A floor of a big hall has dimensions 30 m 60 cm and 23 m 40 cm. It is to be paved with square tiles of same size. What is the minimum number of tiles required ?
221
Calculating Minimum Tiles for the Hall Floor
The problem asks for the minimum number of identical square tiles needed to pave a large hall floor. The hall has dimensions 30 m 60 cm by 23 m 40 cm. To find the minimum number of tiles, we need to use the largest possible square tile size that can fit perfectly into the floor dimensions.
1. Convert Hall Dimensions to a Single Unit
First, let's convert the hall's dimensions into a single, consistent unit, preferably centimeters (cm), to simplify calculations.
- Length of the hall = 30 m 60 cm
- Since 1 meter = 100 cm, 30 meters = \(30 \times 100 = 3000\) cm.
- Total Length = \(3000 \text{ cm} + 60 \text{ cm} = 3060 \text{ cm}\).
- Width of the hall = 23 m 40 cm
- Since 1 meter = 100 cm, 23 meters = \(23 \times 100 = 2300\) cm.
- Total Width = \(2300 \text{ cm} + 40 \text{ cm} = 2340 \text{ cm}\).
So, the dimensions of the hall floor are 3060 cm by 2340 cm.
2. Determine the Largest Possible Square Tile Size
To use the minimum number of tiles, the size of each square tile must be maximized. The side length of the square tile must perfectly divide both the length (3060 cm) and the width (2340 cm) of the hall floor without any remainder. Therefore, the side length of the largest possible square tile is the Greatest Common Divisor (GCD) of 3060 and 2340.
3. Calculate the GCD of the Dimensions
We can find the GCD using the Euclidean algorithm:
- Divide 3060 by 2340: \(3060 = 1 \times 2340 + 720\)
- Divide 2340 by the remainder 720: \(2340 = 3 \times 720 + 180\)
- Divide 720 by the remainder 180: \(720 = 4 \times 180 + 0\)
The last non-zero remainder is 180. Thus, the GCD(3060, 2340) = 180 cm.
This means the largest possible square tile that can pave the floor has a side length of 180 cm.
4. Calculate the Number of Tiles Required
Now, we calculate how many tiles fit along the length and width of the hall.
- Number of tiles along the length = \(\frac{\text{Length of hall}}{\text{Side of tile}} = \frac{3060 \text{ cm}}{180 \text{ cm}}\)
- Number of tiles along the length = \(17\)
- Number of tiles along the width = \(\frac{\text{Width of hall}}{\text{Side of tile}} = \frac{2340 \text{ cm}}{180 \text{ cm}}\)
- Number of tiles along the width = \(13\)
The total minimum number of tiles required is the product of the number of tiles along the length and the number of tiles along the width.
- Total tiles = (Number of tiles along length) \(\times\) (Number of tiles along width)
- Total tiles = \(17 \times 13\)
- Total tiles = \(221\)
Conclusion
Therefore, the minimum number of square tiles required to pave the hall floor is 221.
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