Problem 29 - Precalculus Mathematics for Calculus

Question

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 \( x \) be the number of hours the assistant worked. Since the plumber works twice as long, the plumber worked \( 2x \) hours.

Step 2: Set Up Equation for Total Cost

The total labor cost is calculated by adding the plumber's and assistant's charges: \( 25x + 45(2x) = 4025 \).

Step 3: Simplify the Equation

Simplify the equation: \( 25x + 90x = 4025 \) which results in \( 115x = 4025 \).

Step 4: Solve for Assistant's Hours

Solve for \( x \) by dividing both sides of the equation by \( 115 \): \( x = \frac{4025}{115} \).

Step 5: Calculate Assistant's Hours

Perform the division: \( x = 35 \). This means the assistant worked 35 hours.

Step 6: Calculate Plumber's Hours

Since the plumber worked twice as long as the assistant, the plumber worked \( 2 \times 35 = 70 \) hours.

Key Concepts

linear equations

Linear equations play a crucial role in solving many practical problems, such as determining the time both the plumber and his assistant worked in this scenario. To understand linear equations, it's best to think about their form: they are equations where each term is either a constant or the product of a constant and a single variable. In our exercise, we introduced variables to express the relationship between the plumber's and assistant's hours and the total cost of the labor.

We start by defining variables: let's say the assistant worked for \( x \) hours. Knowing that the plumber worked twice as long, we defined the plumber's hours as \( 2x \). This setup leads us to form a linear equation to express the total cost: \( 25x + 45(2x) = 4025 \).

• Understanding Each Component: The term \( 25x \) represents the assistant's labor charge, while \( 45(2x) \) represents the plumber's labor charge.
• Formulating the Equation: Combining these charges gives us the total charge, resulting in a linear equation.

By solving this equation, we find out how many hours each worked, showing the power of linear equations in problem-solving contexts.

problem solving

Problem solving involves identifying the challenge, breaking it down into comprehensible parts, and using logical procedures to find a solution. In this exercise, we aimed to find how long the plumber and his assistant worked together, which is a classic problem that involves logical deduction through steps.

The key here is to translate a real-world scenario into a mathematical model. We start by defining the problem and identifying what's given and what's needed. We know:

• The plumber charges \( \\(45 \) per hour.
• The assistant charges \( \\)25 \) per hour.
• The total labor cost is \( \$4025 \).
• The plumber worked twice as long as the assistant.

Through this breakdown, we can systematically set up an equation by expressing these relationships mathematically. This logical and methodical approach makes problem solving a structured task, allowing us to reach the solution efficiently. Once the equation is set up, solving it reveals the desired information on each worker's hours.

algebra

Algebra is a branch of mathematics that uses symbols and variables to solve equations and model real-world situations. In this problem, algebra helps us make sense of the relationship between different rates of charges and the time worked.

Using symbols, we assign values to unknown quantities: the hours worked in our case. By expressing the labor charges for both the plumber and his assistant as algebraic expressions, we can set up the equation \( 25x + 90x = 4025 \). Here, algebra simplifies the calculation of complex scenarios into manageable portions.

• Simplification: By combining like terms to form \( 115x = 4025 \), algebra allows us to simplify the equation.
• Solving the Equation: Algebraic techniques involve performing operations (such as division) to isolate variables and solve for their values.

In doing this, algebra provides a reliable framework for tackling a wide array of problems, demonstrating its critical role in both academic and real-world applications.