A room measures 960 cm by 768cm. Find the quantity and the size of the largest square tiles that will be required to cover the floor.

Answer

To solve the problem of finding the quantity and size of the largest square tiles required to cover a room measuring 960 cm by 768 cm, we can follow these steps:

### Step 1: Calculate the area of the room
The area of the room can be calculated using the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given the dimensions of the room:
\[ \text{Area} = 960 \, \text{cm} \times 768 \, \text{cm} \]

### Step 2: Find the dimensions of the largest square tile
To find the size of the largest square tile that can fit perfectly in the room, we need to calculate the greatest common factor (GCF) or greatest common divisor (GCD) of the two dimensions (960 cm and 768 cm).

### Step 3: Calculate the GCD of 960 and 768
We can use the Euclidean algorithm to find the GCD:
1. Divide 960 by 768, which gives a quotient of 1 and a remainder of 192.
2. Now, divide 768 by 192, which gives a quotient of 4 and a remainder of 0.

Since the remainder is now 0, the GCD is the last non-zero remainder, which is 192. Therefore, the size of the largest square tile is 192 cm x 192 cm.

### Step 4: Calculate the area of one tile
The area of one tile can be calculated as:
\[ \text{Area of one tile} = 192 \, \text{cm} \times 192 \, \text{cm} \]

### Step 5: Calculate the number of tiles required
To find the number of tiles required to cover the entire area of the room, we divide the area of the room by the area of one tile:
\[ \text{Number of tiles} = \frac{\text{Area of the room}}{\text{Area of one tile}} = \frac{960 \times 768}{192 \times 192} \]

### Step 6: Perform the calculations
1. Calculate the area of the room:
\[ \text{Area of the room} = 960 \times 768 = 737280 \, \text{cm}^2 \]
2. Calculate the area of one tile:
\[ \text{Area of one tile} = 192 \times 192 = 36864 \, \text{cm}^2 \]
3. Calculate the number of tiles:
\[ \text{Number of tiles} = \frac{737280}{36864} = 20 \]

### Conclusion
The size of the largest square tile required is 192 cm x 192 cm, and the total number of tiles needed to cover the floor is 20.

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