91Ó°ÊÓ

Real estate sales: A real estate agency has fixed monthly costs associated with rent, staff salaries, utilities, and supplies. It earns its money by taking a percentage commission on total real estate sales. During the month of July, the agency had total sales of \(\$ 832,000\) and showed a net income (after paying fixed costs) of \(\$ 15,704\). In August total sales were \(\$ 326,000\) with a net income of only \(\$ 523\). a. Use a formula to express net income as a linear function of total sales. Be sure to identify what the letters that you use mean. b. Plot the graph of net income and identify the slope and vertical intercept. c. What are the real estate agency's fixed monthly costs? d. What percentage commission does the agency take on the sale of a home? e. Find the horizontal intercept and explain what this number means to the real estate agency.

Step by step solution

01

Define Variables and Construct the Linear Function

Let \( S \) represent the total sales for a month and \( N(S) \) be the net income as a function of \( S \). Assume the commission rate is \( c \) and fixed monthly costs are \( F \). The net income formula is: \[ N(S) = c \cdot S - F \] Given July's data: \( N(832,000) = 0.03 \cdot 832,000 - F = 15,704 \) and August's data: \( N(326,000) = 0.03 \cdot 326,000 - F = 523 \).

02

Calculate Commission Rate and Fixed Cost

Using the equations derived: 1. \( 0.03 \times 832,000 - F = 15,704 \)2. \( 0.03 \times 326,000 - F = 523 \)Calculate using simultaneous equations:\( 24,960 - F = 15,704 \Rightarrow F = 24,960 - 15,704 = 9,256 \).Verifying with August:\( 9,780 - 9,256 = 523 \), confirming the commission rate \( c \) is 3% and fixed cost \( F \) is \$9,256.

03

Plot and Analyze the Graph

The linear equation is \( N(S) = 0.03S - 9,256 \). Plot this graph with \( S \) on x-axis and \( N(S) \) on y-axis. The slope is the commission rate \( 0.03 \). The intercept \(-9,256\) is where sales revenue equals zero, indicating total fixed costs.

04

Determine Fixed Costs and Commission Percentage

The vertical intercept is \(-9,256\), corresponding directly to monthly fixed costs \( F \). Thus, fixed monthly costs are \$9,256. From the slope, the commission rate on sales is 3%.

05

Find the Horizontal Intercept and Interpret

Set the net income function to zero: \( 0.03S - 9,256 = 0 \). Solving for \( S \):\( S = \frac{9,256}{0.03} = 308,533.33 \). The horizontal intercept represents the sales needed to cover fixed expenses, equating to a break-even point.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Net Income

Net income is the profit a business makes after all expenses are subtracted from revenue. It's a critical measure of financial health and indicates how much money is left over after all operating costs are covered. In this scenario, the net income of a real estate agency is the amount they earn after deducting their fixed monthly costs from their revenue, which comes from commissions on real estate sales.
The linear function that represents net income is expressed as: \[ N(S) = c \cdot S - F \] - Here, \( N(S) \) stands for net income as a function of total sales \( S \). - \( c \) represents the commission rate, while \( F \) represents fixed monthly costs.
This formula helps to understand how net income changes in response to variations in sales, which is crucial for planning and scaling business operations.

Fixed Costs

Fixed costs are the regular, monthly expenses that do not change regardless of the sales volume. In this context, fixed costs include expenses such as rent, staff salaries, utilities, and supplies that the agency must pay monthly, no matter how many houses they sell.
Knowing the fixed costs is vital because it allows the agency to determine how much they need to make in sales to break even or to earn a profit. For the real estate agency, the calculated fixed costs are $9,256 per month. This number plays a crucial role as all revenues first go towards covering these fixed costs before contributing to net income.

Commission Rate

Commission rate is a percentage of the total sales that an agent earns as income for making a sale. In the real estate agency example, this commission is the primary source of revenue.
The agency earns a commission of 3% on total sales.
This figure, represented by \( c \) in the linear net income function, influences how effectively the agency can convert sales into profit after covering its fixed costs.
For instance, if the total sales for July are $832,000, the commission derived would be \(0.03 \times 832,000 = 24,960\) before deducting fixed costs. Understanding this concept helps businesses set sales targets and project earnings.

Break-even Point

The break-even point is a critical financial threshold where total sales equal total expenses, resulting in zero profit or loss. This means that all income received from sales is used to cover the fixed costs. For the real estate agency, finding this point involves setting the net income equation to zero and solving for \( S \).
To calculate: \[ 0.03S - 9,256 = 0 \]Solving this equation:\[ S = \frac{9,256}{0.03} = 308,533.33 \]
Thus, they need sales of approximately $308,533.33 to cover all fixed costs without gaining any profit.

• Below this sales level, the agency operates at a loss.
• Above it, the agency earns a profit.

Understanding the break-even point assists in planning strategies around sales targets and financial management.