Understanding Profit and Loss Percentages Through Mathematical Examples
When dealing with business transactions, understanding the relationship between cost price (CP) and selling price (SP) is crucial for determining profit or loss percentages. This article explores the concept through various examples, using mathematical formulas and logical reasoning.
Example 1: 15 Bottles and 12 Bottles
Imagine that the cost of 15 bottles is equal to the selling price of 12 bottles. We need to find the profit or loss percentage.
Let's denote the cost price of one bottle as (C). Therefore, the cost price of 15 bottles is:
Total Cost Price (CP) 15C
According to the problem, this cost is equal to the selling price of 12 bottles, where the selling price of one bottle is (S). The selling price for 12 bottles is:
Total Selling Price (SP) 12S
Equating the total cost and total selling price gives us:
15C 12S
We can solve this equation to express (S) in terms of (C):
S (frac{15C}{12} frac{5C}{4})
The profit per bottle is calculated as:
Profit per bottle S - C (frac{5C}{4} - C frac{5C}{4} - frac{4C}{4} frac{C}{4})
Now, we find the profit percentage using the formula:
Profit percent (left[frac{frac{C}{4}}{C}right] times 100 frac{1}{4} times 100 25%)
Example 2: Cost of 15 Selling Price of 20
In this example, the cost of 15 items is equal to the selling price of 20 items. We need to find the loss percentage.
Let the cost price of one item be (frac{15}{15} 1). So, the cost price of one item is (frac{4}{3}).
The selling price of one item is (frac{20}{15} frac{4}{3}).
To calculate the loss, we find the difference between the cost price and the selling price:
Loss CP - SP (frac{4}{3} - frac{3}{4} frac{16}{12} - frac{9}{12} frac{7}{12})
The loss percentage is:
Loss percent (left[frac{frac{7}{12}}{frac{4}{3}}right] times 100 frac{7}{12} times frac{3}{4} times 100 frac{7}{16} times 100 frac{700}{16} approx 43.75%)
Example 3: Profit on 30 Articles
For a set of 30 articles, the cost price of 30 articles is (X). The cost price of one article is (frac{X}{30}).
The selling price of 25 articles is (X), so the selling price of one article is (frac{X}{25}).
The profit per article is calculated as:
Profit per article SP - CP (frac{X}{25} - (frac{X}{30} frac{3 - 25X}{750} frac{5X}{750} frac{X}{150})
The total profit for 30 articles is:
Profit (frac{X}{150} times 30 2)
Therefore, the profit is Rs 2.
Example 4: Cost Price and Selling Price Relationship
Suppose the cost price and selling price of an article are (frac{1}{30}) and (frac{1}{25}) respectively. We need to find the profit percentage.
Profit Selling Price - Cost Price (frac{1}{25} - (frac{1}{30} frac{30 - 25}{750} frac{5}{750} frac{1}{150})
Profit percentage (left[frac{frac{1}{150}}{frac{1}{30}}right] times 100 frac{1}{150} times 30 times 100 frac{30}{150} times 100 frac{100}{5} 20%)
Example 5: Cost Price of 16 Articles and Selling Price of 14 Articles
The cost price of 16 articles is Rs 16, and this is equal to the selling price of 14 articles. The cost price of 14 articles is Rs 14.
Here, SP CP, which indicates a gain.
Gain Rs 16 - Rs 14 Rs 2.
Using the formula for profit percentage:
Profit (frac{text{Gain} times 100}{text{CP}} frac{2 times 100}{14} frac{200}{14} 14.285%)
Example 6: Cost Price and Selling Price for 14 and 16 Articles
Let the cost price of each article be Rs 1.
So, the cost price of 14 articles is Rs 14.
The selling price of 14 articles is Rs 16.
Since the selling price of 14 articles is equal to the cost price of 16 articles, the profit is:
Profit Rs 16 - Rs 14 Rs 2.
Using the profit percentage formula:
Profit (frac{text{Profit} times 100}{text{CP}} frac{2 times 100}{14} frac{200}{14} frac{100}{7} 14.285%)
Conclusion
This article has demonstrated various scenarios involving profit and loss percentages through mathematical examples. Understanding these concepts is vital for making informed business decisions and optimizing profitability.