Problem 40 - Prealgebra

Question

cost of Lumber Trish is building a fence. She buys six fence posts at the lumberyard, each measuring \(5 \mathrm{ft}\) 4 in. The lumber costs $$ 3$ per foot. How much will Trish spend?

Step-by-Step Solution

Answer

Trish will spend $96.

Step 1: Convert Measurements to Feet

Each fence post is 5 feet 4 inches long, which needs to be converted entirely into feet. Since 1 foot equals 12 inches, we first convert the 4 inches to feet: \(\frac{4}{12} = \frac{1}{3}\) feet. So, each fence post is \(5 + \frac{1}{3} = \frac{16}{3}\) feet long.

Step 2: Calculate Total Length of Lumber

There are 6 fence posts, each measuring \(\frac{16}{3}\) feet. To find the total length of all the posts, multiply the length of one post by the number of posts: \(6 \times \frac{16}{3} = \frac{96}{3} = 32\) feet. This means that Trish buys 32 feet of lumber.

Step 3: Calculate Total Cost

Each foot of lumber costs \(3\) dollars, and Trish is buying 32 feet. Therefore, the total cost is \(32 \times 3 = 96\) dollars.

Key Concepts

Linear Measurements

Understanding linear measurements is crucial when working with materials like wood. When measuring the length of objects, you commonly use units such as feet and inches. Sometimes, these measurements need to be converted to a uniform unit for easier calculations.
For example, if you have a length of 5 feet and 4 inches, it's useful to convert everything into feet. We know 1 foot equals 12 inches, so 4 inches can be converted to feet by dividing by 12. This gives \( \frac{4}{12} = \frac{1}{3} \) feet. Add this to the original 5 feet, which results in \( 5 + \frac{1}{3} = \frac{16}{3} \) feet.
Converting measurements into a single unit simplifies calculations and helps avoid errors.

Cost Calculation

Cost calculations help determine how much money you need to complete a project. When buying materials priced per unit length, like wood, you need to know both the total length required and the cost per unit.
Let's say you're buying lumber where each foot costs $3. If you have calculated that you'll need 32 feet, multiply the total length by the cost per foot to find the total cost.

• Total length: 32 feet
• Cost per foot: 3 dollars

Multiply these to get \( 32 \times 3 = 96 \) dollars, which is the total amount you'll spend.
Always remember to double-check each calculation to ensure accurate budgeting.

Multiplication

Multiplication is a fundamental arithmetic operation that helps solve a variety of problems, like calculating total amounts and costs.
In this exercise, multiplication is used to find both the total length of the lumber and the total cost.

• First, calculate total length: Given 6 fence posts, each \( \frac{16}{3} \) feet, the total length is \( 6 \times \frac{16}{3} = 32 \) feet.
• Second, find the total cost: Multiply the total feet (32) by the cost per foot (3) to get \( 32 \times 3 = 96 \) dollars.

Always ensure multiplication is done correctly, especially when dealing with fractions or decimals, as mistakes can impact the final outcome. Multiplication should be clear and checked twice for accuracy.