Problem 29 - 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 plumber worked for 70 hours and the assistant worked for 35 hours.

Step 1: Define Variables

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

Step 2: Set Up the Equation

The cost for the plumber is \( 45 \times 2x \), and the cost for the assistant is \( 25 \times x \). The total cost is given as \( 4025 \). Set up the equation: \[ 45 \times 2x + 25 \times x = 4025 \] which simplifies to \[ 90x + 25x = 4025. \]

Step 3: Combine Like Terms

Combine the terms to simplify the equation: \( 90x + 25x = 115x \), so \[ 115x = 4025. \]

Step 4: Solve for x

Solve the equation \( 115x = 4025 \) by dividing both sides by 115 to find \( x \): \[ x = \frac{4025}{115}. \]

Step 5: Calculate x

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

Step 6: Determine Plumber's Hours

Since the plumber worked twice as long as the assistant, calculate the plumber's hours: \( 2x = 2 \times 35 = 70 \). The plumber worked for 70 hours.

Key Concepts

Linear Equations

Linear equations are foundational in algebra, often appearing in various forms. In this exercise, a linear equation is used to find the number of hours worked by a plumber and his assistant. A linear equation is expressed as a mathematical statement with an equal sign, combining variables and constants through addition, subtraction, multiplication, or division.
In this context, the linear equation \( 90x + 25x = 4025 \) was formed based on the hourly rates and total hours worked by the plumber and his assistant. This is a first-degree equation, meaning it involves variables raised to the power of one. Solving such equations requires isolating the variable on one side of the equation.

• Identify coefficients: Here, the coefficients are 90 and 25, which are the rates per hour for different workers.
• Combine like terms: Simplifying 90x + 25x gives 115x.
• Isolate the variable: Divide both sides of the equation by 115 to solve for x.

Understanding this fundamental structure helps in solving various real-world problems efficiently.

Word Problems

Word problems translate real-life scenarios into mathematical expressions. This exercise exemplifies a classic word problem found in algebra, where verbal explanations must be turned into math equations.
This problem involves determining the hours worked by a plumber and his assistant to match a given total cost. It's important to:

• Identify key information: The plumber's rate, the assistant's rate, the fact that the plumber works twice as much as the assistant, and the total charge.
• Translate words to math: Assign variables to unknowns. Here, 'x' represents assistant's hours, converting the word problem to a computable equation.
• Logical connections: Relate plumber’s work as twice the assistant’s using multiplication.

Breaking down the word problem helps in accurately setting up the equation, leading to the correct answer and reinforcing problem-solving skills.

Problem Solving

Problem solving in mathematics involves a variety of processes to reach a solution. In solving this exercise, several key strategies were employed to find the hours the plumber and his assistant worked.
First, we defined variables: Assigning 'x' to the assistant's hours helped structure the problem logically. Then, setting up the equation with known hourly rates and relationships allowed the problem to be written in mathematical form. Solving this equation entailed combining like terms, an essential algebraic step to simplify the expression, followed by straightforward arithmetic operations to find the solution.

• Define and assign variables to unknowns.
• Understand relationships between different elements of the problem.
• Execute arithmetic operations correctly and efficiently.

Finally, verifying the solution ensures correctness. Calculating that the total hours worked result in the given total cost confirms the accuracy of our method.
Mastering these strategies boosts analytical thinking, enabling you to tackle a wide array of problems with confidence.