Problem 67 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. Writing Your employer offers you a choice of wage scales: a monthly salary of \(\$ 3000\) plus commission of \(7 \%\) of sales or a salary of \(\$ 3400\) plus a \(5 \%\) commission. Write a short paragraph discussing how you would choose your option. At what sales level would the options yield the same salary?

Step-by-Step Solution

Answer

Both wage scales would yield the same salary when the sales level is \$20,000. Choosing the best option would then depend on whether you expect to make more or less than this sales level. If you expect to make more than 20,000 in sales, you would earn more with the first salary option, otherwise, the second option would be beneficial.

Step 1: Write the equation for the first salary option

The first salary option includes a $3000 base salary and a 7% commission. Therefore, the total salary for the first option can be expressed as \(S1 = 3000 + 0.07x\), where \(x\) is the amount of sales.

Step 2: Write the equation for the second salary option

The second salary option includes a $3400 base salary and a 5% commission. Therefore, the total salary for the second option can be expressed as \(S2 = 3400 + 0.05x\), where, as before, \(x\) is the amount of sales.

Step 3: Set the two equations equal to each other

To find at what sales level the two options yield the same salary, set the equations equal to each other. That is: \(3000 + 0.07x = 3400 + 0.05x\).

Step 4: Solve for x

Solving for \(x\) will give the sales level at which the salaries from the two options equalize. To do that, first simplify the equation by subtracting \(3400\) and \(0.07x\) from both sides respectively, which will result in \(0.02x = 400\). Finally, to solve for \(x\), divide both sides of the equation by \(0.02\). The result is \(x = 20000\).

Key Concepts

Commission-Based Salary

At its core, a commission-based salary is compensation that varies with performance, typically the revenue generated from sales. In practical terms, employees on commission receive a certain percentage of the deals they close or the sales they make. This form of payment is common in sales-driven industries where the motivation to sell more is directly linked to higher earnings.

Understanding commission-based salaries requires assessing two main components: the fixed base salary and the commission rate. The base salary is a guaranteed amount you will earn regardless of sales performance. This provides a level of security. The commission rate, expressed as a percentage, is the variable component that increases with sales. A higher commission rate usually translates to greater potential earnings if sales are robust, while the security of a higher base salary is favorable if sales are uncertain or fluctuating.

To make an informed decision between two commission-based salary structures, it's crucial to consider personal sales expectations and choose the option that optimizes potential earnings.

Linear Equations

A staple of algebra, linear equations are used to solve problems involving relationships with a constant rate of change. They are called 'linear' because their graphed representations are straight lines. The general form of a linear equation in one variable is usually written as ax + b = 0, where a and b are constants, and x is the variable we aim to solve for.

In the context of commission-based salary structures, we use linear equations to model the total earnings as a function of sales. This is because the relationship between sales and commission earned is typically direct and proportional, a perfect fit for a linear model. When comparing two salary options with different commission rates and base salaries, setting linear equations equal to each other helps us determine at what sales volume the earnings would be identical, also known as the 'break-even point'.

Algebraic Problem Solving

The essence of algebraic problem solving is to find unknown values by using mathematical operations and relationships. It often involves a systematic approach and a series of steps—identifying variables, translating words into equations, and manipulating these equations to solve for the desired quantity.

In the exercise scenario, you are comparing two salary options. Algebraic problem solving enters the picture when these options are expressed in terms of linear equations with the unknown variable being the sales amount needed to equate the total salary from both options. This process involves setting the two equations equal to solve for the unknown variable, simplifying the equation by combining like terms, and finally isolating the variable to find its value. This methodical approach helps reveal the break-even point of sales, enabling you to make an informed decision about which wage scale to choose.