Problem 33 - Algebra 2

Question

For Exercises \(33-35,\) use the following information. A small corporation decides that 8\(\%\) of its profits would be divided among its six managers. There are two sales managers and four nonsales managers. Fifty percent would be split equally among all six managers. The other 50\(\%\) would be split among the four nonsales managers. Let \(p\) represent the profits. Write an expression to represent the share of the profits each nonsales manager will receive.

Step-by-Step Solution

Answer

Each nonsales manager receives \( \frac{p}{60} \) of the profits.

Step 1: Calculate Total Profit for Managers

First, we need to calculate the total profit reserved for the managers. Since 8% of the corporation's profit is given to the managers, this amount in terms of profit \( p \) is \( 0.08p \).

Step 2: Divide Profits Among Managers

Half of the total profits for the managers is split equally among all six managers. Therefore, \( 0.5 \times 0.08p = 0.04p \) will be equally divided among 6 managers.

Step 3: Calculate Share from Equal Division

Each manager receives from the equal division: \( \frac{0.04p}{6} \). Simplifying this, each manager gets: \( \frac{p}{150} \).

Step 4: Divide Remaining Profits to Nonsales Managers

The remaining 50\% of the manager profits, also \( 0.04p \), is divided only among the four nonsales managers.

Step 5: Calculate Additional Share for Nonsales Managers

Each nonsales manager receives from this division: \( \frac{0.04p}{4} \), which simplifies to \( \frac{p}{100} \).

Step 6: Calculate Total Share for Each Nonsales Manager

Add both shares together for each nonsales manager: \( \frac{p}{150} + \frac{p}{100} = \frac{2p}{300} + \frac{3p}{300} = \frac{5p}{300} = \frac{p}{60} \).

Key Concepts

Corporate Finance

In corporate finance, the management of profits is crucial for ensuring long-term business sustainability and growth. When a corporation makes decisions about how to allocate its profits, it must consider various factors, such as investments, dividends, and compensation. For a small corporation, deciding to allot a portion of its profits to managerial compensation indicates an investment in its human resources.
This investment aims to incentivize managers to perform better and align their goals with those of the company. Typically, distributing profits can lead to improved morale and retention among employees, as they feel more directly rewarded for the company's success. In the exercise, 8% of the corporation's profits are distributed among the managers, reflecting the value placed on managerial contributions to the business.

Managerial Compensation

Managerial compensation is a significant component of corporate finance structures. It involves providing pay packages, which may include base salary, bonuses, and profit-sharing mechanisms. In this exercise scenario, the corporation uses a profit-sharing approach as part of the managerial compensation.
Profit sharing can be advantageous as it directly links employee pay to the company's performance. This alignment can motivate managers to enhance the company's profitability, knowing that their compensation is partially dependent on the company's success. The decision to give 8% of the profits to managers underscores the importance of providing equitable compensation that reflects the managers’ roles and contributions. The managers' share calculated in different ways ensures both fairness and motivation across various managerial roles.

Profit Sharing

Profit sharing is a method used by companies to distribute a portion of their profits among employees, often as an incentive mechanism. In this exercise, the corporation decides to share 50% of the allocated manager's profits equally among all managers, while the remaining 50% goes specifically to nonsales managers. This allocation ensures that all managers benefit from the company's success, but acknowledges the different roles by rewarding nonsales managers distinctly.
Profit sharing can enhance teamwork and company loyalty as employees work together towards a shared financial goal. It can serve to lower turnover and increase job satisfaction, fostering a productive and motivated workforce. By integrating profit sharing into their compensation strategy, companies can encourage a collective focus on profit generation and sustainable business practices.

Mathematical Expressions

Mathematical expressions are crucial in breaking down complex problems into understandable terms. In this exercise, algebraic expressions help in calculating the share of profits for each nonsales manager.
The corporation divides 8% of its total profits among the managers, which is initially calculated as \(0.08p\).

• The first key step involves calculating the equal split among all managers: \(0.5 \times 0.08p = 0.04p\).
• Each manager receives \(\frac{0.04p}{6}\) from this share, simplified to \(\frac{p}{150}\).
• The remaining half of the profits, another \(0.04p\), is divided among the four nonsales managers.
• Each nonsales manager receives \(\frac{0.04p}{4}\), again simplified to \(\frac{p}{100}\).
• Adding these together, each nonsales manager gets \(\frac{p}{150} + \frac{p}{100}\), which simplifies to \(\frac{p}{60}\).

By using these expressions, we can see clearly how profit distribution is calculated, providing a transparent and methodical approach to managerial compensation.