Problem 54 - College Algebra

Question

You are choosing between two different window washing companies. The first charges \(\$ 5\) per window. The second charges a base fee of f \(\$ 40\) plus \(\$ 3\) per window. How many windows would you need to have for the second company to be preferable?

Step-by-Step Solution

Answer

The second company is preferable when washing more than 20 windows.

Step 1: Define Variables

Let the number of windows be represented by the variable \( x \). This will help us set up equations for each company's pricing structure.

Step 2: Express the Cost for Company A

The first company charges \( \$5 \) per window. Therefore, the total cost for \( x \) windows would be \( 5x \).

Step 3: Express the Cost for Company B

The second company charges a base fee of \( \\(40 \) plus \( \\)3 \) per window. Hence, the total cost for \( x \) windows is \( 40 + 3x \).

Step 4: Set Inequality

We want to find the number of windows \( x \) for which the second company is cheaper than the first. This means setting up the inequality: \( 40 + 3x < 5x \).

Step 5: Solve the Inequality

First, subtract \( 3x \) from both sides to get: \( 40 < 2x \). Then, divide both sides by 2 to solve for \( x \): \( x > 20 \).

Step 6: Conclusion

The second company becomes preferable when you need at least 21 windows washed since 21 windows satisfy the condition \( x > 20 \).

Key Concepts

Variable Definitions

In algebraic inequalities, understanding what each variable represents is crucial. Here, the variable \( x \) stands for the number of windows needing washing. This simple choice of variable helps us translate the problem into a form we can work with mathematically.
Whenever you start with algebra problems, properly identify the entities involved. Assign a letter to each measurable quantity in the scenario. In our case, "windows" is the key quantity you're counting. Hence, defining \( x \) helps in setting up equations for each company's cost structure.

Linear Equations

Linear equations are a fundamental part of solving practical problems like cost comparisons. In this exercise, we used linear equations to represent the pricing structures of two window washing companies.
The first company charges a consistent \( \\(5 \) per window, which forms the linear equation \( 5x \). For the second company, the pricing includes a constant fee of \( \\)40 \), plus \( \$3 \) per window. This is represented by the linear equation \( 40 + 3x \).

• Notice the structure of both equations: they are in the form of \( ax + b \), where \( a \) is a constant rate per window, and \( b \) represents any fixed fees.
• Linear equations are beneficial because they simplify by adding or subtracting amounts to isolate the variable \( x \).

Cost Comparison

To decide which company offers a better deal, you need to compare their respective costs. Through cost comparison, we can determine who provides the most value for a given number of windows.
For Company A, the equation \( 5x \) means you pay \( \$5 \) per window, with no additional fees. For Company B, it's \( 40 + 3x \), factoring in a base fee. The challenge is finding when Company B becomes cheaper than Company A.

• To compare costs effectively, focus on setting up inequalities. This involves looking at when \( 40 + 3x < 5x \), which will help find the break-even point.
• This inequality highlights when one company’s pricing structure outweighs the other’s, making cost comparison an essential problem-solving step.

Problem Solving Steps

The process for solving the exercise included several systematic steps, crucial for success. Let's summarize those for ease of understanding and future application.
First, we defined the variable \( x \) as the number of windows. Then, expressed each company’s costs with separate equations. The next was to set up the inequality \( 40 + 3x < 5x \) to find when Company B becomes cheaper.
Once the inequality was established, we solved it by performing basic algebra:

• Subtract \( 3x \) from both sides to simplify the inequality to \( 40 < 2x \).
• Then, divide both sides by 2, leading to \( x > 20 \).

Through these steps, it becomes clear that for \( x \) greater than 20, Company B is more cost-effective. This method shows how breaking a problem down into steps aids in systematically finding a solution.