A rectangular plot is half as long again as it is broad and its area is $\frac{2}{3}$ hectares. Then, its length is

### Concept & Area Conversion The problem relies on interpreting the phrase "half as long again as it is broad", which establishes a ratio between the length and breadth. Additionally, it requires knowing the conversion factor between hectares and square meters: $$ 1 \text{ hectare} = 10000 \text{ m}^2 $$ ### Step-by-Step Solution 1. Let the breadth of the rectangular plot be $b$ meters. 2. The phrase "half as long again as it is broad" means the length $l$ is the breadth plus half of the breadth: $$ l = b + \frac{b}{2} = \frac{3b}{2} $$ 3. The area of the plot is given as $\frac{2}{3}$ hectares. Let's convert this to square meters: $$ \text{Area} = \frac{2}{3} \times 10000 = \frac{20000}{3} \text{ m}^2 $$ 4. We know the area of a rectangle is Length $\times$ Breadth. Substitute the expressions: $$ l \times b = \frac{20000}{3} $$ $$ \left(\frac{3b}{2}\right) \times b = \frac{20000}{3} $$ $$ \frac{3b^2}{2} = \frac{20000}{3} $$ 5. Solve for $b^2$: $$ b^2 = \frac{20000 \times 2}{3 \times 3} = \frac{40000}{9} $$ 6. Take the square root of both sides to find $b$: $$ b = \frac{200}{3} \text{ m} $$ 7. Now, find the length $l$: $$ l = \frac{3b}{2} = \frac{3}{2} \times \frac{200}{3} = \frac{200}{2} = 100 \text{ m} $$ ### Exam Strategy & Shortcut Instead of dealing with large numbers immediately, express the area algebraically as $\frac{3}{2}b^2$. Recognize that $40000$ and $9$ are perfect squares. By maintaining the fractions rather than converting to decimals (like $33.33$), the final square root calculation becomes instantaneous. ### Common Pitfall Misinterpreting "half as long again" is the most common mistake. Students often write $l = \frac{b}{2}$ instead of $l = 1.5b$. Also, forgetting to convert hectares to square meters before applying the length and breadth in meters will yield an incorrect scale. ### Final Answer Therefore, the correct answer is **100 m**.