Problem 21 - Algebra and Trigonometry

Question

Labor Costs A plumber and his assistant work together to replace the pipes in an old house. The plumber charges \(\$ 45\) an hour for his own labor and \(\$ 25\) an hour for his assistant's labor. The plumber works twice as long as his assistant on this job, and the labor charge on the final bill is \(\$ 4025 .\) How long did the plumber and his assistant work on this job?

Step-by-Step Solution

Answer

The assistant worked 35 hours, and the plumber worked 70 hours.

Step 1: Define Variables

Let's define the variables for the time worked. Let \(x\) be the number of hours the assistant works. Since the plumber works twice as long, he works \(2x\) hours.

Step 2: Set Up the Cost Equation

The cost for the assistant is \(25x\) dollars (\\( 25 per hour times \(x\) hours). The cost for the plumber is \(45(2x)\) dollars (\\) 45 per hour times \(2x\) hours). The total labor cost is given as \$ 4025, so we have the equation: \(25x + 45(2x) = 4025\).

Step 3: Simplify the Equation

First, simplify the equation: \(25x + 90x = 4025\). Combine like terms to get \(115x = 4025\).

Step 4: Solve for x

Divide both sides of the equation by 115 to isolate \(x\): \[x = \frac{4025}{115}\]. Calculate \(x\) to find \(x = 35\).

Step 5: Calculate Time Worked by Plumber

Since the plumber works twice as long as the assistant, he works \(2x = 2(35) = 70\) hours.

Key Concepts

Cost Equation

Cost equations are used to determine the total amounts by summing individual components. In this problem, we need to calculate the total labor cost for both the plumber and his assistant. Each has a separate hourly wage multiplied by the hours worked. We need to find the total equation for everyone involved in the work.

To set up the cost equation, consider:

• The assistant earns \(25 per hour, and he works for 'x' hours.
• The plumber earns \)45 per hour, and since he works twice as long, he works '2x' hours.

Thus, the cost equation is: \( 25x + 45(2x) = 4025 \). This equation represents the total cost paid for labor, which helps us figure out how long each person worked.

Variable Definition

Defining variables is a critical step in algebra, as it allows us to express unknowns clearly and solve for them within equations. In this example, 'x' represents the number of hours the assistant worked. This is the starting point for setting up your equations.

Since the plumber works twice as long as his assistant, his working time is represented by '2x'. This relationship between the plumber and assistant’s working time helps us build a cost equation that can be simplified and solved.

Using variables efficiently simplifies complex problems and makes solving them easier.

Solving Linear Equations

Solving linear equations involves isolating the variable to determine its value. Once the cost equation is set up as \( 25x + 45(2x) = 4025 \), we can simplify it.

To simplify:

• Distribute the coefficients correctly: \( 25x + 90x \).
• Combine like terms to get \( 115x = 4025 \).

Next, isolate 'x' by dividing both sides by 115: \[ x = \frac{4025}{115} \].

Calculate to find \( x = 35 \), which means the assistant worked for 35 hours. This clear step-by-step process of isolating is a standard approach in algebra for solving equations.

Labor Cost Calculations

In terms of labor cost calculations, once you have the time worked for each individual, you can confirm the total cost as a check. With 'x' known, we determine each person's hours:

The assistant worked for 35 hours at \(25 per hour. So, his cost is \( 25 \times 35 \), totaling \)875.

Similarly, the plumber worked for \( 2 \times 35 = 70 \) hours at \(45 per hour. So, his cost is \( 45 \times 70 \), totaling \)3150.

Both costs added together, \( 875 + 3150 \), equal the given total cost of $4025.

Labor cost calculations verify that the hours worked and the rate charged meet the total amount billed, ensuring accuracy in problem-solving.