84 Is 42 Of What Number

84 is 42% of What Number? A Deep Dive into Percentage Calculations

Finding the whole from a percentage and a part is a fundamental concept in mathematics, with applications spanning various fields from finance and statistics to everyday life. This article explores how to solve the problem "84 is 42% of what number?" We'll delve into the underlying principles, provide multiple methods for solving this type of problem, and offer practical examples to solidify your understanding. We'll also explore the broader context of percentage calculations and their importance.

Understanding the Basics of Percentages

Before we tackle the specific problem, let's refresh our understanding of percentages. A percentage is simply a fraction expressed as a part of 100. For instance, 42% can be written as 42/100 or 0.42. The key to understanding percentage problems lies in recognizing the relationship between the part, the whole, and the percentage.

Key Terms:

• Part: This is the portion of the whole that we are considering. In our problem, the part is 84.
• Whole: This is the total amount or the 100% value we are trying to find. This is the unknown in our problem.
• Percentage: This is the ratio of the part to the whole, expressed as a percentage. In our problem, the percentage is 42%.

Method 1: Using the Formula

The most straightforward way to solve "84 is 42% of what number?" is to use the standard percentage formula:

Part = Percentage × Whole

We can rearrange this formula to solve for the whole:

Whole = Part / Percentage

Now let's plug in the values from our problem:

• Part = 84
• Percentage = 42% = 0.42 (Remember to convert the percentage to a decimal by dividing by 100)

Whole = 84 / 0.42

Performing the calculation:

Whole = 200

Therefore, 84 is 42% of 200.

Method 2: Using Proportions

Another effective approach involves setting up a proportion. A proportion expresses the equality of two ratios. We can represent the problem as:

84/x = 42/100

Where 'x' represents the whole number we're trying to find.

To solve this proportion, we can cross-multiply:

84 × 100 = 42 × x

8400 = 42x

Now, divide both sides by 42 to isolate 'x':

x = 8400 / 42

x = 200

Again, we find that 84 is 42% of 200.

Method 3: Working with Decimals

This method involves converting the percentage directly to a decimal and setting up an equation. We already know that 42% is equal to 0.42. We can express the problem as:

0.42 * x = 84

To solve for x, divide both sides of the equation by 0.42:

x = 84 / 0.42

x = 200

This method provides the same result: 84 is 42% of 200.

Real-World Applications of Percentage Calculations

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

Finance:

• Calculating interest: Banks and other financial institutions use percentages to calculate interest on loans and savings accounts.
• Analyzing investments: Investors use percentages to track the performance of their investments and assess their returns.
• Determining discounts: Sales and discounts are often expressed as percentages, allowing consumers to easily calculate the price reduction.
• Tax calculations: Taxes are calculated as a percentage of income or the value of goods and services.

Statistics:

• Analyzing data: Percentages are used to present data in a concise and easily understandable format. For example, survey results are often presented as percentages.
• Calculating probabilities: Percentages play a significant role in expressing probabilities and likelihoods.

Everyday Life:

• Calculating tips: Restaurant tips are usually calculated as a percentage of the bill.
• Understanding sales: Sales and discounts are common in retail, and understanding percentages helps consumers make informed decisions.
• Comparing prices: Percentages can be used to compare the prices of different products.

Expanding Your Understanding: More Complex Percentage Problems

While the problem "84 is 42% of what number?" is relatively straightforward, percentage problems can become more complex. Consider these variations:

• Finding the percentage: What percentage of 200 is 84? (This would involve using the formula: Percentage = (Part / Whole) × 100)
• Finding the part: What is 42% of 250? (This would involve direct multiplication: Part = Percentage × Whole)
• Problems involving multiple percentages: A product is discounted by 20% and then by an additional 10%. What is the final price? (This requires a multi-step calculation, accounting for the sequential discounts.)

Mastering these basic percentage calculations forms the foundation for tackling more advanced problems involving compound interest, discounts, tax calculations, and other real-world applications.

Tips for Success with Percentage Calculations

• Convert percentages to decimals: Always convert percentages to decimals (by dividing by 100) before performing calculations. This simplifies the arithmetic.
• Identify the known and unknown values: Carefully identify the part, the whole, and the percentage in the problem. This will help you choose the correct formula or method.
• Use a calculator: For complex calculations or when dealing with larger numbers, a calculator can be a valuable tool. However, understanding the underlying principles is always crucial.
• Check your work: After solving a percentage problem, check your answer to ensure it makes logical sense within the context of the problem.

Conclusion

The problem "84 is 42% of what number?" serves as an excellent introduction to the world of percentage calculations. By understanding the underlying principles and applying the appropriate formulas or methods, you can confidently solve a wide range of percentage problems. Remember to practice regularly, explore different problem types, and apply your knowledge to real-world scenarios to solidify your understanding and improve your problem-solving skills. Proficiency in percentage calculations is a valuable skill with broad applications in various fields.