Problem 52 - Elementary Algebra

Question

Joe bought four 8 -foot 2 -by-4 boards for \(\$ 24.00\). How much did he spend per linear foot?

Step-by-Step Solution

Answer

Joe spent \$0.75 per linear foot.

Step 1: Calculate the Total Length

First, we need to find out the total length of all the boards combined. Joe bought four boards, each of 8 feet. Therefore, the total length is calculated by multiplying the number of boards by the length of each board: \[ \text{Total Length} = 4 \times 8 = 32 \text{ feet} \]

Step 2: Determine the Cost per Linear Foot

To find the cost per linear foot, we divide the total amount Joe spent by the total length of the boards. Joe spent \$24.00 in total. Using this information, we perform the following calculation: \[ \text{Cost per Linear Foot} = \frac{24.00}{32} = 0.75 \text{ per foot} \]

Key Concepts

Linear Measurement

Linear measurement is a fundamental concept that deals with measuring the length of objects. In the context of the original exercise, each board that Joe bought has a linear measurement of 8 feet.

• Understanding Length: Length is the longest dimension of an object, stretching from one end to the other. It's an essential aspect of our everyday life, whether measuring a piece of wood or the distance between two cities.
• Feet as a Unit: Feet is a standard unit of measure in the U.S. customary system. When working with construction materials, like boards, linear measurement frequently uses feet to describe their length.

By understanding linear measurement, you can easily calculate the total length of multiple objects by simply multiplying their individual lengths. In Joe's case, measuring total length helps in further calculations needed for cost analysis.

Cost Calculation

Cost calculation is about determining how much money you need to pay or have spent on items. In Joe's problem, he spent a total of $24.00 on boards. The goal is to find out the expense per linear foot, which spreads the total cost evenly over the length of the boards.

• Total Cost: This is the amount of money spent on all the boards combined. Here, it is $24.00.
• Cost Distribution: To find the cost per foot, you divide the total cost by the total feet of the boards. This calculation helps you understand the expense per unit length.

When you perform a cost calculation, you're essentially figuring out how your money is distributed over the products or services you're paying for. This understanding is especially useful when you're budgeting or comparing the costs of different products.

Basic Algebra Problem

Basic algebra often involves understanding how to manipulate numbers and variables to solve problems. In the exercise at hand, you need to perform a couple of simple calculations to find the desired result.

Solving the Problem

The exercise requires an application of a simple division operation, a foundational algebraic concept.

• Problem Setup: Start with the total cost of \(24.00 and the total length of 32 feet.
• Division Step: Divide the total cost by the total length to find the cost per foot: \[ \frac{24.00}{32} = 0.75 \]
• Result Interpretation: The result, 0.75, means that Joe pays \)0.75 for each foot of the board.

Understanding this simple algebraic operation is key to solving many real-life problems efficiently. It's about breaking down the problem into manageable parts and performing the necessary calculations.