A real estate agent agrees to sell an apartment complex according to the following commission​ schedule: ​$40 comma 000 plus 30​% of the selling price in excess of ​$900 comma 000. Assuming that the complex will sell at some price between ​$900 comma 000 and ​$1 comma 000 comma 000 ​inclusive, over what range does the​ agent's commission​ vary? How does the commission vary as a percent of selling​ price?

To solve this problem, let's break it down step by step.

### 1. Understanding the Commission Formula:
The agent's commission is structured as follows:
- A flat fee of \$40,000.
- Plus, 30% of the selling price that exceeds \$900,000.

Let the selling price be denoted by $ S $.

The commission $ C $ can be expressed as:
\[
C = 40,000 + 0.30(S - 900,000)
\]

### 2. Range of Selling Price:
The problem states that the selling price $ S $ is between \$900,000 and \$1,000,000, inclusive. Thus:
\[
900,000 \leq S \leq 1,000,000
\]

### 3. Calculating the Range of Commission:
Now, we'll calculate the commission at the minimum and maximum values of $ S $.

- **Minimum Selling Price (\$900,000):**
  \[
  C = 40,000 + 0.30(900,000 - 900,000) = 40,000 + 0 = 40,000
  \]

- **Maximum Selling Price (\$1,000,000):**
  \[
  C = 40,000 + 0.30(1,000,000 - 900,000) = 40,000 + 0.30(100,000) = 40,000 + 30,000 = 70,000
  \]

So, the agent's commission varies from \$40,000 to \$70,000.

### 4. Commission as a Percent of Selling Price:
Next, we want to determine how the commission $ C $ varies as a percentage of the selling price $ S $.

- The percentage commission $ P $ is given by:
\[
P = \left(\frac{C}{S}\right) \times 100
\]

Let's calculate $ P $ at the minimum and maximum selling prices:

- **For $ S = 900,000 $:**
  \[
  P = \left(\frac{40,000}{900,000}\right) \times 100 \approx 4.44\%
  \]

- **For $ S = 1,000,000 $:**
  \[
  P = \left(\frac{70,000}{1,000,000}\right) \times 100 = 7\%
  \]

### 5. Summary of Results:
- **Commission Range:** The agent's commission varies from \$40,000 to \$70,000.
- **Percentage of Selling Price:** The commission varies from approximately 4.44% to 7% of the selling price.

### Do you want more details or have any questions?

Here are some related questions:
1. What would the commission be if the selling price were \$950,000?
2. How does the commission percentage change if the selling price exceeds \$1,000,000?
3. What is the formula for the commission if the selling price is less than \$900,000?
4. How would the commission change if the percentage over \$900,000 were 25% instead of 30%?
5. Can you derive a general formula for commission as a function of selling price?

**Tip:** Understanding how to break down a problem into components (like flat fees and percentages) can help in analyzing complex commission structures.

Explore how real estate agent commissions are calculated based on a structured formula. Learn the range of commission amounts and how they vary as a percentage of the selling price between $900,000 and $1,000,000. Detailed step-by-step solution included.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation