Problem 54 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

More than 20 windows.

Step 1: Define Variables

Let \( x \) represent the number of windows that need washing. We need to find when Company 2 is more cost-effective than Company 1.

Step 2: Equation for Company 1

The total cost \( C_1 \) for Company 1 is \$5 per window: \[ C_1 = 5x \]

Step 3: Equation for Company 2

The total cost \( C_2 \) for Company 2 includes a base fee of \\(40 and \\)3 per window: \[ C_2 = 40 + 3x \]

Step 4: Set Inequality

Set the inequality to find when Company 2 is cheaper than or equal to Company 1: \[ 40 + 3x < 5x \]

Step 5: Simplify the Inequality

Move the terms involving \( x \) to one side: \[ 40 < 2x \]Then divide both sides by 2: \[ x > 20 \]

Step 6: Interpret the Result

The result \( x > 20 \) indicates that when the number of windows is more than 20, Company 2 is the cheaper option.

Key Concepts

Cost Comparison

When comparing the cost between two services, it's crucial to consider both the fixed and variable components of pricing. In the context of the window washing companies, one charges a simple per-window price, while the other includes both a base fee and a per-window rate. Understanding these different pricing models is key to determining which company is more cost-effective for your needs.

Here are some tips for effective cost comparison:

• Identify all fixed costs, like the base fee for the second company.
• Calculate the variable costs, i.e., the per-window rate.
• Use algebraic expressions to compare total costs at different service levels.

In this exercise, the company's costs are compared via inequality. The aim is to figure out the threshold at which the second company's variable savings outweigh its fixed cost, making it the cheaper option. This makes it essential to perform a detailed cost analysis based on the number of windows you're planning to clean.

Linear Equations

Linear equations are foundational in algebra, allowing us to represent relationships with a constant rate of change. In this problem, each company’s pricing model can be expressed as a linear function of the number of windows. Linear equations have the general form:

• For Company 1: The cost is entirely variable and grows linearly as you add more windows, expressed as \(C_1 = 5x\).
• For Company 2: The cost is a combination of a fixed base rate and a variable part, giving it the equation \(C_2 = 40 + 3x\).

Linear equations allow us to easily calculate the total cost for any number of windows. In practical scenarios, these equations help you make predictions and comparisons, such as determining break-even points where costs are equivalent, or thresholds like in this exercise, where a particular service becomes cheaper.

Inequality Solving

Solving inequalities is a vital skill for comparing outcomes and making decisions based on variable scenarios. Inequalities differ from equations in that they show relationships where one side is greater or less than the other. In this window washing problem, you're asked to find when the cost from the second company becomes cheaper than the first, which involves setting up and solving an inequality.

The process involves:

• Setting up the inequality based on the total cost equations: \(40 + 3x < 5x\).
• Simplifying the inequality: Move \(3x\) to the right to get \(40 < 2x\).
• Solving for \(x\) by dividing both sides by 2 to find \(x > 20\).

This final inequality tells us that more than 20 windows need to be washed for the second company to be cheaper. Understanding inequality solving is crucial for making data-driven decisions when comparing different variable cost scenarios.