Final answer:
To determine the side length of the smaller tile, set the total area covered by 600 smaller tiles equal to the area covered by 400 slightly larger tiles, and solve the resulting equation.
Step-by-step explanation:
The student is asking about the side length of a square tile needed to cover a rectangular counter. In the first scenario, the counter is covered with 600 square tiles. In the second scenario, 400 of a different size, where each side of these tiles is 1 cm longer, are enough to cover the same counter. We can use the area covered by the tiles in both scenarios to find out the side length of the smaller tile.
Let x be the side length of the smaller tile in cm. Therefore, the area of each smaller tile is x^2 cm^2. Since 600 tiles cover the counter, the total area they cover is 600x^2 cm^2.
For the larger tiles, each has the side length of (x+1) cm, so the area of each larger tile is (x+1)^2 cm^2. With 400 larger tiles covering the same area, the total area is 400(x+1)^2 cm^2. Since the total area covered by the tiles in both scenarios is the same, we can set the equations equal to each other and solve for x:
600x^2 = 400(x+1)^2
After expanding and simplifying the equation, we find that x equals the side length of the smaller tile in cm. This problem demonstrates the mathematical concept of area and how to solve equations that include squares of binomials.