Problem 42 - 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 on the lumber.

Step 1: Convert Fence Post Length

First, convert the measurements from feet and inches into just feet. Each fence post is 5 feet and 4 inches. Since there are 12 inches in a foot, we convert 4 inches into feet:\[ 4 \text{ inches} = \frac{4}{12} \text{ feet} = \frac{1}{3} \text{ feet} \]So, each fence post is:\[ 5 + \frac{1}{3} = \frac{15}{3} + \frac{1}{3} = \frac{16}{3} \text{ feet} \].

Step 2: Calculate Total Length of Lumber

Next, multiply the length of one fence post by the number of posts to find the total length of lumber needed. There are 6 fence posts.\[ \frac{16}{3} \text{ feet per post} \times 6 \text{ posts} = \frac{16 \times 6}{3} = \frac{96}{3} = 32 \text{ feet} \].

Step 3: Calculate Total Cost

Now, calculate the total cost by multiplying the total length by the cost per foot of lumber. Each foot costs $3.\[ 32 \text{ feet} \times 3 \text{ dollars per foot} = 96 \text{ dollars} \].

Key Concepts

Unit Conversion

Unit conversion is a fundamental skill in mathematics, especially in prealgebra word problems. In this scenario, Trish's fence posts are measured in feet and inches, and we need a single unit for uniformity. Normally, we deal with feet for better clarity and simplicity when calculations involve costs per foot. To convert inches to feet, consider that 1 foot equals 12 inches.

• Divide the number of inches by 12 to find its equivalent in feet.
• In Trish's problem, she has 4 inches to convert, which results in \( \frac{4}{12} = \frac{1}{3} \) feet.

Combining this with the whole feet measurement (5 feet), we use addition:

• Add the converted inches to the whole feet to get the fence post length: \( 5 + \frac{1}{3} = \frac{16}{3} \) feet.

Understanding these conversions ensures we avoid errors and helps in aligning the dimensions with other necessary calculations.

Multiplication of Fractions

When dealing with prealgebra word problems, multiplication of fractions is crucial, especially when calculating totals. In this exercise, once the unit conversion is complete, you often need to multiply fractions to find larger sums or products.Consider the multiplication of the length of one fence post by the total number of posts. Here's how it's tackled:

• Trish has \( \frac{16}{3} \) feet per post.
• She needs to multiply this by 6 posts.
• Using fraction multiplication, multiply the numerator by 6: \( \frac{16}{3} \times 6 = \frac{96}{3} \).

Finally, simplify \( \frac{96}{3} \) to get 32 feet as the total length. Always ensure to simplify the fractions after multiplication. Simplification makes it easier to interpret the answers in real-world contexts like buying materials.

Math Problem Solving

Developing strong math problem-solving skills is essential when approaching prealgebra problems. It involves breaking down a problem, converting units, and performing calculations carefully.Break the problem into manageable parts:

• Understand the problem statement: Ensure you know what is being asked.
• Translate words into mathematical expressions: Recognize costs and measurements.
• Follow a structured approach: Convert, compute, and check your computations for any errors.

In math problem solving, physical representations like equations can help. For instance, calculatingthe total cost needs equations: Total cost = (total length) × (cost per foot). With 32 feet and $3 per foot, calculated as:

• \( 32 \times 3 = 96 \) dollars.

This systematic breakdown ensures every dollar is accounted for, and post-calculation checks confirm logical accuracy. Having a consistent strategy enhances clarity and leads to correct resolutions in problem scenarios like Trish's.