Understanding Equivalences in Percentages and Algebraic Expressions

When dealing with mathematical expressions and percentages, it is crucial to maintain clear and logical representations to avoid confusing statements. In this article, we will explore the concept of percentage calculations and algebraic manipulations, and provide a detailed explanation of how to solve a specific problem: When 80 is equivalent to 100 of what number?

Let's start with a common misunderstanding: 120 80 is not a valid equation. The statement is incorrect and should not be used. If you meant to say that 120 of some number x is equal to 80, you would interpret it in terms of algebraic operations. Let's break it down step-by-step.

Step-by-Step Solution

1. **Understanding the Expression** In your problem, you are looking for the value of x where 120% of x equals 80. To solve this, you can transform the percentage to a fraction and set up an equation.

2. **Translating to Mathematical Operations** 120% can be written as ( frac{120}{100} ), which simplifies to ( 1.2 ). Hence, ( 1.2 cdot x 80 ). To find x, divide both sides by 1.2:

[ x frac{80}{1.2} 66overline{6} ]

So, ( 120% ) of ( 66overline{6} ) is equal to 80.

Another Approach

3. **Alternative Method** If you are given that 80 of a number is 100, and you want to find the number, you can follow these steps:

a. Start by assuming the number is x.

b. Write the equation: ( 80% cdot x 100 ).

c. Convert 80% to a fraction: ( frac{80}{100} frac{4}{5} ).

d. Solve for x by multiplying both sides by the reciprocal of ( frac{4}{5} ), which is ( frac{5}{4} ):

[ x 100 cdot frac{5}{4} 125 ]

Therefore, 80% of 125 is 100.

General Explanation

4. **General Explanation** In general, when you are dealing with percentages and algebraic expressions, it's important to understand the relationship between the values. In the examples provided, the key is to correctly interpret and translate the given statements into mathematical operations, and then solve them step by step.

5. **Conclusion** In conclusion, the number that, when 100% of it, equals 125, given that 80% of that number is 100, is 125. This problem demonstrates the importance of clear and consistent mathematical representation and manipulation.

By practicing such problems, you can enhance your understanding of percentages and algebraic expressions, making sure to avoid common pitfalls and ensuring accurate solutions.