Problem 47 - College Algebra

Question

An automobile repair shop charged a customer \(\$ 448\) listing \(\$ 63\) for parts and the remainder for labor. If the cost of labor is \(\$ 35\) per hour, how many hours of labor did it take to repair the car?

Step-by-Step Solution

Answer

It took 11 hours to repair the car.

Step 1: Calculate the Cost of Labor

First, subtract the cost of the parts from the total billed amount to find the total cost of labor. This can be done by doing the calculation: \[448 - 63 = 385 \]So, \$385 is the total cost of labor.

Step 2: Calculate the Hours of Labor

Next, divide the total labor cost by the cost per hour of labor to get the total working hours. The calculation would be: \[385 \div 35 = 11\]Therefore, it took 11 hours to repair the car.

Key Concepts

Linear Equations

Linear equations are fundamental tools in algebra that help us solve problems by finding unknown values. In the exercise given, the primary task is to find out how many hours of labor were required for car repairs using a linear equation model. A linear equation is typically expressed in the form of \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the unknown variable.
In our scenario:

• "\(x\)" represents the number of hours worked.
• "\(\(35x\)" is the total cost of labor (since each hour of work costs \)35).
• "\($63\)" is another constant representing the cost of parts.
• The equation to find the number of hours can then be built as \(35x + 63 = 448\).

To solve for \(x\), perform algebraic operations to isolate \(x\). Once all constants are moved to one side, you just divide to find \(x\). This systematic way of setting up the problem ensures accuracy and clarity in finding the solution.

Cost Calculation

Cost calculation is an essential skill when dealing with financial aspects of any task or problem, especially in word problems involving purchases or services. In the repair shop scenario, we need to calculate how much was spent on labor after accounting for the parts.
Start by separating known constants, like the cost of the parts, from the total cost. This helps in simplifying the problem:

• The total cost billed is \(\(448\).
• Cost directly attributed to parts is \(\)63\).

Subtracting the cost of parts from the total cost helps us find the total labor cost: \[448 - 63 = 385\].
The result tells us that $385 was spent exclusively on labor. Accurate cost calculation ensures the financial aspects of tasks, such as billing or budgeting, are correctly managed.

Division Operations

Division operations are a key arithmetic skill that allows us to distribute costs evenly or determine per-unit value. In this word problem, division helps us determine how many hours the mechanic worked on the car by using the cost of labor per hour.
Given:

• The total labor cost is \(\(385\).
• The rate of labor is \(\)35\) per hour.

By applying division, we calculate the number of labor hours as follows: \[385 \div 35 = 11\].
This indicates that the mechanic worked for 11 hours on the car. Division, as seen here, simplifies understanding by providing a meaningful breakdown of total costs or values into manageable units, such as hours in labor-cost scenarios.