48 Is 75 Of What Number

48 is 75% of What Number? A Deep Dive into Percentage Calculations

Solving percentage problems is a fundamental skill with wide-ranging applications in everyday life, from calculating discounts and taxes to understanding statistics and financial data. This article will explore the question, "48 is 75% of what number?", providing a step-by-step solution, discussing different approaches, and highlighting the broader implications of percentage calculations. We'll also delve into related concepts to solidify your understanding.

Understanding Percentages

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

Understanding this fundamental relationship is key to solving percentage problems. We can express any percentage as a fraction or a decimal, and vice-versa. This flexibility allows us to use different methods to solve problems involving percentages.

Method 1: Using the Percentage Formula

The most straightforward method to solve "48 is 75% of what number?" is using the basic percentage formula:

Part = Percentage × Whole

In this problem:

• Part = 48 (the given part of the whole)
• Percentage = 75% = 0.75 (expressed as a decimal)
• Whole = x (the unknown number we need to find)

Substituting these values into the formula, we get:

48 = 0.75 × x

To solve for x, we divide both sides of the equation by 0.75:

x = 48 / 0.75

x = 64

Therefore, 48 is 75% of 64.

Method 2: Using Proportions

Another effective method is to set up a proportion. A proportion is an equation stating that two ratios are equal. We can set up a proportion based on the given information:

48/x = 75/100

This proportion states that the ratio of 48 to the unknown number (x) is equal to the ratio of 75 to 100 (75%). To solve for x, we can cross-multiply:

48 × 100 = 75 × x

4800 = 75x

x = 4800 / 75

x = 64

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

Method 3: Working with Fractions

Since 75% is equivalent to the fraction ¾, we can also solve the problem using fractions:

48 = (¾) × x

To isolate x, we multiply both sides by the reciprocal of ¾, which is ⁴⁄₃:

x = 48 × (⁴⁄₃)

x = (48 × 4) / 3

x = 192 / 3

x = 64

This method reinforces the interchangeable nature of percentages, decimals, and fractions in solving percentage problems.

Real-World Applications

Understanding percentage calculations is crucial in many real-world scenarios:

• Sales and Discounts: Calculating discounts on sale items. For example, if a $80 item is discounted by 25%, you can use these methods to find the final price.

• Taxes and Tips: Determining the amount of tax or tip to add to a bill.

• Financial Calculations: Calculating interest earned on savings accounts or interest paid on loans.

• Data Analysis: Interpreting statistical data presented as percentages. For example, understanding market share percentages or survey results.

• Recipe Scaling: Adjusting ingredient quantities in recipes based on the number of servings.

Advanced Percentage Problems

While the problem "48 is 75% of what number?" is relatively straightforward, more complex percentage problems require multiple steps. Let's consider some variations:

Finding the Percentage Increase or Decrease

Suppose a value increased from 60 to 75. To find the percentage increase, we first calculate the difference: 75 - 60 = 15. Then we divide this difference by the original value and multiply by 100: (15/60) * 100 = 25%. Therefore, there's a 25% increase.

Similarly, if a value decreased from 80 to 60, the percentage decrease is calculated as: (20/80) * 100 = 25%.

Successive Percentage Changes

When dealing with successive percentage changes (e.g., a 10% increase followed by a 10% decrease), the final result is not simply the sum or difference of the percentages. You must calculate each change sequentially to find the overall effect.

Tips for Solving Percentage Problems

• Identify the known and unknown values: Clearly define what you know (part, percentage, or whole) and what you need to find.

• Choose the appropriate method: Select the method (formula, proportion, or fractions) that you find most comfortable and efficient.

• Convert percentages to decimals or fractions: This simplifies the calculations.

• Check your work: Always verify your answer to ensure it makes sense in the context of the problem.

Conclusion

Solving percentage problems, including finding the whole when given a part and a percentage, is a valuable mathematical skill applicable across various fields. By mastering the different approaches – using the percentage formula, proportions, or fractions – you can confidently tackle a range of percentage calculations in your daily life and professional endeavors. Remember to practice regularly to reinforce your understanding and develop proficiency in this essential skill. The problem "48 is 75% of what number?" serves as a excellent starting point for understanding this fundamental aspect of mathematics. Through repeated practice and application, you’ll strengthen your ability to confidently navigate the world of percentages.