Problem 50 - College Algebra Essentials

Question

A repair bill on a sailboat came to \(\$ 1603,\) including \(\$ 532\) for parts and the remainder for labor. If the cost of labor is S63 per hour, how many hours of labor did it take to repair the sailboat?

Step-by-Step Solution

Answer

The total hours of labor it took to repair the sailboat is the solution to Equation in Step 3

Step 1: Calculate the total Labor Cost

Subtract the cost of parts from total bill to determine the cost of labor. In mathematical form, it will be \(1603 - 532\)

Step 2: Calculate Hours of labor

Divide the amount spent on labor by the hourly labor rate to find out the number of hours. If the result is \(X\), then the equation should be \( \frac{X}{63}\)

Step 3: Final Calculation

Plug in the values obtained into the equation derived in Step 2 and solve it to obtain the number of hours

Key Concepts

Basic Algebra

When tackling word problems in algebra, it's important to translate the given situation into mathematical expressions. In this exercise, you're given a total repair bill, a specific cost for parts, and the hourly labor rate. The goal is to find out how many hours of labor were involved. Understanding the problem starts by defining variables. Often, you can solve these problems by simple arithmetic operations. Here, you need to subtract the cost of parts from the total bill and then use division to find the hours of labor:

• First, calculate the labor cost: Subtract the part cost (\(\\(532\) from the total bill \(\\)1603\)
• Second, calculate the hours spent: Divide the total labor cost by the hourly labor rate \(\$63\)

This approach highlights how basic algebra is used to structure and solve real-world problems.

Cost Analysis

Cost analysis involves breaking down a problem to understand where and how money is spent. In our sailboat repair scenario, differentiating between parts cost and labor cost is key to understanding the financial breakdown. This form of analysis helps to see exactly how resources are allocated:

• Recognize the part cost of \(\\(532\) is already clear and separate.
• You calculate the remaining as labor cost: \(\\)1603 - \\(532 = \\)1071\).
• Remember that labor cost is directly linked to the number of working hours and the rate per hour.

By correctly analyzing costs, you pinpoint the factors that significantly contribute to the total, and understand the cost-efficiency of tasks or services.

Mathematical Reasoning

Mathematical reasoning allows you to deduce new information based on the provided data and a logical approach. When evaluating the repair bill problem, reason logically from both the known and unknown quantities. Begin with whatever clear numerical facts you have, such as:

• Total bill amount (\(\\(1603\))
• Parts cost (\(\\)532\))
• Hourly rate for labor (\(\$63\))

From here, use reasoning to derive the labor hours:

• Calculate the remaining labor cost by subtracting the parts cost from the total bill.
• Determine the number of labor hours by dividing this labor cost by the hourly rate.

Through logical reasoning, you derive the labor hours and ensure the solution is consistent with the context of the problem.