Calculated as \( \frac{\text{Loss}}{CP} \times 100\% \). It is always calculated on the Cost Price.
- Selling Price with Loss: \( SP = CP \times (1 - \frac{\text{Loss \%}}{100}) \).
- Understanding the relationship between CP, SP, Profit/Loss, and their percentages is crucial for solving problems in this area of arithmetic.
- A difference in selling price due to a change in profit percentage directly relates to that percentage difference applied to the original cost price.
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