- Calculate sales for Salary + Commission: \(2000 + 0.03S = 3500 \implies S = \$50,000\).
- Calculate sales for Straight Commission: \(0.07S = 3850 \implies S = \$55,000\).
- Calculate sales for Graduated Commission: \(2000 + 0.08(S - 40000) = 3600 \implies S = \$60,000\).
- Compare results: \(\$60,000 > \$55,000 > \$50,000\).
- Final Answer: \(\boxed{\text{The graduated commission employee}}\).
Explanation
- Data and problem analysis
To determine which employee had the largest sales (\$S\$) in January, we calculate the sales for each compensation structure based on their earnings given in the table. The earnings are provided in thousands of dollars, so we convert them into dollars by multiplying by \$1,000.
- The Salary + Commission employee earned \$3.5 (i.e., \$\$3,500\$). The formula for their earnings is \$\$2,000 + 3\% \times S = 3,500\$.
- The Straight Commission employee earned \$3.85 (i.e., \$\$3,850\$). The formula for their earnings is \$7\% \times S = 3,850\$.
- The Graduated Commission employee earned \$3.6 (i.e., \$\$3,600\$). The formula for their earnings is \$5\% \times 40,000 + 8\% \times (S - 40,000)\$, provided \$S > 40,000\$.
- Calculating sales for each employee
We solve for \$S\$ for each employee:
- For the Salary + Commission employee:
\[ 2,000 + 0.03 \cdot S = 3,500 \]
\[ 0.03 \cdot S = 1,500 \]
\[ S = \frac{1,500}{0.03} = 50,000 \]
The sales are \$\$50,000\$.
- For the Straight Commission employee:
\[ 0.07 \cdot S = 3,850 \]
\[ S = \frac{3,850}{0.07} = 55,000 \]
The sales are \$\$55,000\$.
- For the Graduated Commission employee:
First, calculate the earnings from the first \$\$40,000\$:
\[ 0.05 \cdot 40,000 = 2,000 \]
Since \$\$3,600 > \$2,000\$, we know sales \$S\$ exceeded \$\$40,000\$. Now, solve for \$S\$:
\[ 2,000 + 0.08 \cdot (S - 40,000) = 3,600 \]
\[ 0.08 \cdot (S - 40,000) = 1,600 \]
\[ S - 40,000 = \frac{1,600}{0.08} = 20,000 \]
\[ S = 40,000 + 20,000 = 60,000 \]
The sales are \$\$60,000\$.
- Comparison and Conclusion
Comparing the sales amounts:
- Salary + Commission: \$\$50,000\$
- Straight Commission: \$\$55,000\$
- Graduated Commission: \$\$60,000\$
The largest amount is \$\$60,000\$, which corresponds to the Graduated Commission employee. Thus, the graduated commission employee had the largest dollar amount in sales in January. The correct option is the last one. \(\boxed{\text{The graduated commission employee}}\).
Examples
Understanding how different compensation structures work is crucial in business management and financial planning. For instance, a salesperson might choose between a steady base salary or a higher commission rate based on their confidence in hitting high sales targets. By calculating the total sales required to reach a certain income goal across different pay models, employees can make informed career decisions, and businesses can design incentive programs that align with their revenue targets.