Problem 58 - Elementary Algebra Within Reach

Question

Car Repair The bill for the repair of your car was \(\$ 553\). The cost for parts was \(\$ 265\). The cost for labor was \(\$ 48\) per hour. How many hours did the repair work take?

Step-by-Step Solution

Answer

The repair work took 6 hours.

Step 1: Identify Given Variables

The cost of repair is \(553\), the cost of parts is \(265\) and the labor cost is \(48\) per hour.

Step 2: Subtract the Cost of Parts from the Total Cost

Subtract the cost of the parts from total cost to find out how much was spent on labor. So, the total labor cost will be \(553-265 = 288\) dollars.

Step 3: Calculate the Hours of Labor

We know labor cost is \$48 per hour. So, divide total labor cost by the cost per hour to find out the number of hours worked. Number of hours will be \(\frac{288}{48}=6\) hours.

Key Concepts

Linear Equations

Understanding linear equations is essential when solving word problems in algebra. A linear equation is an equation that models a linear relationship between two quantities. In our car repair example, we want to determine the number of hours (\( h \)) it takes to complete the repair. The problem gave us a total bill and the breakdown of costs, and we can use this information to form a linear equation:
\[ 48h + 265 = 553 \]
This equation signifies that the total cost (\( 553 \)) is the sum of the labor cost (\( 48h \)) and the cost of the parts (\( 265 \)).

• The coefficient (\( 48 \)) represents the cost of one unit of labor which in this case is an hour.
• The constant term (\( 265 \)) is the cost of the car parts.

Solving linear equations like this involves rearranging it to isolate the variable, allowing us to find the unknown value. In our example, that unknown is the number of hours (\( h \)) needed for the repair.

Problem Solving

Problem-solving is a key skill taught in algebra that allows you to use logical steps to arrive at an answer. Let's break down the approach using our example word problem.
When tackling a problem:

• Start by understanding what you know: identify the given values (total cost, cost of parts, labor cost per hour).
• Then, figure out what you need: in this case, the number of hours worked.

The next step is to succinctly model what you have with an equation. A well-defined equation incorporates all variables and knowns. Solving it step by step can then reveal the unknown.

Rechecking your calculations can help ensure accuracy, much like in our car repair task where every step leads logically to the next. Begin with finding the total labor cost then dividing by the hourly cost, which gives you the completed amount of time necessary for the car repair.

Mathematical Operations

Mathematical operations, such as addition, subtraction, multiplication, and division, are the building blocks of algebra. They help break down a problem into manageable parts. In the context of our exercise:

• Subtraction was used to find the total labor cost by deducting the cost of parts from the total bill (\( 553 - 265 = 288 \)).
• Division was then applied to determine how many hours worked by dividing the total labor cost by the hourly rate (\( \frac{288}{48} = 6 \)).

These operations all have their own rules and applications, which must be applied correctly to solve equations:

• Subtraction isolates a part from a whole.
• Division breaks down a total into smaller equal parts.

In this example, understanding these operations allowed the determination of the unknown number of hours necessary to complete the repair. Mastering these operations will support you in solving more complex algebraic problems as your skills develop.