Answer:
The fewest square tiles he can use without cutting any of them = 12 square tiles
Step-by-step explanation:
The given dimensions of Mr. Pete's rectangular patio = 108 inches by 144 inches
The area of the patio = 108 in. × 144 in. = 15,552 in.²
We note that when square tiles are used the must fit both the length and the with of the rectangular patio
Therefore, we have;
The dimensions of the square tiles that the fewest number of tiles is the highest common factor, HCF, of 108 inches and 144
The HCF of 108 and 144 are found from by the prime factors as follows;
108 = 2² × 3³ = 2² × 3² × 3
144 = 2⁴ × 3² = 2² × 3² × 4
From which we have;
The HCF of 108 and 144 = 2² × 3² = 36
Therefore, the size of the sides of the square tiles that will give the fewest number of tiles is 36 inches
The number of the 36 inches square tiles that fit into the length of the rectangular patio = 144 inches/(36 inches/tile) = 4 tiles
The number of the 36 inches square tiles that fit into the breadth of the rectangular patio = 108 inches/(36 inches/tile) = 3 tiles
The total number of tiles he will use = (4 × 3) tiles = 12 tiles = The fewest square tiles he can use without cutting any of them.
