Answered

The table below shows the earnings, in thousands of dollars, for three different commissioned employees.





\begin{tabular}[t]{l}
Employee \#1

\$2,000 + 3\% on all sales
&


Employee \#2

7\% on all sales
&


Employee \#3

5\% on the first \$40,000 + 8\% on anything over \$40,000
\\

\hline December & 4.4 & 5.6 & 5.2 \\

\hline January & 3.5 & 3.85 & 3.6 \\

\hline February & 4.7 & 4.9 & 4.4 \\

\hline & & & \\

\hline

\end{tabular}



Which employee did not have the same dollar amount in sales for the month of February as the other two employees?


AEmployee #1
BEmployee #2
CEmployee #3
DThey each had the same dollar amount in sales.

Verified Answer :

  • Calculate February sales for Employee #1: \(2 + 0.03S_1 = 4.7 \Rightarrow S_1 = 90\).
  • Calculate February sales for Employee #2: \(0.07S_2 = 4.9 \Rightarrow S_2 = 70\).
  • Calculate February sales for Employee #3: \(0.05(40) + 0.08(S_3 - 40) = 4.4 \Rightarrow S_3 = 70\).
  • Identify that Employee #1 has different sales compared to the others, meaning the correct answer is \(\boxed{a}\).

Explanation

  1. Data and Problem Analysis

We are provided with the earnings models for three employees based on their monthly sales \(S\) (in thousands of dollars). The models are:

  • Employee #1: \(E_1 = 2 + 0.03S\)
  • Employee #2: \(E_2 = 0.07S\)
  • Employee #3: \(E_3 = 0.05(40) + 0.08(S - 40)\) for \(S > 40\)

The February earnings (in thousands of dollars) are:

  • Employee #1: \(E_1 = 4.7\)
  • Employee #2: \(E_2 = 4.9\)
  • Employee #3: \(E_3 = 4.4\)

Our goal is to calculate the monthly sales (\(S\)) for each employee during the month of February to determine which employee did not have the same dollar amount in sales as the others.

  1. Calculating Sales for Employee #1

To find the sales \(S_1\) for Employee #1, we use their earnings model:
\[2 + 0.03S_1 = 4.7\]
Subtracting 2 from both sides gives:
\[0.03S_1 = 2.7\]
Dividing by 0.03, we get:
\[S_1 = \frac{2.7}{0.03} = \frac{270}{3} = 90\]
So, Employee #1 generated \(90\) thousand dollars in sales.

  1. Calculating Sales for Employee #2

To find the sales \(S_2\) for Employee #2, we use their earnings model:
\[0.07S_2 = 4.9\]
Dividing by 0.07, we get:
\[S_2 = \frac{4.9}{0.07} = \frac{490}{7} = 70\]
So, Employee #2 generated \(70\) thousand dollars in sales.

  1. Calculating Sales for Employee #3

To find the sales \(S_3\) for Employee #3, we use their earnings model:
\[0.05(40) + 0.08(S_3 - 40) = 4.4\]
First, simplify the expression:
\[2.0 + 0.08(S_3 - 40) = 4.4\]
Subtracting 2.0 from both sides gives:
\[0.08(S_3 - 40) = 2.4\]
Dividing by 0.08, we get:
\[S_3 - 40 = \frac{2.4}{0.08} = \frac{240}{8} = 30\]
Adding 40 to both sides gives:
\[S_3 = 70\]
So, Employee #3 generated \(70\) thousand dollars in sales. Note that \(S_3 = 70 > 40\), which satisfies the condition for this model.

  1. Comparing Sales and Conclusion

We have calculated the February sales for each employee:

  • Employee #1: \(S_1 = 90\)
  • Employee #2: \(S_2 = 70\)
  • Employee #3: \(S_3 = 70\)

Comparing these values, Employee #2 and Employee #3 both had sales of \(70\) thousand dollars, while Employee #1 had sales of \(90\) thousand dollars. Therefore, Employee #1 did not have the same dollar amount in sales as the other two employees. The correct option is a.

Examples

In a real-world office setting, tracking commissioned sales is essential for evaluating performance and calculating payroll accurately. By using the earnings model provided for each employee, managers can reverse-engineer the sales figures to ensure that commissions are paid according to the agreed-upon structure, effectively verifying the consistency of sales data across different teams. This type of algebraic modeling is fundamental in business analytics and financial accounting.

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