Profit Percentage Calculation Using Article Quantities
This solution details how to calculate the profit percentage when the total cost price (CP) of a certain number of articles is equivalent to the total selling price (SP) of a different number of articles. We'll use the given ratio \(y:z = 5:4\) to find the profit percentage.
Understanding the Core Concept
The problem hinges on equating the total cost value with the total selling value based on the number of articles involved. We need to find the relationship between the cost price of one article and the selling price of one article to determine the profit.
Defining Variables and Initial Equation
Let's define:
- CP = Cost Price of one article
- SP = Selling Price of one article
- y = Number of articles whose total CP is considered
- z = Number of articles whose total SP is considered
The problem states:
Cost Price of y articles = Selling Price of z articles
This translates to the equation:
\(CP \times y = SP \times z\)Applying the Given Ratio
We are given the ratio \(y:z = 5:4\). This can be written as:
\(\frac{y}{z} = \frac{5}{4}\)
Now, let's rearrange the initial equation to find the relationship between CP and SP:
\(\frac{CP}{SP} = \frac{z}{y}\)
Substitute the value of the ratio \(\frac{z}{y}\) (which is the inverse of \(\frac{y}{z}\)):
\(\frac{z}{y} = \frac{4}{5}\)
So, the relationship becomes:
\(\frac{CP}{SP} = \frac{4}{5}\)This tells us that the cost price of 4 articles is equal to the selling price of 5 articles. We can express the Selling Price (SP) in terms of the Cost Price (CP):
\(SP = \frac{5}{4} \times CP\)Calculating Profit
Profit is calculated as the Selling Price minus the Cost Price:
Profit = \(SP - CP\)
Substitute the expression for SP we found:
Profit = \(\left( \frac{5}{4} \times CP \right) - CP\)
To simplify, we can factor out CP:
Profit = \(CP \times \left( \frac{5}{4} - 1 \right)\)
Profit = \(CP \times \left( \frac{5}{4} - \frac{4}{4} \right)\)
Profit = \(CP \times \left( \frac{1}{4} \right)\)
So, the profit is \(\frac{1}{4}\) of the Cost Price.
Calculating Profit Percentage
The formula for profit percentage is:
\(Profit \% = \left( \frac{Profit}{CP} \right) \times 100\%\)Now, substitute the profit value we calculated:
Profit % = \(\left( \frac{\frac{1}{4} \times CP}{CP} \right) \times 100\%\)
The CP term cancels out:
Profit % = \(\frac{1}{4} \times 100\%\)
Profit % = \(25\%\)
Final Answer Summary
By equating the cost price of y articles to the selling price of z articles and using the ratio \(y:z = 5:4\), we determined that the selling price is \(\frac{5}{4}\) times the cost price. This results in a profit of \(\frac{1}{4}\) of the cost price, which translates to a profit percentage of 25%.
Important Questions from Profit and Loss
Ram bought a chair at 35% discount on its market price, Had Ram got no discount, he would have had to pay ₹245 more. How much did Ram pay for the chair?
A shopkeeper bought an item for Rs. 4,500 and sold it at a loss of 5%. From this money, he bought another item and sold it at a profit of 10%. What is his overall profit?
A man sells two articles at Rs. 9,180 each. He gains 8% on one article and loses 15% on the other. His overall profit or loss is:
A shopkeeper selling an article for ₹46 loses 8%. In order to gain 6%, what should be the selling price of the article?
Rahul purchased 80 items from the market. 25% items of the total items were defective and the remaining items were sold at 50% profit. What will be the overall profit percentage?
