Final answer:
The ratio of the length of the tile to the length of the board is 4/9. The ratio of the area of the tile to the area of the board is 16/81, after converting the board's measurements to the same unit (feet) as the tile's measurements.
Step-by-step explanation:
To find the ratio of the length of the tile to the length of the board, we first need to make sure that both lengths are in the same unit of measurement. We know that 1 yard (yd) equals 3 feet (ft). Therefore, we can convert the length of the board from yards to feet. Since the board has a length of 1/4 yd, which is the same as 1/4 × 3 ft, this means that the board's length is 3/4 ft.
With both lengths now in feet, the ratio of the length of the tile to the length of the board is 1/3 ft to 3/4 ft. To find this ratio, you would divide 1/3 by 3/4, which is the same as multiplying 1/3 by the reciprocal of 3/4 (which is 4/3). After simplification, the ratio is 4/9.
To find the ratio of the area of the tile to the area of the board, we must find the area of each. The area of a rectangle is found by multiplying its length by its width. Thus, for the tile, the area is (1/3 ft × 1/3 ft) = 1/9 ft2, and for the board, the area is (3/4 ft × 3/4 ft) = 9/16 ft2. The ratio of the areas is 1/9 ft2 to 9/16 ft2, or 1/9 ÷ 9/16, which simplifies to 16/81.