Problem 56 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Correspond to those in Exercises \(15-27\), Section \(5.1 .\) Use the results of your previous work to help you solve these problems. Cantwell Associates, a real estate developer, is planning to build a new apartment complex consisting of one-bedroom units and two- and three-bedroom townhouses. A total of 192 units is planned, and the number of family units (two- and three-bedroom townhouses) will equal the number of one-bedroom units. If the number of one-bedroom units will be 3 times the number of threebedroom units, find how many units of each type will be in the complex.

Step-by-Step Solution

Answer

In the apartment complex, there will be \(48\) one-bedroom units, \(32\) two-bedroom townhouses, and \(16\) three-bedroom townhouses.

Step 1: Set up the equations based on the given information

We can create three equations based on the information given: 1. x + y + z = 192 (Total number of units) 2. x = y + z (The number of one-bedroom units = the number of family units) 3. x = 3z (The number of one-bedroom units = 3 times the number of three-bedroom units) We now have a system of three equations with three variables (x, y, and z).

Step 2: Solve the equations for x, y, and z

Now, let's substitute the expressions from equations (2) and (3) in equation (1): x + y + z = 192 (y + z) + y + z = 192 y + 2z = 64 (since x = 3z) Now we can see that the third equation can help us simplify even further: x = 3z y = 64 - 2z Now we have expressions for both x and y in terms of z. Let's put these expressions in equation (2) to solve for z. x = y + z 3z = (64 - 2z) + z 3z = 64 - z Now, we can solve for z: 4z = 64 z = 16 Now we can find the values of x and y: x = 3z = 3 * 16 = 48 y = 64 - 2z = 64 - 2 * 16 = 32 So, there will be 48 one-bedroom units, 32 two-bedroom townhouses, and 16 three-bedroom townhouses in the complex.

Key Concepts

Systems of Equations

In mathematical modeling, a system of equations is a set of equations with multiple variables. The goal is often to find values for these variables that satisfy all of the equations simultaneously. In our real estate problem, we started with three unknowns: the number of one-bedroom units (\( x \)), two-bedroom townhouses (\( y \)), and three-bedroom townhouses (\( z \)).

By setting up equations according to the problem description:

• \( x + y + z = 192 \) (total units)
• \( x = y + z \) (the number of one-bedroom units equals the number of two- and three-bedroom units combined)
• \( x = 3z \) (one-bedroom units are three times the number of three-bedroom units)

These equations allowed us to apply algebraic methods to pinpoint specific values for \( x \), \( y \), and \( z \). Systems of equations are a powerful tool in algebraic problem-solving and are often used to model real-world scenarios where multiple conditions must be met.

Real Estate Mathematics

In real estate projects, understanding the allocation and the relationships between different types of units is crucial. For Cantwell Associates' apartment complex, it was necessary to balance the number of one-bedroom and family units, which included both two- and three-bedroom townhouses.

This type of problem requires careful planning and consideration of constraints:

• The total number of units is fixed (192 units).
• Specific relationships exist, such as the equality of one-bedroom units to the total of family units.
• Proportional relationships, like the one-bedroom units being three times the number of three-bedroom units, guide the solutions.

Such real estate mathematics problems demand precision and clarity to ensure that developers meet their design goals and satisfy any given requirements or constraints. Being able to model and solve such problems effectively can lead to successful development projects.

Algebraic Problem-Solving

Algebraic problem-solving involves using various algebraic techniques to find solutions to problems like the one presented in the real estate example. Here, we used substitution and simplification - two common strategies.

The method involved:

• First, setting up equations to express the known conditions and constraints.
• Then, substituting known expressions from one equation into another to reduce complexity.
• Finally, isolating each variable to solve the equations systematically.

For instance, substituting \( x = 3z \) into the equation \( x = y + z \) allowed us to express \( y \) in simpler terms. Continuing the process enabled us to solve for the number of each type of unit. Such steps highlight how algebraic techniques provide a structured way to handle problem-solving scenarios, making complex real-world problems more manageable.