To Cover A Rectangular Patio, 540 Tiles Of 600 M² Each Were Used. How Many Square Tiles With Sides Of 20 Cm Will Be Needed To Cover The Same Patio?

To solve the problem of how many square tiles are needed to cover a rectangular patio, we will break it down step by step. The initial scenario involves 540 tiles, each with an area of 600 square meters. Our goal is to determine how many smaller, square tiles with sides of 20 cm are required to cover the same patio area.

Understanding the Initial Area

First, let's calculate the total area of the patio using the information provided. We know that 540 tiles, each with an area of 600 m², were used. Therefore, the total area of the patio can be calculated by multiplying these two values:

Total Area = Number of Tiles × Area per Tile
Total Area = 540 × 600 m²
Total Area = 324,000 m²

So, the total area of the rectangular patio is 324,000 square meters. This is the area we need to cover with the new, smaller tiles. This initial calculation provides a crucial baseline for our subsequent steps. Understanding the total area allows us to compare it with the area of the smaller tiles and accurately determine how many are needed. The concept of area is fundamental in this problem, bridging the gap between the larger tiles initially used and the smaller tiles we aim to use. By establishing this foundation, we ensure that our calculations are rooted in a clear understanding of the physical space we are working with.

Calculating the Area of the Smaller Tiles

Now that we know the total area of the patio, the next step is to determine the area of the smaller, square tiles we intend to use. Each of these tiles has a side length of 20 cm. To find the area of a square, we use the formula:

Area = Side × Side

In our case, the side length is 20 cm. So, the area of each smaller tile is:

Area = 20 cm × 20 cm
Area = 400 cm²

Therefore, each square tile has an area of 400 square centimeters. However, we need to ensure that our units are consistent. The total area of the patio is in square meters, while the area of the tile is in square centimeters. To proceed, we must convert the area of the tile from square centimeters to square meters. This conversion is crucial because it aligns our units, allowing for accurate comparisons and calculations. The relationship between square meters and square centimeters is that 1 square meter is equal to 10,000 square centimeters. Understanding this conversion factor is essential for correctly scaling measurements between different units, which is a common requirement in many practical problems involving area and dimensions. By converting the tile area to square meters, we can directly compare it with the patio's total area, ensuring that our final calculation of the number of tiles needed is precise and meaningful.

Converting Square Centimeters to Square Meters

To convert the area from square centimeters to square meters, we use the conversion factor: 1 m² = 10,000 cm². Thus, we divide the area in square centimeters by 10,000 to get the equivalent area in square meters:

Area in m² = Area in cm² / 10,000
Area in m² = 400 cm² / 10,000
Area in m² = 0.04 m²

So, each square tile has an area of 0.04 square meters. This conversion is a pivotal step in our calculation. By expressing both the patio area and the individual tile area in the same unit (square meters), we can accurately determine how many tiles are needed to cover the patio. The process of unit conversion is fundamental in mathematical and practical applications, ensuring that we are comparing like measurements. In this context, it prevents us from making errors due to mismatched scales. By meticulously converting the tile area to square meters, we set the stage for the final calculation, which will reveal the precise number of tiles required to cover the 324,000-square-meter patio.

Determining the Number of Tiles Needed

Now that we have the total area of the patio in square meters (324,000 m²) and the area of each tile in square meters (0.04 m²), we can calculate how many tiles are needed to cover the patio. To do this, we divide the total area of the patio by the area of each tile:

Number of Tiles = Total Area / Area per Tile
Number of Tiles = 324,000 m² / 0.04 m²
Number of Tiles = 8,100,000

Therefore, 8,100,000 square tiles, each with a side length of 20 cm, are needed to cover the rectangular patio. This final calculation represents the culmination of our step-by-step approach. By systematically determining the total area of the patio, calculating the area of the smaller tiles, and ensuring consistent units through conversion, we have arrived at a precise answer. The magnitude of the number—8,100,000 tiles—underscores the importance of accurate measurements and conversions in such calculations. This result not only answers the initial question but also illustrates the practical application of area calculations in real-world scenarios, such as tiling a patio. The process highlights how breaking down a complex problem into smaller, manageable steps can lead to a clear and accurate solution.

Conclusion

In summary, to cover a rectangular patio that was initially covered by 540 tiles of 600 m² each, you would need 8,100,000 square tiles with sides of 20 cm. This solution demonstrates the importance of understanding area calculations and unit conversions in practical applications. From start to finish, the solution has involved careful consideration of areas, conversion factors, and mathematical principles, showcasing the interconnectedness of mathematical concepts in solving real-world problems. The initial step of calculating the total patio area established a solid foundation for subsequent calculations. Converting the area of the smaller tiles from square centimeters to square meters ensured that all measurements were in a consistent unit, which is crucial for accurate comparisons and calculations. Finally, dividing the total patio area by the area of each smaller tile provided the exact number of tiles needed. This methodical approach not only answers the specific question at hand but also provides a framework for tackling similar problems in the future. Understanding these steps and the principles behind them is invaluable in various fields, from construction and design to everyday problem-solving.