Cost Calculation For Carpeting A Rectangular Room

Introduction

In this article, we will delve into a practical problem involving area calculation and cost estimation. Specifically, we'll tackle the scenario of carpeting a rectangular room. This is a common task in home improvement and interior design, and understanding the steps involved can save you both time and money. Our main goal is to determine the total cost of carpeting a room, considering its dimensions and the price of carpet tiles. We will go through the process step-by-step, from calculating the area of the room to figuring out the number of carpet tiles needed and the final cost. Whether you're a homeowner planning a renovation or simply interested in the application of mathematical concepts in everyday life, this guide will provide you with a clear and comprehensive understanding of the calculations involved. So, let's embark on this journey of mathematical problem-solving and discover how to efficiently and cost-effectively carpet a room.

Problem Statement

We are presented with a rectangular room that has a length of 5.6 meters and a width of 4 meters. The task at hand is to carpet this room entirely using square carpet tiles. These tiles measure 40 cm by 40 cm and are sold in packs of 20 for £16 per pack. Our objective is to determine the total cost required to completely carpet the room. This problem involves several steps: calculating the area of the room, converting units to ensure consistency, determining the area of each carpet tile, finding the number of tiles needed, calculating the number of packs to purchase, and finally, determining the total cost. By breaking down the problem into these smaller steps, we can systematically arrive at the solution. This type of problem is not only relevant in practical scenarios like home renovation but also helps in developing problem-solving skills and understanding the application of mathematical concepts in real-world situations. The ability to accurately calculate areas and costs is a valuable skill in many aspects of life, from planning a home improvement project to managing a budget.

Step 1: Calculate the Area of the Room

The initial step in determining the cost of carpeting the room is to calculate the room's total area. The room is rectangular, and the area of a rectangle is calculated by multiplying its length by its width. In this case, the room has a length of 5.6 meters and a width of 4 meters. Therefore, the area of the room can be calculated as follows:

Area = Length × Width Area = 5.6 m × 4 m Area = 22.4 square meters

This calculation gives us the total area that needs to be covered with carpet tiles. Understanding this area is crucial for the subsequent steps, as it allows us to determine the number of carpet tiles required. The area calculation is a fundamental concept in geometry and is widely used in various real-world applications, such as construction, interior design, and landscaping. By accurately calculating the area, we can ensure that we purchase the correct amount of materials, avoiding wastage and unnecessary costs. This step highlights the importance of basic mathematical principles in practical problem-solving. Now that we have the total area of the room, we can move on to the next step, which involves converting units to ensure consistency in our calculations. This is a critical step, as using different units can lead to errors in the final result.

Step 2: Convert Units (Meters to Centimeters)

To proceed with our calculations, it's essential to have all measurements in the same units. We have the dimensions of the carpet tiles in centimeters (40 cm x 40 cm), while the room dimensions are in meters (5.6 m x 4 m). To avoid confusion and errors, we need to convert the room dimensions from meters to centimeters. We know that 1 meter is equal to 100 centimeters. Therefore, we can convert the length and width of the room as follows:

Length in cm = Length in meters × 100 Length in cm = 5.6 m × 100 Length in cm = 560 cm

Width in cm = Width in meters × 100 Width in cm = 4 m × 100 Width in cm = 400 cm

Now we have the dimensions of the room in centimeters: 560 cm in length and 400 cm in width. This conversion is crucial because it allows us to work with consistent units when calculating the area of the carpet tiles and determining how many tiles are needed to cover the room. Unit conversion is a fundamental skill in mathematics and science, as it ensures that we are comparing and combining quantities that are expressed in the same terms. This step demonstrates the practical application of unit conversion in a real-world scenario, highlighting its importance in ensuring accurate calculations and problem-solving. With the room dimensions now in centimeters, we can proceed to the next step, which involves calculating the area of each carpet tile.

Step 3: Calculate the Area of One Carpet Tile

Now that we have the room dimensions in centimeters, the next step is to calculate the area of a single carpet tile. We know that each carpet tile is a square measuring 40 cm by 40 cm. The area of a square is calculated by multiplying the length of one side by itself. In this case, the area of one carpet tile can be calculated as follows:

Area of one tile = Side × Side Area of one tile = 40 cm × 40 cm Area of one tile = 1600 square centimeters

This calculation gives us the area that one carpet tile will cover. This information is crucial for determining how many tiles we need to cover the entire room. Understanding the area of individual components is essential in many practical situations, such as tiling a floor, laying out paving stones, or even calculating the amount of material needed for a sewing project. By accurately calculating the area of each carpet tile, we can ensure that we purchase the correct number of tiles, minimizing waste and cost. This step reinforces the importance of basic geometric principles in everyday problem-solving. With the area of one carpet tile calculated, we can now move on to the next step, which involves determining the total number of tiles needed to cover the entire room.

Step 4: Calculate the Number of Carpet Tiles Needed

To determine the number of carpet tiles required to cover the entire room, we need to divide the total area of the room by the area of one carpet tile. We have already calculated the area of the room in square centimeters (from the conversion in Step 2) and the area of one carpet tile in square centimeters (in Step 3). However, before we divide, we need to calculate the area of the room in square centimeters:

Area of the room = Length × Width Area of the room = 560 cm × 400 cm Area of the room = 224,000 square centimeters

Now we can calculate the number of carpet tiles needed:

Number of tiles = (Area of the room) / (Area of one tile) Number of tiles = 224,000 sq cm / 1600 sq cm Number of tiles = 140

Therefore, we need 140 carpet tiles to completely cover the room. This calculation is a crucial step in the process, as it directly impacts the cost of the project. By accurately determining the number of tiles needed, we can avoid overspending on unnecessary materials. This step highlights the importance of careful measurement and calculation in practical problem-solving. The concept of dividing a total area by the area of individual units is widely applicable in various fields, such as construction, landscaping, and interior design. With the number of tiles needed calculated, we can now move on to the next step, which involves determining the number of packs of carpet tiles to purchase, considering that they are sold in packs of 20.

Step 5: Calculate the Number of Packs to Purchase

The carpet tiles are sold in packs of 20, and we need 140 tiles to cover the room completely. To determine the number of packs we need to purchase, we will divide the total number of tiles required by the number of tiles in each pack. If the result is not a whole number, we will need to round up to the nearest whole number since we cannot purchase a fraction of a pack. The calculation is as follows:

Number of packs = (Total number of tiles) / (Tiles per pack) Number of packs = 140 tiles / 20 tiles/pack Number of packs = 7 packs

In this case, the result is a whole number, which means we need to purchase exactly 7 packs of carpet tiles. If the result had been a decimal, for example, 7.2, we would have rounded up to 8 packs to ensure we have enough tiles to cover the entire room. This step is crucial for cost-effectiveness, as purchasing too few packs would leave us short of tiles, while purchasing too many would result in unnecessary expense. The concept of dividing a total quantity by a unit size and rounding up to the nearest whole number is a common practice in many practical situations, such as purchasing materials for a project or ordering supplies for an event. With the number of packs needed calculated, we can now move on to the final step, which involves determining the total cost of carpeting the room.

Step 6: Calculate the Total Cost

We have determined that we need to purchase 7 packs of carpet tiles to cover the room completely. Each pack of carpet tiles costs £16. To calculate the total cost, we simply multiply the number of packs needed by the cost per pack. The calculation is as follows:

Total cost = (Number of packs) × (Cost per pack) Total cost = 7 packs × £16/pack Total cost = £112

Therefore, the total cost to completely carpet the room is £112. This is the final answer to our problem. By systematically working through each step, from calculating the area of the room to determining the number of packs needed and the final cost, we have successfully solved the problem. This process demonstrates the practical application of mathematical concepts in real-world scenarios. Understanding how to calculate costs for home improvement projects can save you money and ensure that you stay within your budget. This example highlights the importance of basic mathematical skills in everyday life, from planning a renovation to managing personal finances. We have now successfully completed the calculation and determined the total cost of carpeting the room.

Conclusion

In conclusion, we have successfully calculated the total cost of carpeting a rectangular room with a length of 5.6 meters and a width of 4 meters. By following a step-by-step approach, we were able to break down the problem into manageable parts and arrive at the solution. We began by calculating the area of the room, then converted the measurements from meters to centimeters to ensure consistency in our calculations. Next, we determined the area of a single carpet tile and used this information to calculate the total number of tiles needed to cover the room. Since the tiles are sold in packs of 20, we calculated the number of packs to purchase, rounding up to the nearest whole number to ensure we had enough tiles. Finally, we multiplied the number of packs by the cost per pack to arrive at the total cost. The final answer is that it will cost £112 to completely carpet the room. This exercise demonstrates the practical application of mathematical concepts such as area calculation, unit conversion, and cost estimation in real-world scenarios. The ability to perform these calculations is valuable in various situations, from home improvement projects to financial planning. By understanding the steps involved, individuals can make informed decisions and manage their resources effectively. This problem-solving approach can be applied to many other situations, highlighting the importance of mathematical literacy in everyday life. Furthermore, this exercise underscores the significance of attention to detail and accuracy in calculations, as even small errors can lead to significant discrepancies in the final result. Overall, this article has provided a comprehensive guide to calculating the cost of carpeting a room, emphasizing the practical relevance of mathematical skills.