A homeowner is deciding on the size of tiles to use to fully tile a rectangular wall in her bathroom that is 80 inches by 40 inches. The tiles are squares and come in three side lengths: 8 inches, 4 inches, and 2 inches. State if you agree with each statement about the tiles. Explain your reasoning.
a. Regardless of the size she chooses, she will need the same number of tiles.
b. Regardless of the size she chooses, the area of the wall that is being tiled is the same.
c. She will need two 2-inch tiles to cover the same area as one 4-inch tile. @ she will need 4-inch tiles to cover the same area as the one 8-inch tile. ①
d. If she choses the einch tiles she will need a quarter as many tiles as she would with a lich tiles.
The Correct Answer and Explanation is :
Let’s assess each statement based on the details of the tiling situation for the bathroom wall, which measures 80 inches by 40 inches, and the available tile sizes: 8 inches, 4 inches, and 2 inches square tiles.
a. Regardless of the size she chooses, she will need the same number of tiles.
- This statement is false. The number of tiles needed depends on the area each tile covers, which varies with tile size.
- The area of the wall is (80 \text{ inches} \times 40 \text{ inches} = 3200 \text{ square inches}).
- For 8-inch tiles, each tile covers (8 \text{ inches} \times 8 \text{ inches} = 64 \text{ square inches}). Number of 8-inch tiles needed: (\frac{3200}{64} = 50).
- For 4-inch tiles, each tile covers (4 \text{ inches} \times 4 \text{ inches} = 16 \text{ square inches}). Number of 4-inch tiles needed: (\frac{3200}{16} = 200).
- For 2-inch tiles, each tile covers (2 \text{ inches} \times 2 \text{ inches} = 4 \text{ square inches}). Number of 2-inch tiles needed: (\frac{3200}{4} = 800).
- As the tile size decreases, the number of tiles required increases.
b. Regardless of the size she chooses, the area of the wall that is being tiled is the same.
- This statement is true. The area of the wall to be tiled does not change and remains 3200 square inches, irrespective of the tile size used.
c. She will need two 2-inch tiles to cover the same area as one 4-inch tile. @ she will need 4-inch tiles to cover the same area as the one 8-inch tile.
- This statement is false. Let’s analyze each part:
- One 4-inch tile covers 16 square inches.
- Two 2-inch tiles together cover (2 \times 4 = 8 \text{ square inches}). Four 2-inch tiles would be needed to cover the same area as one 4-inch tile.
- One 8-inch tile covers 64 square inches.
- Four 4-inch tiles together cover (4 \times 16 = 64 \text{ square inches}). Four 4-inch tiles are needed to cover the same area as one 8-inch tile, not just one 4-inch tile.
d. If she chooses the 8-inch tiles she will need a quarter as many tiles as she would with 4-inch tiles.
- This statement is true. As calculated:
- For 8-inch tiles: 50 tiles.
- For 4-inch tiles: 200 tiles.
- (\frac{200}{50} = 4). Hence, when using 8-inch tiles, only a quarter as many tiles are needed compared to 4-inch tiles.
Each tile choice results in different numbers of tiles needed to cover the same total area due to the variation in the area each tile covers.