At Lakeside Park restaurant, servers earn an average of $$\$ 840$$ less per month than chefs. The restaurant employs 4 chefs and 18 servers. Let \(c\) represent the average monthly pay of a chef. Which of the following functions correctly shows the relationship between the monthly payroll and the wages of these employees? A. \(4 c+18 c-840\) B. \(4(c-840)+18 c\) C. \(22 c-840\) D. \(4 c+18(c-840)\)

Algebraic expressions are a cornerstone of algebra and provide a way to represent real-world problems with mathematical symbols and numbers. An expression can include variables, constants, and operating symbols, such as addition and multiplication. In the context of our GED math practice exercise, we defined the variable 'c' to represent the average monthly pay of a chef. The average monthly pay of a server, which is $840 less than that of a chef, is represented as \(c - 840\). It's important to understand that algebraic expressions can be manipulated according to the standard rules of algebra to simplify them or to solve for variables.

When we encounter algebraic expressions like the server's pay \(c - 840\), it's a reminder that algebra serves as a tool for translating real-world scenarios into a mathematical language that can be analyzed and solved. This is critical in fields ranging from business to engineering, making it a valuable skill for students to master.

Solving equations is about finding the value of the variable that makes the equation true. In our exercise, however, we are not solving for 'c' directly, but creating an equation that represents the total monthly payroll. An equation sets two expressions equal to each other, which allows us to establish relationships and solve for unknowns. For example, if we had to solve for 'c' given a total payroll, we would set up an equation with our established expression and manipulate it to isolate 'c'.

In our case, though, we are piecing together the payroll for chefs and servers to form the correct equation. Understanding how to combine and simplify algebraic expressions is fundamental to solving equations. This also involves recognizing patterns and common mistakes, such as negative signs and the distributive property, which are often tested in GED math practice exercises.

Monthly payroll calculation is a practical application of algebra that businesses use to manage their finances. In the exercise from the textbook, we use algebra to calculate the payroll for chefs and servers collectively. By multiplying the number of chefs by their average pay, and the number of servers by their average pay (which is less than that of the chefs), we get the total monthly payroll.

Payroll calculations often require careful attention to precise details – such as the differences in pay and the number of employees – to ensure the accuracy of the result. This practice can extend beyond simple calculations to more complex issues such as incorporating taxes, benefits, and overtime. Understanding the mathematics behind payroll can help students not only solve textbook problems but also manage real-life financial situations effectively.