a square kitchen floor can be exactly covered withe square tiles that are either 9 inches or 15 inches on a side. what is the minimym area in square feet that the kitchen can have? how many of each size tile would be needed to cover the minimum size floor?
To find the minimum area of the kitchen floor, we need to find the greatest common divisor (GCD) between the tile sizes, 9 inches and 15 inches.
The GCD of 9 and 15 is 3.
To convert the measurements to feet, we divide by 12 (as there are 12 inches in a foot):
9 inches = 9/12 feet = 0.75 feet
15 inches = 15/12 feet = 1.25 feet
Now, we divide the side length of each tile size by the GCD:
0.75 feet / 3 = 0.25 feet
1.25 feet / 3 = 0.4166666667 feet (rounded to 10 decimal places)
We can see that these values can be simplified further.
0.25 feet = 1/4 feet
0.4166666667 feet = 5/12 feet
So, the simplified side lengths are:
1/4 feet and 5/12 feet.
To find the minimum area of the kitchen floor, we multiply these side lengths:
(1/4 feet) * (5/12 feet) = 5/48 square feet
Hence, the minimum area of the kitchen floor is 5/48 square feet.
To determine how many tiles of each size are needed to cover the minimum floor size, we divide the minimum floor area by the area of each tile size.
For the 9-inch tiles:
Area of 9-inch tile = (9 inches)^2 = 81 square inches
Area of 9-inch tile in square feet = (81 square inches) / (144 square inches/square foot) = 0.5625 square feet
(5/48 square feet) / (0.5625 square feet) ≈ 0.089
So, we would need approximately 0.089 or about 1 tile of the 9-inch size to cover the minimum size floor.
Similarly, for the 15-inch tiles:
Area of 15-inch tile = (15 inches)^2 = 225 square inches
Area of 15-inch tile in square feet = (225 square inches) / (144 square inches/square foot) = 1.5625 square feet
(5/48 square feet) / (1.5625 square feet) ≈ 0.032
So, we would need approximately 0.032 or about 0.33 (rounded up) tiles of the 15-inch size to cover the minimum size floor.
Therefore, to cover the minimum size floor, we would need 1 tile of the 9-inch size and 1 tile of the 15-inch size.
To find the minimum area of the square kitchen floor, we need to determine the side length of the square floor in inches.
Since the square tiles can be either 9 inches or 15 inches on a side, we need to find the greatest common divisor (GCD) of these two numbers.
The GCD of 9 and 15 can be found using the Euclidean algorithm:
15 = 9 * 1 + 6
9 = 6 * 1 + 3
6 = 3 * 2 + 0
The remainder of the final step is 0, which means that 3 is the GCD of 9 and 15.
Now, to find the side length of the square kitchen floor, we divide the GCD (3 inches) into the smaller dimension of the tiles (9 inches):
9 / 3 = 3 inches
Therefore, the side length of the square kitchen floor is 3 inches.
To convert this to square feet, we divide by 12 (since there are 12 inches in a foot):
3 / 12 = 0.25 feet
So, the minimum area of the kitchen floor is 0.25 square feet.
To determine how many tiles of each size are needed to cover the minimum floor, we divide the side length of the floor by the side length of each tile.
For the 9-inch tiles:
3 inches (side length of floor) / 9 inches (side length of tile) = 1/3 tiles
For the 15-inch tiles:
3 inches (side length of floor) / 15 inches (side length of tile) = 1/5 tiles
Since we can't have fractions of tiles, we need to determine the least common multiple (LCM) of 3 and 5 to find the number of tiles needed.
The LCM of 3 and 5 is 15, so we multiply the above fractions by 15 to get whole numbers:
1/3 * 15 = 5 tiles of the 9-inch size
1/5 * 15 = 3 tiles of the 15-inch size
Therefore, to cover the minimum size floor, you would need 5 tiles of the 9-inch size and 3 tiles of the 15-inch size.
minimum area for square is either 9*9= 81 or 15*15=225. 81 is less so 81 sq. inches
each tile has a side length of 9 inches and you'd have a box of nine rows and nine columns of tiles like this
__
|__|
so each end has nine tiles * 9 inches per tile, so that's 81 inches of tile on each side, 4 sides total so you get 81 times 4 is equal to 324 in of tile (perimeter)