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Understanding percentage calculation is a fundamental skill that allows us to determine what a certain fraction of a whole is. For example, when you hear that something is reduced by 25%, it means that it is 25% less than the original amount. To calculate a percentage, you multiply the original value by the percentage (expressed as a decimal value). For instance, to find 5% of a number, you multiply it by 0.05. This foundational prealgebra concept is useful in a wide range of real-world scenarios, such as calculating discounts, interest rates, and yes, salary raises.

Using percentages, we can easily compare increases or decreases across different values. While the percentage might be consistent (like a 5% salary raise), the actual impact in terms of real numbers can differ, especially if the original values are not the same. This highlights the importance of looking beyond percentages and understanding the actual numbers involved.

Computing a salary raise based on a percentage involves a straightforward mathematical operation. When an employee receives a salary raise of a certain percentage, you calculate the actual amount of the raise by multiplying the current salary by the percentage expressed as a decimal. Then, you add the result to the initial salary to find the new salary. For example, if an employee earning \(42,000 receives a 5% raise:

\begin{align*} \text{Raise amount} &= \text{Original Salary} \times \text{Percentage as a decimal} \ &= \)42,000 \times 0.05 \ &= \(2,100. \text{New Salary} &= \text{Original Salary} + \text{Raise amount} \ &= \)42,000 + \(2,100 \ &= \)44,100.\text{

Apply the same method to compute new salaries after the raises for other employees and compare them to determine who has the highest salary post-raise.}

Comparing numbers is a core mathematical concept that includes identifying which of a set of numbers is the largest or smallest, arranging numbers in ascending or descending order, or determining equality. In real-world terms, comparing numbers can help us decide the best deal, the most efficient option, or as in our case, which employee has the highest salary after a raise.

To effectively compare numbers, align their places value by value, and start comparing from the highest place value to the lowest. For example, when comparing whole numbers like salaries, you would compare the tens of thousands, then the thousands, and so on. In many situations, especially involving percentages and finances, having a clear understanding of how to compare numbers accurately is crucial for making informed decisions.

Prealgebra serves as the groundwork for building mathematical understanding that is necessary for algebra and beyond. It includes concepts such as operations with whole numbers, decimals, fractions, and the basics of geometry. Working through problems using these prealgebra concepts provides the tools for solving more complex equations in the future.

The concepts of percentage and number comparison fall under prealgebra and are critical for students to master before moving on to algebra. These topics are not only essential academically but are also incredibly practical. They are frequently applied in calculating finances, such as determining changes in wages, which is what our salary raise problem entails. By mastering these foundational concepts, students can approach more intricate mathematical problems with confidence and a solid understanding.