11.85 Is 75 Of What Number

11.85 is 75% 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 and financial calculations. This article delves deep into how to solve the problem "11.85 is 75% of what number?" We'll explore multiple methods, providing a comprehensive understanding of the underlying concepts and demonstrating practical applications. Beyond the immediate solution, we'll equip you with the tools to confidently tackle similar percentage problems.

Understanding the Problem: Deconstructing the Percentage Equation

The core of this problem lies in understanding the relationship between percentages, parts, and wholes. The statement "11.85 is 75% of what number?" can be translated into a mathematical equation:

11.85 = 0.75 * x

Where:

• 11.85 represents the part (or percentage amount).
• 0.75 represents the percentage (75% expressed as a decimal; divide the percentage by 100).
• x represents the whole number we need to find.

Method 1: Algebraic Solution – Solving for the Unknown

This is the most straightforward approach, leveraging basic algebra to isolate and solve for the unknown variable (x).

1. Rewrite the equation: We start with our equation: 11.85 = 0.75 * x

2. Isolate x: To isolate x, divide both sides of the equation by 0.75:

   x = 11.85 / 0.75

3. Calculate the result: Performing the division gives us:

   x = 15.8

Therefore, 11.85 is 75% of 15.8.

Method 2: Using Proportions – A Ratio Approach

Proportions offer another effective way to solve percentage problems. We set up a proportion relating the part to the whole:

Part / Whole = Percentage / 100

Substituting our known values:

11.85 / x = 75 / 100

Now, we solve for x using cross-multiplication:

1. Cross-multiply: 11.85 * 100 = 75 * x

2. Simplify: 1185 = 75x

3. Solve for x: x = 1185 / 75

4. Calculate: x = 15.8

Again, we arrive at the solution: 11.85 is 75% of 15.8.

Method 3: The Unitary Method – Finding 1%

The unitary method involves first calculating the value of 1% and then scaling up to find the desired percentage.

1. Find 1%: If 75% is 11.85, then 1% would be:

   11.85 / 75 = 0.158

2. Find 100%: To find 100% (the whole), we multiply the value of 1% by 100:

   0.158 * 100 = 15.8

Therefore, 11.85 is 75% of 15.8.

Verifying the Solution

It's always good practice to verify your answer. We can do this by calculating 75% of 15.8:

0.75 * 15.8 = 11.85

This confirms our solution is correct.

Practical Applications of Percentage Calculations

Understanding percentage calculations has wide-ranging practical applications:

1. Finance and Budgeting:

• Calculating interest: Determining simple or compound interest earned on savings or owed on loans.
• Analyzing discounts: Calculating the final price after a percentage discount.
• Tracking expenses: Determining the percentage of income spent on different categories.
• Investment returns: Measuring the performance of investments in terms of percentage gains or losses.

2. Retail and Sales:

• Calculating markups: Determining the selling price of a product after adding a percentage markup to the cost price.
• Analyzing sales figures: Tracking sales growth or decline as percentages.
• Calculating discounts and sales tax: Applying percentages to determine final prices.

3. Science and Engineering:

• Determining concentration: Calculating the concentration of solutions in chemistry.
• Analyzing data: Representing data using percentages in statistical analysis.
• Calculating efficiency: Measuring the efficiency of processes or machines using percentages.

4. Everyday Life:

• Calculating tips: Determining the appropriate tip amount in restaurants.
• Understanding sales tax: Calculating the total cost of purchases, including sales tax.
• Analyzing surveys and polls: Interpreting results presented in percentages.

Beyond the Basics: Handling More Complex Percentage Problems

While the problem "11.85 is 75% of what number?" provides a solid foundation, real-world scenarios often involve more complexity. Here are some considerations:

• Percentage Increase and Decrease: These involve calculating the percentage change between two values. For example, finding the percentage increase from 100 to 125.
• Successive Percentage Changes: This involves applying multiple percentage changes sequentially. For example, a 10% increase followed by a 5% decrease.
• Percentage Points vs. Percentages: It's crucial to understand the difference. A change from 20% to 25% is a 5 percentage point increase, but a 25% increase relative to the original 20%.

Mastering these nuances will allow you to tackle a wider range of percentage-related problems.

Conclusion: Building Confidence in Percentage Calculations

Solving percentage problems, such as "11.85 is 75% of what number?", requires a clear understanding of the underlying concepts and a systematic approach. Whether using algebraic methods, proportions, or the unitary method, consistent practice and a thorough understanding of the different techniques will build your confidence and ability to solve a wide range of percentage problems encountered in various aspects of life. Remember to always check your work to ensure accuracy. With consistent practice, you'll become proficient in solving these types of problems, applying this valuable skill to numerous situations.