85 Of What Number Is 68

Decoding the Percentage Puzzle: 85% of What Number is 68?

Finding the whole when you know a percentage and a part is a common mathematical problem encountered in various real-world scenarios, from calculating sales tax and discounts to determining the original price of a sale item. This article will delve into the solution of the question: "85% of what number is 68?", providing a step-by-step approach, exploring different methods of solution, and presenting practical applications to solidify your understanding.

Understanding the Problem

The core of this problem lies in understanding the relationship between percentages, parts, and the whole. We're given that 68 represents 85% of an unknown number. Our goal is to find this unknown number, often referred to as the "whole" or "base". This requires a clear grasp of percentage calculations and the ability to manipulate algebraic equations.

Method 1: Using Algebraic Equations

This method involves setting up an algebraic equation to represent the given information and solving for the unknown variable. Let's denote the unknown number as 'x'. We can translate the problem into an equation:

0.85x = 68

Here, 0.85 represents 85% as a decimal (85/100). To solve for 'x', we follow these steps:

1. Divide both sides of the equation by 0.85: This isolates 'x' on one side of the equation.

   x = 68 / 0.85

2. Perform the division: Using a calculator or manual long division, we find the value of 'x'.

   x ≈ 80

Therefore, 85% of 80 is 68.

Method 2: Using Proportions

Proportions provide another effective approach to solving percentage problems. We can set up a proportion using the known percentage and the corresponding part:

85/100 = 68/x

This proportion states that the ratio of 85 to 100 is equal to the ratio of 68 to the unknown number 'x'. To solve for 'x', we cross-multiply:

85x = 68 * 100

85x = 6800

Now, divide both sides by 85:

x = 6800 / 85

x ≈ 80

This confirms our previous result: 85% of 80 is 68.

Method 3: Working Backwards from the Percentage

This intuitive approach involves understanding that if 68 represents 85%, then we can find the value of 1% by dividing 68 by 85:

68 / 85 ≈ 0.8

This means 1% of the unknown number is approximately 0.8. To find the whole number (100%), we simply multiply this value by 100:

0.8 * 100 = 80

Again, we arrive at the solution: 85% of 80 is 68. This method is particularly helpful for mental calculations or estimations.

Practical Applications

Understanding percentage calculations is crucial in numerous real-life situations:

• Sales and Discounts: Imagine a store offering a 15% discount on an item. If the discounted price is $76.50, you can use the same methods above to determine the original price.
• Taxes: Calculating sales tax or income tax often involves finding a percentage of a larger amount.
• Finance: Interest calculations on loans or investments heavily rely on percentage computations.
• Data Analysis: Percentages are commonly used to represent proportions and trends in data analysis.
• Surveys and Polls: Survey results are frequently presented as percentages, reflecting the proportion of responses within a particular category.
• Grading Systems: Many academic grading systems utilize percentages to represent a student's performance.

Expanding the Understanding: Beyond the Basics

While the problem "85% of what number is 68?" provides a fundamental understanding of percentage calculations, expanding this understanding involves considering more complex scenarios:

• Multiple Percentages: Imagine calculating the final price after applying multiple discounts or taxes sequentially. Understanding the order of operations and the impact of compounding percentages is essential.
• Percentage Increases and Decreases: These involve calculating the new value after increasing or decreasing a base value by a certain percentage. This is frequently encountered in economic and financial contexts.
• Percentage Change: Determining the percentage change between two values is a crucial skill for comparing data over time or across different categories. This involves calculating the difference between the two values, dividing it by the original value, and multiplying by 100.
• Compound Interest: This involves calculating interest on both the principal amount and accumulated interest from previous periods. Compound interest calculations are complex but extremely important in finance and investments.

Tips for Mastering Percentage Problems

• Practice Regularly: Consistent practice is key to mastering percentage calculations. Try solving various problems with different values and percentages.
• Understand the Concepts: Don't just memorize formulas; understand the underlying concepts of percentages, proportions, and algebraic equations.
• Use Multiple Methods: Experiment with different methods (algebraic equations, proportions, working backwards) to find the approach that suits you best.
• Check Your Answers: Always check your solutions to ensure accuracy. You can do this by substituting your answer back into the original problem or using a different method to verify your results.
• Utilize Online Resources: There are numerous online resources and calculators available to assist you with percentage calculations and to help you understand the concepts better.

Conclusion

The question, "85% of what number is 68?" serves as an excellent example of a common percentage problem. By utilizing algebraic equations, proportions, or the method of working backwards, we consistently find the answer to be 80. This seemingly simple problem underscores the importance of understanding percentage calculations, a fundamental skill applicable in diverse real-world scenarios. Through consistent practice and a clear grasp of the underlying mathematical principles, you can master percentage problems and confidently tackle more complex percentage-based calculations in various fields. Remember, understanding the "why" behind the calculations is as important as knowing "how" to solve them. This understanding will equip you to confidently tackle various percentage-related challenges, laying a solid foundation for success in mathematics and beyond.