Problem 16 - Algebra A Combined Function

Question

A plumber gave an estimate for the renovation of a kitchen. Her hourly pay is \(\$ 27\) per hour and the plumbing parts will cost \(\$ 80 .\) If her total estimate is \(\$ 404\), how many hours does she expect this job to take?

Step-by-Step Solution

Answer

The job is expected to take 12 hours.

Step 1: Understand the Problem

The plumber has given an estimate of a total cost for renovating a kitchen, which includes her hourly pay and the cost of materials. We need to determine how many hours she expects to work based on the total cost estimate.

Step 2: Set Up the Equation

Let the number of hours be represented by \( h \). The cost of the plumber's labor is \( 27h \) dollars, and the cost of parts is \( 80 \) dollars. The total estimated cost is \( 404 \) dollars. So, the equation is: \( 27h + 80 = 404 \).

Step 3: Solve for the Number of Hours

Subtract the cost of parts from the total estimate to find the labor cost alone: \( 404 - 80 = 324 \).

Step 4: Calculate the Hours

Divide the labor cost by the hourly rate to find the number of hours: \( 324 \div 27 = 12 \).

Step 5: Conclusion

The plumber expects to work for 12 hours on the kitchen renovation.

Key Concepts

Setting Up Equations

Setting up equations is a crucial stage in solving algebra word problems, as it involves translating the problem statement into a mathematical expression. For the plumber's problem, we are given specific costs: her hourly pay is \( \\( 27 \) per hour, and the cost of materials is \( \\) 80 \). The total estimate cost for the renovation is \( \\( 404 \).

To find the number of hours she plans to work, we need to establish what part of the total cost is variable and what part is fixed. The variable portion is her hourly pay, dependent on the hours \( h \). Thus, the cost for labor is \( 27h \). The fixed portion is the \( \\) 80 \) cost of materials.

By setting up the equation \( 27h + 80 = 404 \), we express the total cost as the sum of both the labor and materials. This equation helps us isolate the variable, \( h \), and solve for it to find exactly how long the job should take.

Solving Equations

Solving equations involves manipulating the equation to uncover the value of the unknown variable. Once our equation \( 27h + 80 = 404 \) is set, the next step is to isolate \( h \) by performing operations that simplify the equation.

First, we handle the fixed cost, which is \( \\( 80 \). By subtracting this from the total \( \\) 404 \), we find the variable cost which can be attributed to labor alone. So, \( 404 - 80 = 324 \).

Now, the modified equation becomes \( 27h = 324 \). The next and final step is to solve for \( h \) by dividing both sides of the equation by the hourly rate, \( 27 \). Therefore, \( 324 \div 27 = 12 \). This shows that the plumber plans to work for 12 hours to complete the kitchen renovation.

Understanding Problem Statements

Understanding problem statements is an essential skill, especially in algebra word problems. It enables you to translate written text into a mathematical framework. For this particular problem, discerning key pieces of information is a must. The estimate includes both labor (hourly pay) and a fixed cost (materials).

Initially, the problem gives us three main figures: the hourly wage \( \\( 27 \) per hour, a fixed cost of \( \\) 80 \), and a total estimate of \( \$ 404 \). Understanding these values and the expected outcome guides how you approach setting up equations for the solution.

To solve effectively, scrutinize each part of the statement to ask relevant questions like, "What does the total cost include?" and "What part of the total cost varies?" This analysis leads to a logical setup and prevents errors in solving the problem. It's about picking apart complex-sounding paragraphs to unveil something manageable and solvable with mathematical reasoning.