Problem 33 - Prealgebra

Question

Carl is having a mechanic fix his car. The mechanic said that the job was going to cost at least \(\$ 375\) for parts and labor. If the cost of the parts was \(\$ 150,\) and the mechanic charges \(\$ 60\) an hour, how many hours is the mechanic planning on working on the car?

Step-by-Step Solution

Answer

The mechanic is planning on working around 4 hours.

Step 1: Understanding the Problem

First, we need to identify what is being asked. We know that the total cost for fixing the car is at least \( \\(375 \). The cost of parts is \( \\)150 \), which means the total amount that will be spent on labor is \( \\(375 - \\)150 \). We know the mechanic charges \( \$60 \) per hour.

Step 2: Calculate Labor Cost

The total cost minus the cost of parts gives us the labor cost. So, calculate: \[ \text{Labor Cost} = 375 - 150 = 225 \] The mechanic will charge \( \$225 \) for the labor.

Step 3: Determine Hours of Work

Since the mechanic charges \( \$60 \) per hour, divide the labor cost by the hourly wage to find out how many hours the mechanic plans to work: \[ \text{Hours Worked} = \frac{225}{60} = 3.75 \] However, since mechanics typically round to the nearest whole hour, the mechanic is likely planning on charging for 4 hours.

Key Concepts

Problem Solving

One fundamental concept in prealgebra is problem solving, particularly when dealing with real-life scenarios like calculating costs. In the given problem, Carl is facing a practical situation where he needs to estimate the cost of repairing his car. At its core, problem solving in mathematics involves

• Understanding what is being asked
• Decomposing the problem into manageable parts
• Using relevant mathematical operations to find the solution

This particular exercise requires us to figure out how many hours of labor Carl will be charged based on given costs. Recognizing the relationship between total cost, part cost, and labor cost is crucial.
By clearly identifying these components, we can then solve the problem systematically using basic arithmetic operations.

Step-by-Step Solutions

Step-by-step solutions are vital because they break down complex problems into simple, achievable tasks. This approach makes it easier for you to tackle each part of the problem without feeling overwhelmed. Let's revisit how the solution for Carl's car repair cost was tackled.
First, understanding the problem allowed us to discern that the total cost of the repair is a sum of part and labor costs. By calculating the labor cost, we used basic arithmetic: subtracting the part cost from the total cost to find out what will be spent on labor.
The next step involved using this labor cost to determine how many hours the mechanic would work. By dividing the labor cost by the hourly rate, we found the planned hours of work. By following each logical step, even the most complex problems can be made easier to understand and solve.

Basic Arithmetic Operations

Basic arithmetic operations are the foundation of mathematics and problem-solving in prealgebra. In this exercise, we primarily used subtraction and division. Let's break down how these operations were applied:

• Subtraction: We subtracted the cost of parts (\(150\)) from the total estimated cost (\(375\)) to find the amount allocated to labor costs, which was \(225\).
• Division: Once the labor cost was found, we divided it by the hourly wage (\(60\)) to figure out the number of hours the mechanic planned to work, which calculated to 3.75 hours.

This division indicated that the mechanic would likely charge for 4 hours, rounding to the nearest whole number. Understanding how to use these basic operations is key to unlocking more challenging math concepts.