48 Is 120 Of What Number

48 is 120% of What Number? A Comprehensive Guide to Percentage Calculations

This article delves into the solution of the mathematical problem: "48 is 120% of what number?" We'll not only provide the answer but also explain the underlying concepts of percentages, demonstrating different methods for solving similar problems. Understanding percentages is crucial in various aspects of life, from finance and budgeting to sales and statistics. This guide aims to equip you with the knowledge and skills to confidently tackle percentage calculations.

Understanding Percentages

Before diving into the solution, let's solidify our understanding of percentages. A percentage is a fraction or ratio expressed as a number out of 100. The symbol "%" represents "per cent," meaning "out of one hundred." For example, 50% means 50 out of 100, which is equivalent to the fraction 50/100 or the decimal 0.5.

Percentages are used to express proportions or ratios, making it easier to compare and understand different quantities. They are prevalent in various contexts, including:

• Finance: Interest rates, discounts, taxes, and investment returns are often expressed as percentages.
• Statistics: Data representation and analysis frequently utilize percentages to illustrate proportions within a dataset.
• Sales and Marketing: Sales figures, discounts, and market share are commonly expressed as percentages.
• Everyday Life: Tips, grades, and survey results are often given as percentages.

Method 1: Using Algebraic Equations

This is the most direct and widely used method to solve percentage problems. We can translate the problem "48 is 120% of what number?" into an algebraic equation:

48 = 1.20 * x

Where:

• 48 represents the final value.
• 1.20 represents 120% (expressed as a decimal by dividing by 100).
• x represents the unknown number we are trying to find.

To solve for 'x', we perform the following steps:

1. Divide both sides of the equation by 1.20: This isolates 'x' on one side of the equation.

   48 / 1.20 = x

2. Calculate the result:

   x = 40

Therefore, 48 is 120% of 40.

Method 2: Using Proportions

This method involves setting up a proportion, relating the given percentage to the whole. We can represent the problem as:

48/x = 120/100

Where:

• 48 is the part.
• x is the whole (the number we're trying to find).
• 120/100 represents 120% as a fraction.

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

1. Cross-multiply: Multiply the numerator of the first fraction by the denominator of the second fraction and vice-versa.

   48 * 100 = 120 * x

2. Simplify:

   4800 = 120x

3. Solve for x: Divide both sides by 120:

   x = 4800 / 120

4. Calculate the result:

   x = 40

Again, we find that 48 is 120% of 40.

Method 3: Working with Fractions

This method involves converting the percentage to a fraction and then solving the equation. 120% can be expressed as the fraction 120/100, which simplifies to 6/5. The problem can then be written as:

48 = (6/5) * x

To solve for x:

1. Multiply both sides by the reciprocal of 6/5, which is 5/6: This cancels out the fraction on the right side.

   (5/6) * 48 = x

2. Simplify and calculate:

   x = (5 * 48) / 6 = 240 / 6 = 40

Once again, we arrive at the answer: 48 is 120% of 40.

Practical Applications of Percentage Calculations

The ability to solve percentage problems is invaluable in numerous real-world scenarios. Here are a few examples:

• Calculating Sales Tax: If a product costs $80 and the sales tax is 6%, you would calculate the tax amount as 0.06 * $80 = $4.80, resulting in a total cost of $84.80.

• Determining Discounts: A 25% discount on a $100 item means you save 0.25 * $100 = $25, leaving a final price of $75.

• Analyzing Financial Statements: Understanding percentages helps in interpreting financial statements, comparing different financial ratios, and tracking performance over time. For example, if a company's profit margin is 15%, it means that 15% of its revenue is profit.

• Interpreting Statistical Data: Percentages are vital in understanding and interpreting statistical data, such as survey results, election polls, and market research findings.

Common Mistakes to Avoid

When working with percentages, be mindful of these common pitfalls:

• Incorrect Decimal Conversion: Ensure you accurately convert percentages to decimals before performing calculations. Remember to divide the percentage by 100. For example, 25% is 0.25, not 25.

• Mixing Percentages and Decimals: Avoid mixing percentages and decimals in the same equation without proper conversion.

• Rounding Errors: Be aware of rounding errors, especially when dealing with multiple calculations. It's best to keep as many decimal places as possible during intermediate calculations and round only at the final answer.

• Incorrect Formula Application: Make sure you are using the correct formula based on the specific problem.

Further Practice Problems

To further solidify your understanding, try solving these practice problems:

1. 60 is 75% of what number?
2. What percentage of 80 is 20?
3. If a price is increased by 10% to $110, what was the original price?
4. A store offers a 30% discount on an item originally priced at $50. What is the sale price?

By mastering these percentage calculation methods and understanding the underlying concepts, you can confidently approach various mathematical problems and real-world situations requiring percentage calculations. Remember to break down the problem, identify the known values, and use the appropriate method to solve for the unknown variable. Regular practice will enhance your proficiency and confidence in tackling any percentage-related challenge.