Three plots having areas 110, 130 and 190 square metres are to be subdivided into flower beds of equal size. If the breadth of a bed is 2 metre, the maximum length of a bed can be

### Concept & Highest Common Factor (HCF) To divide different total areas into equal-sized beds of maximum possible area, find the Highest Common Factor (HCF) of the given total areas. $$ \text{Area of Bed} = \text{Length} \times \text{Breadth} $$ ### Step-by-Step Solution 1. **Find the maximum area of a single flower bed:** - The areas of the plots are $110$, $130$, and $190$ sq. m. - The maximum area of an equal-sized bed must perfectly divide all three plot areas. - $\text{Maximum Area} = \text{HCF}(110, 130, 190)$. - The HCF of $110$, $130$, and $190$ is $10$ (since $11 \times 10 = 110$, $13 \times 10 = 130$, $19 \times 10 = 190$, and $11, 13, 19$ are prime to each other). - So, the maximum area of a flower bed is $10 \text{ sq. m}$. 2. **Find the maximum length of the bed:** - The breadth of the bed is given as $2 \text{ m}$. - $\text{Area} = \text{Length} \times \text{Breadth} \Rightarrow 10 = \text{Length} \times 2$. - $\text{Length} = \frac{10}{2} = 5 \text{ m}$. ### Exam Strategy & Shortcut Recognize instantly that dividing distinct groups into maximum equal components requires the HCF. The trailing zero in 110, 130, 190 makes 10 the obvious HCF. Since Area = $10$ and Breadth = $2$, Length = $5$. This can be solved mentally in 5 seconds. ### Common Pitfall A common mistake is trying to find the LCM instead of the HCF, or forgetting to divide the resulting HCF (area) by the given breadth to find the length. ### Final Answer Therefore, the correct answer is **5 m**.