Problem 95 - Elementary Algebra

Question

A quilter is buying fabric that is 44 in. wide to make a back for a rectangular quilt that is 102 in. wide and 120 in. long. The quilter needs to buy three strips of fabric that are as wide as the quilt plus two additional inches of fabric on each end of the strip. Find the amount of fabric in yards that the quilter should buy. Round \(u p\) to the nearest tenth.

Step-by-Step Solution

Answer

The quilter should buy 8.8 yards of fabric.

Step 1: Determine the Width of Each Strip

The width of each strip of fabric needs to be the width of the quilt plus two additional inches on each end. The width of the quilt is 102 inches, so calculate the total width of each strip:102 in + 2 in + 2 in = 106 in.

Step 2: Calculate Total Fabric Length Needed

The quilter needs three strips that are each 106 inches wide in order to cover the entire length of the quilt. Since each strip is 44 inches wide, determine how many strips are needed to cover the 120-inch length. To do this:120 in / 44 in ≈ 2.727 ≈ 3 strips.So, three strips are needed.

Step 3: Calculate Total Fabric Length in Inches

Since three strips are needed and each strip must be 106 inches wide, calculate the total length of fabric needed in inches:3 strips * 106 in = 318 inches.

Step 4: Convert Inches to Yards

There are 36 inches in a yard, so to convert the total length of fabric from inches to yards, use the following conversion:318 in / 36 in/yard ≈ 8.833 yards.

Step 5: Round to the Nearest Tenth

Finally, round the total length of fabric to the nearest tenth of a yard:8.833 yards ≈ 8.8 yards.

Key Concepts

Unit Conversion

When working on quilting projects, it is essential to understand how to convert units of measurement. In this exercise, we needed to convert inches to yards because fabric is often sold by the yard in stores.
There are 36 inches in a yard, so to convert inches to yards, you divide the number of inches by 36. For example, if we have 318 inches of fabric, we can convert this to yards by calculating \(\frac{318}{36} = 8.833\). This helps us determine that 318 inches is approximately 8.8 yards when rounded to the nearest tenth.

Understanding unit conversion ensures that you buy the correct amount of fabric, thereby avoiding the risk of having too little fabric for your quilting project.

Rectangular Quilts

Quilting projects often involve dealing with rectangular shapes. Essentially, a rectangular quilt has a width and length that must be precisely measured and taken into account when buying fabric.
In our exercise, the quilt's dimensions were 102 inches wide and 120 inches long. It's important to ensure that the fabric strips will cover the entire width and length of the quilt to avoid any gaps in coverage.
Understanding the shape and size of your quilt helps in planning and executing your project by assuring that the fabric pieces fit the intended design and dimensions.

Fabric Calculation

Calculating fabric requirements involves several steps to ensure you have enough material for your quilting project. Here's a structured approach:
1. **Determine the Width of Each Strip**: The width should include the width of the quilt plus additional inches for seams. In this case, 102 inches plus 4 inches (2 inches on each end) gives 106 inches.
2. **Determine the Number of Strips Needed**: If each strip of fabric is 44 inches wide, and your quilt length is 120 inches, you calculate the number of strips by dividing 120 by 44, resulting in approximately 2.727 strips, rounding up to 3 strips.
3. **Total Fabric Calculation**: Multiply the number of strips (3) by their width (106 inches) to get the total fabric length in inches, which is 318 inches.
This step-by-step method ensures that you properly calculate the total fabric needed for a project, thus preventing any shortage or excess.

Algebraic Problem-Solving

Algebra provides tools for solving various problems that involve finding unknown variables. In our quilting example, we tackled the problem using algebraic steps and simple arithmetic:
1. **Identify Variables and Known Values**: Variables include the width and length of the quilt and fabric strips. Known values are the quilt's dimensions and fabric specifications.
2. **Set Up Equations**: Structure our calculations step-by-step as we did, ensuring logical flow from one step to the next.
3. **Problem Solving**: Sequentially solving for smaller parts (width, length, total fabric) ensures we don’t miss any important aspects. Converting units and rounding off final answers are crucial parts, ensuring practical and usable results.
This methodical approach helps break down complex problems into manageable pieces, resulting in accurate solutions.