What Percent Of 75 Is 21

What Percent of 75 is 21? A Comprehensive Guide to Percentage Calculations

This article delves into the question: "What percent of 75 is 21?" We'll not only solve this specific problem but also explore the underlying concepts of percentages, providing you with the tools to tackle similar calculations confidently. We'll cover various methods, from simple formulas to more advanced techniques, ensuring a thorough understanding for all readers, regardless of their mathematical background. By the end, you'll be able to confidently calculate percentages in various contexts.

Understanding Percentages

Percentages are a fundamental concept in mathematics and everyday life. A percentage represents a fraction of 100. The term "percent" literally means "per hundred," indicated by the symbol "%". Understanding this foundational concept is crucial for solving percentage problems.

Key Terminology

Before diving into calculations, let's clarify some key terms:

• Percentage: A fraction or ratio expressed as a number out of 100.
• Base: The total or whole value upon which the percentage is calculated. In our example, 75 is the base.
• Part: The portion of the base that represents the percentage. In our example, 21 is the part.
• Rate: The percentage itself, often represented as a decimal or a fraction. This is what we need to find in our problem.

Method 1: Using the Formula

The most straightforward method to determine what percent of 75 is 21 involves using a simple formula:

(Part / Base) * 100 = Percentage

Let's plug in our values:

(21 / 75) * 100 = Percentage

1. Divide the part by the base: 21 / 75 = 0.28

2. Multiply the result by 100: 0.28 * 100 = 28

Therefore, 21 is 28% of 75.

Method 2: Setting up a Proportion

Another effective method involves setting up a proportion. A proportion is an equation stating that two ratios are equal. We can express the problem as:

21/75 = x/100

Where 'x' represents the unknown percentage.

To solve for 'x', we cross-multiply:

21 * 100 = 75 * x

2100 = 75x

Now, divide both sides by 75:

x = 2100 / 75 = 28

Again, this confirms that 21 is 28% of 75.

Method 3: Using Decimal Equivalents

This method leverages the understanding that percentages can be expressed as decimals. We can convert the percentage to a decimal by dividing by 100. For instance, 28% is equivalent to 0.28.

We can then set up an equation:

0.28 * 75 = x

Solving for x:

x = 21

This confirms that 28% of 75 is indeed 21. This method is useful for checking your answer using a different approach.

Practical Applications of Percentage Calculations

Understanding percentage calculations is crucial in various real-world scenarios. Here are a few examples:

• Sales and Discounts: Calculating discounts offered in stores. For example, a 20% discount on a $100 item.
• Financial Calculations: Determining interest rates on loans or investments.
• Data Analysis: Interpreting statistical data represented as percentages. For example, analyzing survey results or market share.
• Scientific Applications: Expressing experimental results or changes in quantities.

Advanced Percentage Problems and Solving Techniques

While the problem "What percent of 75 is 21?" is relatively straightforward, let's explore some more complex scenarios and techniques to address them:

Finding the Base

Sometimes, you know the percentage and the part but need to find the base. For example:

"20% of what number is 10?"

In this case, we can use the formula:

Base = (Part / Percentage) * 100

Substituting the values:

Base = (10 / 20) * 100 = 50

Therefore, 20% of 50 is 10.

Finding the Part

Conversely, you might know the percentage and the base but need to determine the part:

"What is 15% of 200?"

The formula here is:

Part = (Percentage / 100) * Base

Substituting the values:

Part = (15 / 100) * 200 = 30

Therefore, 15% of 200 is 30.

Troubleshooting Common Mistakes

Here are some common mistakes to watch out for when calculating percentages:

• Incorrectly applying the formula: Double-check that you're using the correct formula for the type of problem you're solving.
• Decimal errors: Carefully handle decimals during calculations to avoid rounding errors.
• Units: Ensure that all your units are consistent (e.g., using percentages and decimals consistently).
• Confusing Part, Base, and Percentage: Clearly identify which value is the part, the base, and the percentage to avoid substitution errors.

Conclusion

This comprehensive guide has explored various methods for calculating percentages, specifically addressing the question, "What percent of 75 is 21?" We've covered the fundamental concepts, practical applications, and advanced techniques, along with common errors to avoid. By mastering these methods, you can confidently tackle a wide range of percentage problems in your academic pursuits, professional life, and everyday decision-making. Remember to practice regularly to build your proficiency and always double-check your work to avoid mistakes. Understanding percentages is a valuable skill that will serve you well across various disciplines.