Problem 54 - Algebra 1: Concepts and Skills

Question

The bill (parts and labor) for the repair of a car is \(\$ 458 .\) The cost of parts is \(\$ 339 .\) The cost of labor is \(\$ 34\) per hour. Which equation could you use to find the number of hours of labor? F) \(34(x+339)=458\) G) \(34+339 x=458\) H) \(34 x+339=458\) J) \(34+x+339=458\)

Step-by-Step Solution

Answer

The correct equation to find the number of hours of labor is H) \(34 x + 339 = 458\).

Step 1: Understand the Problem

The total cost of the car repair bill is $458. This total bill is made up of two components - parts and labor. The cost of parts is a fixed cost of $339. The cost of labor is $34 per hour, but we do not know how many hours of labor were spent, which is represented by 'x'.

Step 2: Formulate the Equation

In this step, we need to setup an equation that represents the given situation. The total cost is the sum of the cost of parts and cost of labor. So, the equation should be of the form cost of parts + cost of labor (hourly labor cost multiplied by number of hours) = total cost. Translated into numerical equation, it would look like: $339 (parts) + $34(x) (labor) = $458 (total).

Step 3: Identify the Right Equation

Looking at the options provided, the correct equation fitting our derived format is \(34 x + 339 = 458\) which corresponds to option H.

Key Concepts

Cost Analysis

Understanding cost analysis is essential in solving mathematical problems involving expenses. It involves breaking down overall costs into simpler, more manageable components. This skill allows us to analyze and comprehend how different factors contribute to the total cost. In the given exercise, the total cost includes both parts and labor. Knowing this, we can isolate each part to understand its contribution.

Cost analysis often includes:

• Identifying each component of cost (e.g., parts and labor).
• Understanding fixed costs, like the $339 for car parts in the exercise.
• Factoring in variable costs, such as the hourly rate for labor.

This approach simplifies complex problems and enables efficient calculation. Now let's see how hourly rate calculation fits into this analysis.

Hourly Rate Calculation

Calculating hourly rates is crucial when dealing with tasks that involve labor costs. To find the total labor expense, multiply the hourly rate by the number of hours worked. In our scenario, the hourly rate for car labor is $34. However, the number of hours worked is unknown, and we represent it as 'x'.

In mathematical terms, the total labor cost is calculated using the formula:

• Hourly Rate \( \times \) Number of Hours = Total Labor Cost
• Given: \( 34 \times x \)

This method helps you isolate labor costs and understand its impact on the total bill. Calculating labor this way helps us solve for unknowns more efficiently, which is a key element of problem-solving in mathematics.

Problem-Solving in Mathematics

Problem-solving in mathematics often requires dissecting a scenario into manageable parts and then recombining those parts to find a solution. Applying problem-solving skills can help in understanding the structure of equations and fitting given information into those equations.

In this exercise, we start with a given total cost and specific known expenses (cost of parts $339), then factor in unknowns (hours of labor) and determine how they fit into the equation. Here's how we approach such problems:

• Read and understand the problem statement thoroughly.
• Identify known and unknown factors.
• Set up equations based on those factors.
• Solve for unknowns by rearranging the equation.

This systematic approach ensures that each component of the problem is given attention, ensuring a thorough and complete solution. Employing these strategies, mathematics becomes a powerful tool for solving real-world problems like cost analysis.