40 percent of the cost price of an article is equal to the 24 percent of its selling price. What is the profit percentage?
66.66 percent
Finding the Profit Percentage: Step-by-Step Calculation
Let's break down how to find the profit percentage when we are given a relationship between the cost price and the selling price of an article.
The problem states that 40 percent of the cost price (CP) is equal to 24 percent of the selling price (SP).
Setting up the Equation based on the given information
We can write this relationship as an equation:
40% of CP = 24% of SP
Converting percentages to decimals (or fractions), we get:
\(\frac{40}{100} \times \text{CP} = \frac{24}{100} \times \text{SP}\)
\(0.40 \times \text{CP} = 0.24 \times \text{SP}\)
Finding the Relationship between SP and CP
To find the profit percentage, we need to understand the ratio of Selling Price (SP) to Cost Price (CP). Let's rearrange the equation to find this ratio:
\(\frac{\text{SP}}{\text{CP}} = \frac{0.40}{0.24}\)
We can simplify this fraction by removing the decimals (multiplying numerator and denominator by 100):
\(\frac{\text{SP}}{\text{CP}} = \frac{40}{24}\)
Now, simplify the fraction \(\frac{40}{24}\) by dividing both the numerator and denominator by their greatest common divisor, which is 8:
\(\frac{\text{SP}}{\text{CP}} = \frac{40 \div 8}{24 \div 8} = \frac{5}{3}\)
So, the relationship is \(\frac{\text{SP}}{\text{CP}} = \frac{5}{3}\), which means SP is \(\frac{5}{3}\) times CP, or \(\text{SP} = \frac{5}{3} \times \text{CP}\).
Calculating the Profit
Profit is defined as the difference between the Selling Price and the Cost Price:
Profit = SP - CP
Substitute the relationship we found for SP:
Profit = \(\frac{5}{3}\) CP - CP
To subtract CP from \(\frac{5}{3}\) CP, think of CP as \(\frac{3}{3}\) CP:
Profit = \(\frac{5}{3}\) CP - \(\frac{3}{3}\) CP
Profit = \((\frac{5}{3} - \frac{3}{3})\) CP
Profit = \(\frac{5-3}{3}\) CP
Profit = \(\frac{2}{3}\) CP
Calculating the Profit Percentage
The profit percentage is calculated using the formula:
Profit Percentage = \(\frac{\text{Profit}}{\text{CP}} \times 100\)
Substitute the profit we found (\(\frac{2}{3}\) CP) into the formula:
Profit Percentage = \(\frac{\frac{2}{3} \times \text{CP}}{\text{CP}} \times 100\)
The CP terms cancel out:
Profit Percentage = \(\frac{2}{3} \times 100\)
Profit Percentage = \(\frac{200}{3}\)
Now, calculate the value:
\(\frac{200}{3} = 66.666...\)
This is approximately 66.67 percent or often written as 66.66 percent in contexts where specific options are provided.
| Calculation Step | Details |
|---|---|
| Given Relation | \(0.40 \times \text{CP} = 0.24 \times \text{SP}\) |
| Ratio SP/CP | \(\frac{\text{SP}}{\text{CP}} = \frac{0.40}{0.24} = \frac{40}{24} = \frac{5}{3}\) |
| Profit | \(\text{SP} - \text{CP} = \frac{5}{3}\text{CP} - \text{CP} = \frac{2}{3}\text{CP}\) |
| Profit Percentage | \(\frac{\text{Profit}}{\text{CP}} \times 100 = \frac{\frac{2}{3}\text{CP}}{\text{CP}} \times 100 = \frac{2}{3} \times 100\) |
| Result | \(66.66...\) percent |
Therefore, the profit percentage is 66.66 percent (approximately).
Revision Table: Key Concepts in Profit & Loss
| Concept | Definition/Formula |
|---|---|
| Cost Price (CP) | The price at which an article is purchased. |
| Selling Price (SP) | The price at which an article is sold. |
| Profit | When SP > CP. Profit = SP - CP. |
| Loss | When CP > SP. Loss = CP - SP. |
| Profit Percentage | \(\frac{\text{Profit}}{\text{CP}} \times 100\). Always calculated on CP unless otherwise specified. |
| Loss Percentage | \(\frac{\text{Loss}}{\text{CP}} \times 100\). Always calculated on CP unless otherwise specified. |
Additional Information: Understanding Percentages in CP and SP Problems
Problems involving percentages of Cost Price (CP) and Selling Price (SP) are common in profit and loss calculations. It's crucial to correctly set up the equation based on whether the percentage refers to CP or SP.
- When a percentage is given as "X% of CP", it means \( \frac{X}{100} \times \text{CP} \).
- When a percentage is given as "Y% of SP", it means \( \frac{Y}{100} \times \text{SP} \).
Equating these expressions allows you to find the relationship between CP and SP, which is the essential first step in calculating profit or loss percentages.
Remember that profit percentage is always calculated with respect to the Cost Price (CP) as the base, unless the question specifically asks for profit percentage on Selling Price (SP).
Important Questions from Profit and Loss
By selling an article for Rs. 2,200, a profit of 10% is earned. If the same article is sold for Rs. 2,600, then what will be the gain percentage?
By selling an article for Rs. 640, a person loses 15% of its selling price. At what price (in Rs.) should he sell it to gain 15% on its cost price?
The selling prices of articles A and B are the same. A is sold at a profit of 28 percent and B is sold at a loss of 24 percent. If the total selling price of the both articles is Rs. 48640, then what is the cost price of A and B, respectively ?
The cost price of an article is Rs. 2800. Profit as a percentage of selling price is 20 percent. What is the actual profit (in Rs.) ?
A trader bought 640 kg of rice. He sold a part of rice at 20% profit and the rest at 5% loss. He earned a profit of 15% in the entire transaction. What is the quantity (in kg) of rice that he sold at 5% loss?
