Problem 38 - Intermediate Algebra

Question

In Problems \(34-37\) recall that the money a business spends to produce a product (or service) is called its cost and the money that it takes in from the sales of a product (or service) is called the revenue. In business and economics, it is important to determine the value at which costs equal the revenue, called the break-even point. Salary Options. A sales clerk can choose from two salary plans: 1\. a straight \(7 \%\) commission 2\. \(\$ 150+2 \%\) commission How much would the clerk have to sell for each plan to produce the same monthly paycheck?

Step-by-Step Solution

Answer

The clerk needs to sell \( \$3000 \) to have the same paycheck with both salary plans.

Step 1: Define Variables

Let the total sales amount be \( x \) dollars. We need to find the sales amount \( x \) that will make the salary equivalent under both plans.

Step 2: Express Plan 1 as an Equation

For the first plan which is a \( 7\% \) commission, the salary is \( 0.07x \).

Step 3: Express Plan 2 as an Equation

For the second plan, the salary is a fixed amount of \( \$150 \) plus a \( 2\% \) commission, which can be expressed as \( 150 + 0.02x \).

Step 4: Set Equations Equal

To find the break-even point, set the equations for the two salary plans equal to each other: \[ 0.07x = 150 + 0.02x \]

Step 5: Solve for x

Start solving the equation: \[ 0.07x - 0.02x = 150 \] Simplify the left side: \[ 0.05x = 150 \] Divide both sides by \( 0.05 \) to solve for \( x \): \[ x = \frac{150}{0.05} = 3000 \]

Step 6: Conclusion

The clerk must make \( \$3000 \) in sales for both salary plans to result in the same paycheck.

Key Concepts

Cost vs Revenue

In the world of business, two essential terms often come up: cost and revenue. Understanding these can help in making informed financial decisions. Cost refers to the total expenses necessary to produce a product or a service. This includes everything from material costs and labor to overhead expenses. Revenue, on the other hand, is the income a business generates from selling its product or service. In simple terms, cost is what you spend to make something, and revenue is what you earn from selling it.

Calculating these correctly is crucial for finding the break-even point of a business. The break-even point is reached when total costs are equal to total revenues. At this point, the business is neither making a profit nor a loss. Achieving this balance is vital to ensure a business is sustainable over time. It helps businesses understand minimum sales requirements and the impact of costs on profitability.

• Costs = Total expenses incurred to produce/sell
• Revenue = Total income from selling
• Break-even point = Costs equal to revenue

Commission Salary

A commission salary is a compensation model commonly used in sales roles, where the earnings are based on the volume of sales made. This model serves as a strong incentive for employees to increase their sales since their salary is directly linked to their performance. In a commission structure, the benefits for both employer and employee are pronounced. Employers only pay high salaries when the employee brings in revenue, while employees can earn more by selling more.

For instance, in our original exercise, we consider two different commission plans:

• Plan 1 is a straight 7% commission, where the employee earns 7% of everything they sell.
• Plan 2 offers a base salary of $150 with an additional 2% commission on all sales.

Finding out when these two plans break even is crucial for the employee to make an informed decision. By comparing different commission plans, employees can choose the one that maximizes their earnings based on their sales performance.

Sales Equation

The sales equation helps in determining the appropriate financial outcomes from different compensation plans. In our exercise, we use algebraic expressions to represent different salary plans. By equating these, we can find the break-even sales amount.

Let's break down the equations from our example:

• For Plan 1: The equation is \( 0.07x \), representing 7% of the total sales.
• For Plan 2: The equation is \( 150 + 0.02x \), which combines a fixed salary with a percentage of sales.

To find the sales amount \( x \) that equates these plans, we set the equations equal to each other:\[ 0.07x = 150 + 0.02x \] This results in a simple algebraic resolution where we solve for \( x \), representing the point at which the employee earns the same under either plan. Understanding these equations is vital for determining optimal sales strategies and compensation models.

Business Algebra Problem

A business algebra problem involves using algebraic methods to solve real-world business scenarios, such as determining break-even points or comparing salary plans. The exercise we looked at is a perfect example of how algebra is employed to make informed business decisions. By defining variables and setting up equations, businesses can perform calculations to answer important questions.

For example, in our exercise:

• Define the variables: Let \( x \) be the total sales amount.
• Set up the equations based on the given salary plans.
• Find where the equations intersect to determine the break-even point.

Business algebra problems provide a systematic approach to solving complex financial issues. This empowers any business, from small startups to large corporations, to strategize their operations, plan budgets, and minimize costs while maximizing profits.