The dimension of the room are 16 meters in length and 8 meters wide.A contracter wants to tile the room using the least number of square tiles.What should the tile dimension be?
GCF(8,16) = 8
so using 8x8 tiles, you will need only 1x2 = 2 tiles
I apologize for the mistake in my previous response. Yes, you're correct that the GCF of 8 and 16 is 8, and using 8x8 tiles would require only 2 tiles to cover the room. Thank you for pointing that out.
To find the tile dimension that would require the least number of square tiles to cover the room, we need to determine the greatest common divisor (GCD) between the length and width of the room.
The GCD represents the largest common factor that divides both the length and width without leaving any remainder. By using the GCD, we can determine the tile dimension that evenly divides the room's length and width.
In this case, the length of the room is 16 meters, and the width is 8 meters. To find the GCD, we can use different methods, such as Prime Factorization or Euclidean Algorithm. Let's use the Euclidean Algorithm.
Step 1: Divide the larger number (in this case, 16) by the smaller number (8) and find the remainder.
16 ÷ 8 = 2 remainder 0
Since the remainder is 0, we have found the GCD between 16 and 8, which is 8.
The GCD of the length and width is 8 meters. Therefore, the tile dimension should be such that one side divides evenly into 8 meters.
Possible tile dimensions could be 1 meter by 1 meter, 2 meters by 2 meters, 4 meters by 4 meters, or 8 meters by 8 meters.
In terms of minimizing the number of tiles used, the tile dimension that would require the least number of tiles is 8 meters by 8 meters. In this case, each tile would cover an area of 64 square meters, and you would only need one tile to cover the entire room.
Please note that this is the most efficient option in terms of using the least number of square tiles. However, practical considerations such as aesthetics, cost, and availability may also influence the final choice of tile dimension.
To find the least number of square tiles needed to cover the room, we need to find the area of the room and the area of each tile, and then divide the two.
The area of the room is:
Length x Width = 16m x 8m = 128m²
Let x be the side length of each square tile.
The area of each tile is:
x²
To find the least number of tiles needed, we want to divide the area of the room by the area of each tile, and get a whole number:
128m² / x² = n, where n is a whole number
We can simplify this equation by multiplying both sides by x²:
128m² = n x²
To find the smallest value of x that makes n a whole number, we need to find the factors of 128 that are perfect squares. The factors of 128 are:
1, 2, 4, 8, 16, 32, 64, 128
The perfect squares among these factors are:
1, 4, 16, 64
We can try each of these values for x, and see which one gives a whole number for n:
For x = 1m, n = 128m² / 1m² = 128 (not a perfect square)
For x = 2m, n = 128m² / 4m² = 32 (a perfect square)
For x = 4m, n = 128m² / 16m² = 8 (a perfect square)
For x = 8m, n = 128m² / 64m² = 2 (not a perfect square)
So the tile dimension should be 4 meters by 4 metres, which gives us 32 square metres for each tile. This means we will need 128m² / 32m² = 4 tiles to cover the room.