48 Is 75 Percent Of What Number

48 is 75 Percent of What Number: A Comprehensive Guide to Percentage Calculations

Solving percentage problems is a fundamental skill applicable across various fields, from everyday budgeting and shopping to complex scientific calculations and financial analysis. Understanding how to determine the original number when given a percentage and its corresponding value is crucial for accurate computations and informed decision-making. This article will delve deep into solving the problem: "48 is 75 percent of what number?", providing a detailed explanation of the process, exploring different approaches, and offering practical examples to solidify your understanding.

Understanding the Fundamentals of Percentages

Before we tackle the specific problem, let's review the basic concepts of percentages. A percentage is simply a fraction expressed as a part of 100. The symbol "%" represents "per hundred." For example, 75% means 75 out of 100, which can also be written as the fraction 75/100 or the decimal 0.75.

Key Terms:

• Percentage: The ratio of a part to a whole, expressed as a number out of 100.
• Base: The whole quantity or the original number. This is the number we are trying to find in our problem.
• Part: The portion of the base represented by the percentage. In our problem, the part is 48.

Method 1: Using the Percentage Formula

The most straightforward method to solve "48 is 75 percent of what number?" is to use the standard percentage formula:

Part = Percentage × Base

In our problem:

• Part = 48
• Percentage = 75% = 0.75 (Remember to convert the percentage to a decimal by dividing by 100)
• Base = x (This is the unknown number we need to find)

Substituting these values into the formula, we get:

48 = 0.75 × x

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

x = 48 / 0.75

x = 64

Therefore, 48 is 75 percent of 64.

Method 2: Using Proportions

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

48/x = 75/100

This proportion states that the ratio of the part (48) to the whole (x) is equal to the ratio of the percentage (75) to 100.

To solve for x, we can cross-multiply:

48 × 100 = 75 × x

4800 = 75x

Now, divide both sides by 75:

x = 4800 / 75

x = 64

Again, we find that 48 is 75 percent of 64.

Method 3: Using the Decimal Equivalent

As mentioned earlier, 75% is equivalent to the decimal 0.75. We can directly use this decimal to solve the problem:

If 0.75 of a number is 48, then the number (x) can be found by dividing 48 by 0.75:

x = 48 / 0.75

x = 64

This method offers a concise and efficient solution.

Practical Applications and Real-World Examples

The ability to solve percentage problems like this one has numerous real-world applications:

• Sales and Discounts: Imagine a store offering a 25% discount on an item. If the discounted price is $60, you can use this method to find the original price.
• Taxes and Interest: Calculating taxes, interest earned on savings, or interest paid on loans frequently involves percentage calculations.
• Surveys and Statistics: Analyzing survey results or understanding statistical data often necessitates working with percentages.
• Scientific Calculations and Data Analysis: Many scientific experiments and data analysis procedures rely on percentage calculations for accurate interpretations.
• Financial Planning and Budgeting: Budgeting, calculating savings rates, and determining investment returns all utilize percentages.

Expanding on Percentage Problems: Variations and Challenges

While the problem "48 is 75 percent of what number?" provides a clear illustration of percentage calculation, let's explore some variations and challenges:

• Finding the Percentage: Instead of finding the base, you might be asked to find the percentage that one number represents of another. For example: "What percentage of 80 is 20?"
• Finding the Part: You may need to determine a specific percentage of a given number. For instance: "What is 20% of 150?"
• Complex Percentage Problems: Some problems involve multiple percentages or sequential percentage changes, requiring a more complex approach and a deeper understanding of percentage principles. For example, calculating the final price after applying multiple discounts successively.
• Percentage Increase and Decrease: Problems often involve percentage increases or decreases, demanding careful attention to the initial value and the change.

Troubleshooting and Common Mistakes

Several common mistakes can occur when solving percentage problems:

• Incorrect Conversion of Percentage to Decimal: Remember to divide the percentage by 100 before using it in calculations.
• Mathematical Errors: Carefully check your calculations to avoid simple arithmetic errors.
• Incorrect Interpretation of the Problem: Ensure you clearly understand what the problem is asking you to find – the base, the part, or the percentage.
• Confusing Percentage Increase/Decrease: Understand the difference between adding or subtracting a percentage from the original value.

Conclusion: Mastering Percentage Calculations

Solving percentage problems is a crucial skill with far-reaching applications. The ability to confidently and accurately tackle problems like "48 is 75 percent of what number?" empowers you to handle numerous situations in your personal and professional life. By mastering different methods—using the percentage formula, proportions, or the decimal equivalent—you equip yourself with the tools to analyze data, interpret results, and make informed decisions. Continuously practicing and tackling variations of percentage problems will further solidify your understanding and improve your proficiency. Remember to review the fundamental concepts, practice regularly, and pay close attention to detail to avoid common mistakes. The more you practice, the more effortless and intuitive these calculations will become.