Problem 77 - College Algebra and Calculus: An Applied Approach

Question

Job Choices You are offered two different jobs. Company A offers an annual salary of \(\$ 30,000\) plus a year-end bonus of \(2.5 \%\) of your total sales. Company \(\mathrm{B}\) offers a salary of \(\$ 24,000\) plus a year-end bonus of \(6.5 \%\) of your total sales. What is the amount you must sell in one year to eam the same salary working for either company?

Step-by-Step Solution

Answer

You must sell $150,000 in one year to earn the same salary working for either company.

Step 1: Understand the Problem

Identify the components of the salary from each company. For Company A, the salary is made up of a fixed component of $30,000 and a variable component of 2.5% of total sales. For Company B, the fixed component is $24,000 and the variable component is 6.5% of total sales. The objective is to find the value of the total sales such that the salaries from both companies are equal.

Step 2: Set Up the Equation

Turn the problem setup into an equation. Set the total salary from Company A equal to the total salary from Company B. This means, $30,000 + 0.025x = $24,000 + 0.065x, where \(x\) represents the total sales. Here, the percentage numbers are converted into decimal for calculation. The fixed salary components are on one side and the sales percentage components are on the other side.

Step 3: Solve the Equation

Rearrange the equation to isolate \(x\) by itself. First subtract $24,000 and subtract 0.025x from both sides, yielding: 0.065x - 0.025x = $30,000 - $24,000. Simplifying both sides obtains 0.04x = $6,000. To obtain the value of \(x\), divide both sides by 0.04 to get \(x = $6,000 / 0.04\).

Step 4: Interpret the Solution

Evaluate the value of \(x\) (i.e., total sales) in the context of the problem. \(x\) represents the total sales necessary to earn the same salary for either company. Plug in the value of \(x\) into the salary structures for both companies to verify the solution.

Key Concepts

Equation Solving

Solving equations is a foundational skill in algebra, and it's all about finding the value of the unknown variable that makes the equation true. When faced with a word problem, we start by translating the words into a mathematical equation. In this case, the equation represents the point at which two different salary offers equate. We assign the variable, often denoted as 'x', to represent the total sales we're trying to find.

After setting up the equation, the key to solving it is to get 'x' by itself on one side of the equation. This involves combining like terms and using inverse operations. For example, if a term is added to 'x', we subtract it from both sides, and vice versa for a term that is subtracted. When a variable is multiplied by a number, we divide by that number to isolate the variable. Through these steps, we aim for a simplified expression where 'x' equals a number, providing us with the solution.

Percentages in Algebra

Percentages are used to describe how one quantity relates to another in terms of parts per hundred. In algebra, we work with percentages by converting them into decimals to perform calculations. To convert a percentage to a decimal, we divide by 100. For instance, a 2.5% bonus is expressed as 0.025 in decimal form. This conversion is crucial because it allows us to incorporate percentages into equations seamlessly.

In the context of salary calculations, percentages often represent a commission or bonus. To calculate this, we multiply the decimal equivalent of the percentage by the total sales (our variable 'x'). Understanding how to translate percentages into algebraic expressions is vital to solving word problems involving salary, discounts, interest rates, and more.

Salary Calculations

Calculating salaries in word problems requires a clear understanding of fixed and variable components. A fixed salary is a set amount, such as the annual salary offered by a company, which doesn't change with sales or performance. In contrast, a variable component, like a sales commission or bonus, depends on other factors—total sales in our scenario.

To calculate the total compensation, we sum the fixed salary and any variable amounts. In algebraic problems, we create expressions that reflect these sums, and by equating the salary structures of different job offers, we can determine what needs to be true for them to balance out—in this case, the amount in sales needed to make the total salaries from two job offers equal. This type of calculation is an invaluable skill, applicable not only in algebra but also in making real-life financial decisions.