Problem 69 - Prealgebra

Question

A wallpaper roll contains a sheet that is 40 feet long and 18 inches wide. What is the minimum number of rolls of wallpaper needed to cover 500 square feet of wall space?

Step-by-Step Solution

Answer

You need 9 rolls of wallpaper to cover 500 square feet.

Step 1: Convert All Measurements to Consistent Units

First, we'll convert the width from inches to feet to match the length's units. Since there are 12 inches in a foot, the width in feet is \( \frac{18}{12} = 1.5 \) feet.

Step 2: Calculate the Area of One Roll

Each roll therefore covers an area of \( 40 \times 1.5 = 60 \) square feet.

Step 3: Determine the Number of Rolls Needed

We need to cover 500 square feet. To find out how many rolls are needed, divide the total area by the area covered by one roll: \( \frac{500}{60} \approx 8.33 \).

Step 4: Round Up to the Nearest Whole Number

Since you can't buy a fraction of a roll, round 8.33 up to the nearest whole number. Therefore, you need 9 rolls to ensure complete coverage.

Key Concepts

Unit Conversion

When solving problems involving measurements, using consistent units is essential. This ensures accuracy and simplifies calculations. In this exercise, we have a wallpaper roll with a width given in inches and a length in feet. To make calculations easier, it’s important to express both dimensions using the same unit.

Since there are 12 inches in a foot, converting inches to feet often simplifies the problem. For example, a width of 18 inches can be converted to feet by dividing by 12:

• 18 inches ÷ 12 = 1.5 feet.

This conversion allows us to easily calculate the area of the wallpaper roll using a single unit, enhancing clarity and preventing errors.

Geometry

In geometry, calculating area is a fundamental skill. It applies when determining the coverage of an object or material, like wallpaper. The area can be found by multiplying the length and width, provided they are in the same unit.

For the wallpaper roll, once the width is converted to feet, the area is calculated by multiplying the length (40 feet) by the converted width (1.5 feet). The formula used here is:

• Area = Length × Width

Thus, the area of one roll is:

• Area = 40 feet × 1.5 feet = 60 square feet

Understanding how to determine area is crucial, especially when dealing with shapes in real-world scenarios.

Problem Solving

Effective problem solving often involves breaking down a complex problem into manageable parts. In this exercise, we need to cover a specified area with wallpaper rolls. To solve it effectively, follow a step-by-step approach:

• Convert measurements to the same units.
• Calculate the area of one roll.
• Determine how many rolls are needed to cover the total area.

We need to cover 500 square feet. By calculating the area covered by a single roll (60 square feet), we then divide the total area needed by the area covered by one roll: \[ \frac{500}{60} \approx 8.33 \]Since purchases are in whole rolls, round up to ensure complete coverage, totaling 9 rolls. This systematic approach ensures accuracy and efficiency.

Real-World Applications

Calculating the area and determining how much material is needed is an essential skill in many real-world tasks. From interior design and construction to everyday projects, such calculations help manage resources effectively.

When you understand these basic concepts, you can make informed decisions about supplies needed in various situations, reducing waste and ensuring efficiency. For instance, knowing how to calculate the area helps in purchasing the right amount of paint, flooring, or wallpaper, as we explored. This not only saves money, but also ensures the project is completed without unnecessary delays.

Applying these principles helps bridge the gap between theoretical math problems and practical, real-world usage, which is invaluable in personal and professional tasks.