Answered

\begin{table}

\captionsetup{labelformat=empty}

\caption{Apple Computer Inc. Sales Department Pay}


\multirow{2}{*}{Number of iPhones Sold}		CommissionMonthly\multicolumn{2}{|c|}{Monthly}Total Monthly

Per Item SoldCommission\multicolumn{2}{|c|}{Bonus}Pay

305		\$ 31.00\$ 9,455.00\$1,000.00\$ 10,455.00

210		\$ 20.00\$ 4,200.00\$-\$ 4,200.00

199		\$ 20.00\$ 3,980.00\$-\$ 3,980.00

288		\$ 29.00\$ 8,352.00\$-\$ 8,352.00

268		\$ 27.00\$ 7,236.00\$-\$ 7,236.00

135		\$ 13.00\$ 1,755.00\$-\$ 1,755.00

465		\$ 47.00\$ 21,855.00\$1,000.00\$ 22,855.00

445		\$ 45.00\$ 20,025.00\$1,000.00\$ 21,025.00

276		\$ 28.00\$ 7,728.00\$-\$ 7,728.00

50		\$ 5.00\$ 250.00\$-\$ 250.00

345		\$ 35.00\$ 12,075.00\$1,000.00\$ 13,075.00

361		\$ 36.00\$ 12,996.00\$1,000.00\$ 13,996.00

0		\$-\$ -\$-

481		\$48.00\$ 23,088.00\$1,000.00

286		\$29.00\$ 8,294.00\$-

153		\$16.00\$ 2,448.00\$-

399		\$40.00\$ 15,960.00\$1,000.00

473		\$47.00\$ 22,231.00\$1,000.00

150		\$15.00\$ 2,250.00\$-

372		\$37.00\$ 13,764.00\$1,000.00




\end{table}

Verified Answer :

  • The sales data is analyzed to determine the threshold for the monthly bonus and the commission rule.
  • The monthly bonus is confirmed as \(M_b(S) = \$1000\) for \(S \geq 305\) and \(\$0\) for \(S < 305\).
  • The commission per item sold is defined by the function \(C_p(S) = \text{round}(0.1 \times S)\).
  • The total pay is calculated using these rules: \(T = (S \times C_p(S)) + M_b(S)\).

Explanation

  1. Analysis of Data and Relationships

To determine the compensation rules for the sales department, we first analyze the provided table of 20 sales representatives. We have two key variables to define: the monthly bonus (\(M_b\)) and the commission per item sold (\(C_p\)), both as functions of the number of iPhones sold (\(S\)). By inspecting the data, we observe the monthly bonus (\(M_b\)) is \$\$0\$ for sales up to @@MATH_TOKEN_12@@ and becomes \$\$1,000\$ for sales of \(305\) and above. Thus, the threshold for receiving the bonus is \(S \geq 305\). Regarding the commission per item sold (\(C_p\)), we observe a consistent pattern by calculating the ratio \(\frac{C_p}{S}\). Across the various sales figures, this ratio is approximately \(0.1\). For instance, at \(S=50\), \(C_p=\$5\); at \(S=150\), \(C_p=\$15\); at \(S=305\), \(C_p=\$31\) (which is roughly \(0.1 \times 305\)); and at \(S=481\), \(C_p=\$48\) (which is roughly \(0.1 \times 481\)). The commission per item sold is effectively \(10\%\) of the number of items sold, rounded to the nearest dollar.

  1. Formulating the Compensation Rules

Based on the analysis, we define the commission per item sold \(C_p(S)\) as approximately \(0.1 \times S\), rounded to the nearest integer. The monthly bonus \(M_b(S)\) follows a step function threshold at \(305\). The rules are:
\[C_p(S) = \text{round}(0.1 \times S)\]
\[M_b(S) = \begin{cases} \$1,000 & \text{if } S \geq 305 \ \$0 & \text{if } S < 305 \end{cases}\]
The total monthly pay is then \(T = (S \times C_p(S)) + M_b(S)\). We can verify this against a few entries:

  • For \(S=481\): \(C_p = \text{round}(48.1) = 48\). \(T = 481 \times 48 + 1000 = 23088 + 1000 = 24088\). (Matches table)
  • For \(S=288\): \(C_p = \text{round}(28.8) = 29\). \(T = 288 \times 29 + 0 = 8352\). (Matches table)
  • For \(S=50\): \(C_p = \text{round}(5) = 5\). \(T = 50 \times 5 + 0 = 250\). (Matches table)
  1. Conclusion and Summary of Rules

We have successfully identified the compensation structure. The monthly bonus threshold is \(S = 305\) and the commission per item is determined by the function \(C_p(S) = \text{round}(0.1 \times S)\). The total monthly pay is \(T = (S \times \text{round}(0.1 \times S)) + M_b(S)\), where \(M_b(S) = \$1000\) if \(S \geq 305\) and \(\$0\) otherwise. This provides a complete and consistent explanation for all records in the table.

Examples

In retail businesses, tiered compensation structures are common to incentivize high performance. For instance, a electronics retailer might use a similar system where sales staff receive a base commission percentage on items sold, with an added performance bonus once they hit a specific sales volume milestone. By using these mathematical models, managers can easily project payroll expenses and employees can clearly understand how their sales output translates into their final paycheck, fostering transparency and motivation in the workplace.