Final answer:
The ratio of the length of the model tile to the actual tile is 1:9, and the ratio of the area of the model tile to the actual tile is 1:20.25.
Step-by-step explanation:
The student asked about the ratio of the lengths of the tiles in a model to the actual tiles and the ratio of their areas. The length of a tile in the model is 1/3 inch, and the length of an actual tile is 1/4 foot. To find the ratio of the lengths, we need to convert one of the measurements so that both are in the same units. Since there are 12 inches in a foot, the actual tile length in inches is (1/4) × 12 = 3 inches. Thus, the ratio of the length of a tile in the model to the actual tile is (1/3) / 3 = 1/9.
For the area, the area of a model tile is (1/3 inch) × (1 inch) = 1/3 square inches. The actual tile area is (1/4 foot) × (3/16 foot) = (1/4) × (3/16) square feet = 3/64 square feet. Converting square feet to square inches by multiplying by (12 × 12) to get 3/64 × 144 = 432/64 = 6.75 square inches for the actual tile's area. The ratio of the model tile area to the actual tile area is (1/3) / 6.75 = 1/20.25 or 1:20.25.