Problem 82 - Elementary Algebra Within Reach

Question

Labor Cost The total cost for a new deck (including materials and labor) was \(\$ 1830\). The materials cost \(\$ 1500\) and the cost of labor was \(\$ 55\) per hour. How many hours did it take to build the deck?

Step-by-Step Solution

Answer

It took 6 hours to build the deck.

Step 1: Understand the problem

The total cost of the deck includes the cost of the materials along with the labor cost. The deck cost a total of \(\$1830\), materials cost \(\$1500\) and the labor \(\$55\) per hour. The goal is to determine the number of hours it took to build the deck.

Step 2: Subtract the cost of materials from the total cost

First, the cost of the materials (\$1500) is subtracted from the total cost (\$1830). This will give the total amount spent on labor. Calculation: \(\$1830 - \$1500 = \$330\)

Step 3: Determine the number of hours

Finally once the labor cost has been determined, this is divided by the rate of labor to find the number of hours. Calculation: \(\$330 ÷ \$55/hour = 6 hours\)

Key Concepts

Cost Analysis

Understanding cost analysis is essential in solving algebra word problems that involve expenses. In problems where you break down total costs into different components, cost analysis helps you identify and isolate these components.
For instance, analyzing the cost of building a deck involves more than just the total price. You need to separate the cost of materials from the labor cost.

• Identify components: Look at your total cost and identify all elements contributing to it. In our example: total cost = materials cost + labor cost.
• Direct expenses: These are costs like materials that you know upfront, they don't vary with the number of hours worked.
• Variable expenses: These are expenses like labor, which depend on hours worked.

By dissecting and analyzing these costs, you can determine where your money goes and understand the problem more deeply.

Problem Solving

Problem solving is a fundamental skill in algebra that helps you approach complex questions with a structured plan. In our deck-building problem, you use specific steps to reach a solution.
A robust approach to problem solving includes:

• Step-by-step process: Start by understanding the problem. Know what is given (like total cost and rate per hour) and what you need to find (like the hours worked).
• Reduction technique: Break the problem into simpler parts. For example, to find out how many hours the laborers worked, first determine how much was spent on labor.
• Logical reasoning: Ensure each step logically follows the previous one. Subtract material cost from total cost to focus solely on labor expenses.

By following these strategies, you tackle algebraic problems with clarity and confidence.

Equation Solving

Equation solving in algebra allows you to find unknowns from given information using mathematical expressions.
In our exercise with the deck, the unknown is the number of labor hours.

• Define the equation: Write down the equation that represents the situation. Here, it becomes \(55h = 330\), where \(h\)\ represents the hours worked.
• Manipulate the equation: Use algebraic operations to solve for the unknown. Divide both sides of the equation by 55 to isolate \(h\).
  
• Solve for the unknown: Calculate \(h\) ), which gives you the answer, 6 hours in this case.
  Equation solving simplifies complex problems by providing a clear method to find unknown values. Practice consistently to improve your skills in tackling word problems with confidence.