Problem 58 - College Algebra with Corequisite Support

Question

When hired at a new job selling electronics, you are given two pay options: Option A: Base salary of $$\$ 10,000$$ a year with a commission of \(9 \%\) of your sales Option B: Base salary of $$\$ 20,000$$ a year with a commission of \(4 \%\) of your sales How much electronics would you need to sell for option \(\mathrm{A}\) to produce a larger income?

Step-by-Step Solution

Answer

Sell more than \( 200,000 \) in electronics for Option A to be better.

Step 1: Define Variables

Let the annual sales amount be represented by \( s \). For option A, the total income will be based on a base salary plus commission. Similarly, for option B, the total income will follow the same computation.

Step 2: Write Income Equations

For Option A, the income equation is \( I_A = 10000 + 0.09s \), and for Option B, the income equation is \( I_B = 20000 + 0.04s \). These equations account for the base salary and the commission based on sales \( s \).

Step 3: Set Up Inequality

To determine when Option A produces more income than Option B, set up the inequality: \( 10000 + 0.09s > 20000 + 0.04s \).

Step 4: Solve the Inequality

Subtract \( 0.04s \) from both sides to focus the inequality on sales-related terms: \( 10000 + 0.05s > 20000 \). Then, subtract 10000 from both sides to simplify further: \( 0.05s > 10000 \). Divide both sides by 0.05 to isolate \( s \): \( s > 200000 \).

Step 5: Conclusion

Option A will yield a higher income than Option B when you sell more than \( 200000 \) worth of electronics in a year.

Key Concepts

Linear Equations

Linear equations are a cornerstone of algebra and help us represent relationships where change is constant. In our exercise, we dealt with linear equations to help Nikki, our hypothetical electronics salesperson, decide which payment option maximizes earnings. A linear equation is structured as follows: \( y = mx + b \). Here, \( m \) represents the slope (or rate), while \( b \) is the y-intercept (or starting value). For a sales context, the equation can express income as a function of sales.

• Option A: \( I_A = 10000 + 0.09s \) is Nikki's earnings combining a base salary of \(10,000 and 9% commission.
• Option B: \( I_B = 20000 + 0.04s \) gives a base salary of \)20,000 plus 4% commission.

This structure allows us to predict and understand how income changes with different sales amounts, making it a powerful tool for financial planning and decision-making.

Commission Calculation

Understanding how commission calculations work is crucial for salespeople. Commission is usually expressed as a percentage of total sales, giving employees a direct incentive to sell more. This exercise illustrates two scenarios, highlighting how different commission rates affect income.

• For Option A: A 9% commission translates to an income boost proportional to 0.09 times the sales amount.
• For Option B: The 4% commission, a lesser boost, equals 0.04 multiplied by sales.

When examining payment options, calculate total income by adding base salary to the computed commission. This helps in assessing not just earning potential but also the risk involved in lower base salaries. Thus, knowing how to calculate and compare commissions ensures you choose the option that maximizes earnings based on sales capabilities.

Income Comparison

Comparing income structures allows salespeople to make informed career decisions. In this task, figure out which payment option results in a higher income depending on sales performance. First, convert the income equations into an inequality: \( 10000 + 0.09s > 20000 + 0.04s \). By solving for sales \( s \), you determine the sales threshold needed for Option A to become more beneficial.

• After simplifying the inequality, you find \( 0.05s > 10000 \), which leads to \( s > 200000 \).
• This means sales exceeding $200,000 result in Option A being more lucrative.

Understanding this comparison is critical because it contextualizes the mathematical solution, showcasing how algebraic principles can solve everyday problems. Therefore, by comparing initial and potential earnings, professionals can strategically choose compensation structures that align with their sales potential.