A room is $12 \frac{1}{4}$ m long and 7 m wide. The maximum length of a square tile to fill the floor of the room with whole number of tiles should be

### Concept & HCF of Dimensions When filling a rectangular area with the largest possible square tiles without cutting any, the side length of the square tile must be the Highest Common Factor (HCF) of the rectangle's length and width. $$ \text{Maximum Tile Side} = \text{HCF}(\text{Length}, \text{Width}) $$ ### Step-by-Step Solution 1. **Convert dimensions into the same smallest unit (cm):** - $\text{Length} = 12 \frac{1}{4} \text{ m} = 12.25 \text{ m} = 1225 \text{ cm}$. - $\text{Width} = 7 \text{ m} = 700 \text{ cm}$. 2. **Find the HCF of 1225 and 700:** - Prime factorization of $1225$: $1225 = 25 \times 49 = 5^2 \times 7^2$. - Prime factorization of $700$: $700 = 7 \times 100 = 7 \times 2^2 \times 5^2$. - The common factors are $5^2$ and $7$. - $\text{HCF} = 5^2 \times 7 = 25 \times 7 = 175$. 3. **Determine the tile length:** - The HCF represents the maximum length of the square tile in cm. - Maximum length = $175 \text{ cm}$. ### Exam Strategy & Shortcut Use the difference method for HCF or check the options. The options are $125, 150, 175, 200$. The width is $700$. $700$ is divisible by $175$ ($175 \times 4 = 700$). Let's check $1225$: $175 \times 7 = 1225$. Both divide perfectly, making $175$ a common factor. Since it's the largest among the valid options, it must be the answer. ### Common Pitfall Leaving the measurements in meters and dealing with decimals or fractions for HCF calculations often leads to arithmetic errors. Converting entirely to centimeters bypasses this. ### Final Answer Therefore, the correct answer is **175 cm**.