Question

Real Estate A real estate agent agrees to sell an apartment complex according to the following commission schedule: \(\$ 45,000\) plus \(25 \%\) of the selling price in excess of \(\$ 900,000\) Assuming that the complex will sell at some price between \(\$ 900,000\) and \(\$ 1,100,000,\) inclusive, over what range does the agent's commission vary? How does the commission vary as a percent of selling price?

Short answer

The agent's commission varies from \(45,000 to \)95,000. As a percentage of the selling price, it varies from 5% to approximately 8.64%.

Step-by-step solution

Identify the Commission Formula

The agent's commission is a sum of a fixed amount of \(45,000 and 25% of the selling price that exceeds \)900,000. Let's denote the selling price as S. The commission C can be expressed as: \[ C = 45000 + 0.25(S - 900000) \]

Determine the Range of Selling Prices

Given that the selling price can range from \(900,000 to \)1,100,000, we need to substitute these values into the commission formula to find the range of the commission:

Calculate the Minimum Commission

For the minimum selling price \(S = 900,000\): \[ C = 45000 + 0.25(900000 - 900000) \] \[ C = 45000 + 0.25 \times 0 \] \[ C = 45000 \]

Calculate the Maximum Commission

For the maximum selling price \(S = 1,100,000\): \[ C = 45000 + 0.25(1100000 - 900000) \] \[ C = 45000 + 0.25 \times 200000 \] \[ C = 45000 + 50000 \] \[ C = 95000 \]

Determine the Commission Range

From the previous steps, the agent’s commission varies from \(45,000 to \)95,000, depending on the selling price.

Calculate Commission as a Percent of Selling Price

To express the commission as a percentage of the selling price, we use the formula: \[ \text{Percent Commission} = \frac{C}{S} \times 100 \] For the minimum selling price \(S = 900,000\): \[ \text{Percent Commission} = \frac{45000}{900000} \times 100 \] \[ \text{Percent Commission} = 5\text{\%} \] For the maximum selling price \(S = 1,100,000\): \[ \text{Percent Commission} = \frac{95000}{1100000} \times 100 \] \[ \text{Percent Commission} \text{ is approximately } 8.64\text{\%} \]

Range of Commission Percentages

Thus, the commission as a percentage of the selling price varies from 5% to approximately 8.64%, depending on the selling price.

Key concepts

headline of the respective core concept

Real estate is a sector that deals with buying, selling, and leasing property such as houses, apartments, and commercial buildings. In this context, a real estate agent helps clients buy or sell properties. Their payment, known as a commission, is often a mix of a fixed fee and a percentage of the property's selling price.
For example, in our exercise, the real estate agent gets a fixed amount of \(45,000 plus 25% of the selling price exceeding \)900,000.
This kind of commission structure motivates agents to secure higher prices for properties, as their commission increases with the selling price. Understanding how to calculate this commission is vital for both agents and clients to ensure a fair transaction.

headline of the respective core concept

A percentage is a way to express a number as a fraction of 100. It is used widely in various fields, including finance, to compute interest rates, commission, discounts, and more.
In our exercise, the commission involves percentages.
The agent gets 25% of the selling price that exceeds \(900,000. This means if the selling price goes above this baseline, the agent earns a quarter of the extra amount as an additional commission.
For instance, if the selling price is \)1,100,000, the excess amount is \(200,000. Hence, the agent earns 25% of \)200,000 which is \(50,000, on top of the fixed \)45,000.

headline of the respective core concept

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols.
In our exercise, the commission calculation is represented by an algebraic expression.
The formula to determine the commission is: \( C = 45000 + 0.25(S - 900000) \) Here, \( C \) stands for the commission, and \( S \) represents the selling price. This expression helps us calculate the commission for any selling price within the given range.
By substituting the selling price \( S \) into the formula, we can find the exact commission the agent will earn.

headline of the respective core concept

The range of a function is the set of all possible output values. In our example, we need to determine the range of the commission based on the selling price.
Given the selling price range from \(900,000 to \)1,100,000, we use the commission formula to find the range of the commission: Minimum Commission: \( C = 45000 + 0.25(900000 - 900000) = 45000 \) Maximum Commission: \( C = 45000 + 0.25(1100000 - 900000) = 95000 \) Thus, the commission varies from \(45,000 to \)95,000.
Additionally, the commission as a percentage of the selling price ranges from 5% to approximately 8.64% depending on the selling price. This range helps both the agent and the client understand the possible earnings from a sale.