What is the minimum number of identical square tiles required to tile a floor of length 6 m 24 cm and width 4 m 80 cm?

### Concept & Minimum Tiles via HCF To minimize the number of identical square tiles needed to cover a rectangular floor, the size of each square tile must be maximized. The maximum side length of the tile is the HCF of the floor's length and width. $$ \text{Minimum Number of Tiles} = \frac{\text{Floor Area}}{\text{Maximum Tile Area}} $$ ### Step-by-Step Solution 1. **Convert dimensions into centimeters:** - $\text{Length} = 6 \text{ m } 24 \text{ cm} = 624 \text{ cm}$. - $\text{Width} = 4 \text{ m } 80 \text{ cm} = 480 \text{ cm}$. 2. **Find the maximum side length of a square tile (HCF):** - Find $\text{HCF}(624, 480)$. - Let's use the difference method: $624 - 480 = 144$. The HCF must be a factor of 144. - Let's factor out common multiples: Both are divisible by $48$ (since $48 \times 10 = 480$ and $48 \times 13 = 624$). - Since $10$ and $13$ are co-prime, the HCF is $48$. - Side of largest square tile = $48 \text{ cm}$. 3. **Calculate the minimum number of tiles:** - $\text{Number of tiles} = \frac{\text{Total Area}}{\text{Tile Area}} = \frac{624 \times 480}{48 \times 48}$. - $\text{Number of tiles} = \left(\frac{624}{48}\right) \times \left(\frac{480}{48}\right) = 13 \times 10 = 130$. ### Exam Strategy & Shortcut Once you find the factors linking the dimensions to their HCF ($624 = 48 \times 13$ and $480 = 48 \times 10$), do not calculate the actual areas. Simply multiply the resulting co-prime quotients: $13 \times 10 = 130$. ### Common Pitfall A common error is finding the HCF and stopping, selecting an option that matches the HCF instead of completing the final step to find the *number* of tiles. Read the question carefully to see if it asks for "size of tile" or "number of tiles". ### Final Answer Therefore, the correct answer is **130**.