To solve this problem, we will apply the concept of the greatest common divisor (GCD) to find the largest square tile size that can exactly cover both the cafeteria and the adjoining entryway.

Part 1: Largest tile size for the cafeteria

The cafeteria has dimensions of 96 feet by 56 feet. The largest possible square tile that can fit both dimensions perfectly is the greatest common divisor (GCD) of 96 and 56.

Step 1: Find the GCD of 96 and 56

We can use the Euclidean algorithm to find the GCD.

• 96 ÷ 56 = 1 remainder 40
• 56 ÷ 40 = 1 remainder 16
• 40 ÷ 16 = 2 remainder 8
• 16 ÷ 8 = 2 remainder 0

Since the remainder is now 0, the GCD is 8.

Thus, the largest square tile that can cover the cafeteria exactly is 8 feet by 8 feet.

Part a) How many tiles should the school purchase for the cafeteria?

To determine how many 8x8 tiles are needed, we calculate the area of the cafeteria and divide by the area of one tile.

• Area of the cafeteria = 96 × 56 = 5,376 square feet
• Area of one tile = 8 × 8 = 64 square feet

Number of tiles needed = $\frac{5376}{64} = 84$ tiles.

So, the school should purchase 84 tiles for the cafeteria.

Part 2: Can they use the same tiles for the entryway?

The adjoining entryway has dimensions of 48 feet by 36 feet. We need to determine if the same 8x8 tiles can be used for this space as well.

Step 2: Find the GCD of 48 and 36

Again, we use the Euclidean algorithm:

• 48 ÷ 36 = 1 remainder 12
• 36 ÷ 12 = 3 remainder 0

The GCD is 12.

So, the largest square tile that can fit exactly in the entryway is 12 feet by 12 feet.

Conclusion:

The tiles that can be used in both the cafeteria and the entryway must have dimensions that divide evenly into both 96x56 and 48x36. Since the cafeteria can only accommodate 8x8 tiles, and the entryway requires 12x12 tiles, the same tile size cannot be used for both the cafeteria and the entryway.

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Summary:

• The largest tile size for the cafeteria is 8 feet by 8 feet.
• The school should purchase 84 tiles for the cafeteria.
• The same tile size cannot be used for both the cafeteria and the entryway, as the entryway requires 12x12 tiles.

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Do you have any questions or would you like more details on any part of this?

Relative Questions:

1. How would the tile sizes change if the cafeteria had different dimensions?
2. How do you calculate the GCD using the Euclidean algorithm in general?
3. If the entryway was 72 feet by 48 feet, what would the largest tile size be for both areas?
4. What if the school wanted tiles that divide both areas evenly, regardless of size?
5. How do factors like grout space or tile patterns affect the number of tiles needed?

Tip:

Always check if the greatest common divisor (GCD) of two dimensions can simplify tile planning efficiently for large areas.