The least number of square tiles required to pave the ceiling of a room, 12 m 95 cm long and 3 m 85 cm broad, is:

Options

A. 14225
B. 14245
C. 14000
D. 12425

Answer

To solve the problem of finding the least number of square tiles required to pave the ceiling of a room measuring 12 m 95 cm in length and 3 m 85 cm in breadth, we can follow these steps:

### Step 1: Convert Measurements to Centimeters
First, we need to convert the dimensions of the room from meters to centimeters.

- Length: 12 m 95 cm = \(12 \times 100 + 95 = 1295\) cm
- Breadth: 3 m 85 cm = \(3 \times 100 + 85 = 385\) cm

### Step 2: Calculate the Area of the Ceiling
Next, we calculate the area of the ceiling using the formula for the area of a rectangle (Area = Length × Breadth).

\[ \text{Area} = 1295 \, \text{cm} \times 385 \, \text{cm} = 498575 \, \text{cm}^2 \]

### Step 3: Find the Size of the Square Tile
To determine the size of the square tile that can be used, we need to find the highest common factor (HCF) of the length and breadth of the room.

- Length = 1295 cm
- Breadth = 385 cm

Using the Euclidean algorithm or prime factorization, we find that:

\[ \text{HCF}(1295, 385) = 35 \, \text{cm} \]

### Step 4: Calculate the Area of One Tile
Now, we calculate the area of one square tile with a side length equal to the HCF.

\[ \text{Area of one tile} = 35 \, \text{cm} \times 35 \, \text{cm} = 1225 \, \text{cm}^2 \]

### Step 5: Calculate the Least Number of Tiles Required
Finally, we find the least number of tiles required by dividing the total area of the ceiling by the area of one tile.

\[ \text{Number of tiles} = \frac{\text{Area of ceiling}}{\text{Area of one tile}} = \frac{498575 \, \text{cm}^2}{1225 \, \text{cm}^2} = 407 \]

Thus, the least number of square tiles required to pave the ceiling of the room is **407 tiles**.

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