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%.
Similar Questions
- A person bought a book at \(3/4^{th}\) of its listed price and sold it at 50% more than its listed price. What is the percentage of gain in the transaction?
- A shopkeeper gives three consecutive discounts 10%, 20% and 25% after which he sells the article at a profit of 8% on the cost price. Had he sold the article after the first discount, how much profit would he have got?
A person sells article X for ₹34,500 and makes a profit of 15%. He sells article Y at a loss of 10%. He neither loses nor gains on the whole because of these two transactions. What is the selling price of article Y?
A shopkeeper sold his goods at the cost price. By using false weights, he gained \(11\frac{1}{9}\%\). What weight did he use for 1 kg?
The ratio of the cost price of two articles X and Y is \(1:2\). A businessman earns a profit of \(p\%\) by selling X and a loss of \(3p\%\) by selling Y. If he loses \(20\%\) in this transaction, then what is the value of \(p\)?
Important Questions from Profit and Loss
Rohit buys 8 pens and 4 pencils for Rs. 2400. He sells pencils at a profit of 20 percent and pens at the loss of 10 percent. If his overall profit is Rs. 240, then what is the sum of the cost price of one pen and one pencil?
A man sells a car to his friend at the loss of 10 %; who in return sells it for Rs. 54000 making a profit of 20 %. What was the initial value of the car?
An article was sold at a loss of 24%. If it were sold for Rs. 1,596 more, then there would have been a gain of 18%, The cost price of the article is:
The cost price of an article is Rs.6,450. If it sold at a profit 16%, how much would be its selling price?
If the selling price of 7 articles is equal to the cost price of 6 articles, then what is the percentage loss?
