Problem 79 - Precalculus: Functions and Graphs

Question

Sales commission A recent college graduate has job offers for a sales position in two computer firms. Job A pays 50,000 dollars per year plus \(10\%\) commission. Job B pays only 40,000 dollars per year, but the commission rate is \(20 \%\). How much yearly business must the salesman do for the second job to be more lucrative?

Step-by-Step Solution

Answer

The salesman must do more than $100,000 in business for Job B to be more lucrative.

Step 1: Define Variables

Let's define some variables to work with: Let \( S \) be the amount of yearly business (sales) the graduate must do. Job A has a base salary of \(50,000 and offers a commission of \(10\%\), while Job B has a base salary of \)40,000 with a \(20\%\) commission rate.

Step 2: Write Equations for Total Income

For Job A, the total income is the base salary plus the commission. That is, Total Income A = \(50000 + 0.10 \times S\). For Job B, the total income is the base salary plus the commission as well. Total Income B = \(40000 + 0.20 \times S\).

Step 3: Set Up the Inequality

To find the amount of yearly business needed for Job B to be more lucrative, we need Total Income B > Total Income A. Therefore, we set up the inequality: \[ 40000 + 0.20S > 50000 + 0.10S \]

Step 4: Solve the Inequality

Subtract \(0.10S\) from both sides of the inequality:\[ 40000 + 0.20S - 0.10S > 50000 \]This simplifies to:\[ 40000 + 0.10S > 50000 \]Subtract 40,000 from both sides:\[ 0.10S > 10000 \]Finally, divide each side by \(0.10\) to solve for \(S\):\[ S > 100000 \]

Step 5: Interpret the Solution

The inequality \( S > 100000 \) means that the salesman must do more than $100,000 in business for Job B to be more lucrative than Job A.

Key Concepts

Linear Inequality

In this scenario, we use a linear inequality to determine when one sales job becomes more profitable than the other. Let's break it down! Linear inequalities are expressions where one side is not equal to the other but is either greater than, or lesser than. They are much like linear equations, but with an inequality sign instead of an equal sign. For our sales commission problem, we need to figure out when the total income from Job B is greater than that from Job A. This involves comparing the two incomes, leading us to set up the inequality:

• \(40000 + 0.20S > 50000 + 0.10S\)

Solving this inequality helps us find out how much the salesman needs to sell to make Job B his best choice.

Base Salary

A base salary is essentially a fixed amount of money an employee earns before commissions or bonuses. It's like a guaranteed income they receive regardless of their sales performance. In the sales commission problem, two different base salaries are offered:

• Job A offers a base salary of \(50,000\) dollars per year.
• Job B offers a base salary of \(40,000\) dollars per year.

The difference in these base salaries significantly impacts the decision-making process since Job A's higher base salary offers more financial security initially, but also less potential upside compared to the higher commission rate of Job B. Keep in mind that a higher base salary doesn't necessarily mean a higher overall income once commissions are factored in.

Commission Rate

The commission rate is the percentage of sales that a salesperson earns on top of their base salary. This means that for every product sold, they earn a specific amount added to their income. In our exercise, the two jobs offer different commission rates:

• Job A pays a \(10\%\) commission on sales.
• Job B pays a \(20\%\) commission on sales.

A higher commission rate is enticing as it allows a salesperson to earn more with higher sales. However, it also means income can be less predictable if sales are low. In our exercise, this means for Job B to be more lucrative overall, the salesman must generate sufficient sales to benefit from the higher commission rate, even though its base salary is lower than Job A.

Total Income Equation

The total income equation helps compute the complete compensation a salesperson can earn. This equation combines both the base salary and the income from commissions. For Job A, the total income equation is:

• \( \text{Total Income A} = 50000 + 0.10S \)

For Job B, the equation is:

• \( \text{Total Income B} = 40000 + 0.20S \)

In these equations, \(S\) represents the amount of sales. By setting these two equations against each other with an inequality \(40000 + 0.20S > 50000 + 0.10S\), we figure out how much the salesman must sell for Job B to provide a higher total income than Job A. This comparison focuses not just on fixed salaries but also on potential earnings from differing commission rates.