Problem Overview: Tiling a Ceiling

This question asks for the minimum number of identical square tiles needed to perfectly cover a rectangular ceiling. The ceiling dimensions are given as length 15 m 17 cm and breadth 9 m 2 cm. To minimize the number of tiles, we must use the largest possible square tile size. The side length of this largest square tile is determined by the Greatest Common Divisor (GCD) of the ceiling's length and breadth.

Step-by-Step Solution Process

1. Convert Dimensions to a Uniform Unit

First, we need to express both the length and the breadth in the same unit. Let's convert everything to centimeters (cm) as it avoids decimals.

• Ceiling Length: 15 m 17 cm
  • Since 1 m = 100 cm, then 15 m = 15 $\times$ 100 cm = 1500 cm.
  • Total Length = 1500 cm + 17 cm = 1517 cm.
• Ceiling Breadth: 9 m 2 cm
  • 9 m = 9 $\times$ 100 cm = 900 cm.
  • Total Breadth = 900 cm + 2 cm = 902 cm.

So, the ceiling measures 1517 cm in length and 902 cm in breadth.

2. Find Largest Square Tile Size Using GCD

The side length of the largest square tile that can perfectly pave the ceiling must be a divisor of both the length (1517 cm) and the breadth (902 cm). The largest such side length corresponds to the Greatest Common Divisor (GCD) of 1517 and 902.

We will use the Euclidean Algorithm to find the GCD:

1. Divide the larger dimension (1517) by the smaller dimension (902):

   $1517 = 1 \times 902 + 615$

2. Now, divide the previous divisor (902) by the remainder (615):

   $902 = 1 \times 615 + 287$

3. Divide the previous divisor (615) by the new remainder (287):

   $615 = 2 \times 287 + 41$

4. Divide the previous divisor (287) by the new remainder (41):

   $287 = 7 \times 41 + 0$

The last non-zero remainder is 41. Therefore, the GCD(1517, 902) = 41.

This means the side length of the largest possible square tile we can use is 41 cm.

3. Calculate Total Number of Tiles Needed

With the ceiling dimensions (1517 cm $\times$ 902 cm) and the tile side length (41 cm), we can now find the number of tiles required. The total number of tiles is the area of the ceiling divided by the area of a single tile.

Number of Tiles = $ \frac{\text{Area of Ceiling}}{\text{Area of one Tile}} $

Number of Tiles = $ \frac{1517 \text{ cm} \times 902 \text{ cm}}{41 \text{ cm} \times 41 \text{ cm}} $

This calculation can be simplified by dividing the dimensions by the tile side length first:

Number of Tiles = $ \left( \frac{1517}{41} \right) \times \left( \frac{902}{41} \right) $

Performing the divisions:

• Number of tiles along the length = $ \frac{1517}{41} = 37 $
• Number of tiles along the breadth = $ \frac{902}{41} = 22 $

Now, multiply these two numbers to get the total number of tiles:

Total Number of Tiles = $ 37 \times 22 $

Total Number of Tiles = $ 814 $

Final Answer

The calculation shows that 814 square tiles, each with a side length of 41 cm, are required to pave the ceiling.