Marcus has a large supply of rectangular wooden tiles. He wants to lay them out to form a square such that there are no gaps, and no tiles overlap. Each tile measures 44 mm wide and 56 mm long
a) What will be the dimensions of the smallest square he can form using these tiles?
b) How many tiles will be required to construct the square?
LCM(44,56) = 7*8*11
So, there will be 14*44 by 11*56 = 616x616 mm, using 14*11 = 154 tiles
To find the dimensions of the smallest square that Marcus can form using these tiles, we need to consider the common factors of the width and length of the tiles.
a) Let's find the greatest common divisor (GCD) of 44 mm and 56 mm. The GCD represents the largest dimension that can be evenly divided into both measurements without a remainder.
Using the Euclidean algorithm, we can find the GCD as follows:
Step 1: Divide 56 mm by 44 mm. The remainder is 12 mm.
Step 2: Divide 44 mm by 12 mm. The remainder is 4 mm.
Step 3: Divide 12 mm by 4 mm. The remainder is 0 mm.
Since we have reached a remainder of 0, the last divisor used (4 mm) is the GCD.
So, the GCD of 44 mm and 56 mm is 4 mm.
To form a square, Marcus needs the width and length to be equal. Therefore, he can arrange the tiles in a square with the dimensions equal to the GCD of 4 mm.
Thus, the dimensions of the smallest square Marcus can form using these tiles are 4 mm by 4 mm.
b) To calculate how many tiles are required to construct the square, we need to divide the width of the square by the width of a single tile and then multiply by the length of the square divided by the length of a single tile.
Since the width and length of the square are both 4 mm, we have:
Number of tiles required = (Width of square / Width of a single tile) * (Length of square / Length of a single tile)
= (4 mm / 44 mm) * (4 mm / 56 mm)
= (1/11) * (1/14)
= 1/154
Therefore, Marcus will need 1/154th of a tile to form the smallest square, which implies he will require 154 tiles.