15 Is 75 Of What Number

15 is 75% of What Number? A Comprehensive Guide to Percentage Calculations

Finding the whole number when you know a percentage and its corresponding part is a fundamental skill in mathematics with broad applications in various fields. This article delves into the detailed process of solving the problem "15 is 75% of what number?". We'll not only provide the solution but also explain the underlying concepts, explore different methods of calculation, and offer practical examples to solidify your understanding. This guide aims to equip you with the knowledge to tackle similar percentage problems confidently.

Understanding Percentages

Before diving into the calculation, let's refresh our understanding of percentages. A percentage is a fraction or ratio expressed as a number out of 100. The symbol "%" represents "per cent" or "out of 100". For instance, 75% means 75 out of 100, which can also be written as the fraction 75/100 or the decimal 0.75.

Understanding this foundational concept is crucial for solving percentage problems. We consistently convert percentages into decimals or fractions to perform calculations efficiently.

Method 1: Using the Percentage Formula

The most straightforward method involves using the basic percentage formula:

Part = Percentage × Whole

In our problem, "15 is 75% of what number?", we know:

• Part: 15
• Percentage: 75% or 0.75 (converting the percentage to a decimal)
• Whole: This is what we need to find (let's represent it with 'x')

Substituting these values into the formula, we get:

15 = 0.75 × x

To solve for 'x', we need to isolate it. We can do this by dividing both sides of the equation by 0.75:

x = 15 / 0.75

x = 20

Therefore, 15 is 75% of 20.

Method 2: Using Proportions

Another effective approach is to use proportions. Proportions represent the equality of two ratios. We can set up a proportion to represent the problem:

15/x = 75/100

This proportion states that the ratio of the part (15) to the whole (x) is equal to the ratio of the percentage (75) to 100.

To solve this proportion, we can cross-multiply:

15 × 100 = 75 × x

1500 = 75x

Now, divide both sides by 75:

x = 1500 / 75

x = 20

Again, we find that 15 is 75% of 20.

Method 3: Working Backwards from the Percentage

This method involves thinking about the problem in reverse. If 75% is equal to 15, we can find the value of 1% by dividing 15 by 75:

1% = 15 / 75 = 0.2

Since 1% represents 0.2, we can find 100% (the whole number) by multiplying 0.2 by 100:

100% = 0.2 × 100 = 20

Therefore, 15 is 75% of 20.

Practical Applications: Real-World Examples

Percentage calculations are essential in various real-life situations. Here are a few examples illustrating their practical applications:

1. Sales and Discounts:

Imagine a store offering a 25% discount on an item originally priced at $80. To find the discounted price, we calculate 25% of $80:

Discount = 0.25 × $80 = $20

Discounted price = $80 - $20 = $60

2. Calculating Tax:

If the sales tax in your area is 6%, and you purchase an item for $50, the tax amount would be:

Tax = 0.06 × $50 = $3

Total cost = $50 + $3 = $53

3. Analyzing Financial Data:

Percentage changes are frequently used to track financial performance. For example, if a company's profit increased from $100,000 to $120,000, the percentage increase would be:

Percentage increase = [(120,000 - 100,000) / 100,000] × 100% = 20%

4. Determining Grades and Scores:

In educational settings, percentages are used extensively. If a student scores 30 out of 40 marks on a test, their percentage score would be:

Percentage score = (30/40) × 100% = 75%

Troubleshooting Common Errors

While percentage calculations are relatively straightforward, some common errors can occur:

• Incorrect conversion of percentages to decimals: Remember to divide the percentage by 100 before using it in calculations. For example, 25% should be converted to 0.25, not 25.

• Mixing up the parts of the formula: Ensure you correctly identify the "part," "percentage," and "whole" in the problem.

• Calculation errors: Carefully perform each step of the calculation to avoid arithmetic mistakes. Use a calculator if needed to enhance accuracy.

• Misinterpreting the question: Read the problem carefully to understand what is being asked. Pay close attention to the wording.

Expanding Your Knowledge: Further Exploration

Once you've mastered the basics, you can explore more advanced percentage calculations, such as:

• Compound interest: This involves calculating interest on both the principal amount and accumulated interest.

• Percentage change: This includes calculating percentage increase or decrease.

• Percentage points: This represents the arithmetic difference between two percentages, unlike percentage change, which indicates relative change.

• Solving more complex percentage word problems: These problems often involve multiple steps and require a strong understanding of the fundamental concepts.

Conclusion

Solving the problem "15 is 75% of what number?" highlights the importance of understanding percentage calculations. By mastering these fundamental skills, you gain a powerful tool applicable across diverse fields. Whether dealing with discounts, taxes, financial data, or grading systems, the ability to work with percentages efficiently and accurately is an invaluable asset. Remember to practice consistently, and you'll soon find these calculations second nature. Utilize the methods described above, practice regularly with different examples, and don't hesitate to seek clarification when needed. This consistent effort will solidify your understanding and build confidence in tackling percentage problems of any complexity.