A rectangular grassy plot measures 110 metres by 65 metres. There is a gravel path of uniform width 2.5 metres all around the inside of the plot. Find the total cost of gravelling this path at the rate of 80 paise (Rs. 0.80) per square metre.

Introduction:
This problem is a classic example of finding the area of a border or path around a rectangle. The rectangular grassy plot has a gravel path of uniform width running all around the inside. We must compute the area of this path and then multiply by the cost per square metre. The question tests understanding of inner and outer dimensions, subtraction of areas, and simple cost calculation based on unit area.

Given Data / Assumptions:

• Outer dimensions of the rectangular plot: 110 m by 65 m. • Gravel path runs all around the inside of the plot. • Width of the path = 2.5 m (uniform on all sides). • Cost of gravelling = 80 paise per square metre = Rs. 0.80 per sq m. • We must find the total cost of gravelling the path only.

Concept / Approach:
The area of the gravel path is equal to the area of the outer rectangle minus the area of the inner rectangle that remains grassy. When a border of width w is taken all around inside a rectangle of dimensions L and B, the inner rectangle has dimensions (L - 2w) and (B - 2w). Once we compute both areas, we subtract to get the area of the path. Finally, we multiply by the cost per square metre to get the total cost.

Step-by-Step Solution:
Step 1: Compute the area of the outer rectangle. Outer length L = 110 m. Outer breadth B = 65 m. Area_outer = L * B = 110 * 65 = 7150 sq m. Step 2: Compute dimensions of the inner rectangle (inside the path). Width of path w = 2.5 m on all sides. Inner length = L - 2w = 110 - 2 * 2.5 = 110 - 5 = 105 m. Inner breadth = B - 2w = 65 - 5 = 60 m. Step 3: Compute the area of the inner rectangle. Area_inner = 105 * 60 = 6300 sq m. Step 4: Compute the area of the gravel path. Area_path = Area_outer - Area_inner = 7150 - 6300 = 850 sq m. Step 5: Compute the cost of gravelling the path. Cost per sq m = Rs. 0.80. Total cost = Area_path * cost per sq m = 850 * 0.80 = Rs. 680.

Verification / Alternative check:
We can also reason by expansion: The path occupies a border of width 2.5 m along the entire boundary. Checking consistency, 2.5 m is reasonable relative to dimensions 110 m and 65 m, and the remaining inner dimensions 105 m and 60 m are positive and realistic. The area difference of 850 sq m is a moderate fraction of the total 7150 sq m, which is plausible for a relatively narrow border. Multiplying 850 by 0.80 gives 680, a clean and consistent number matching one of the options.

Why Other Options Are Wrong:
Option Rs. 480: This would correspond to an area of only 600 sq m at Rs. 0.80 per sq m, which is smaller than the actual path area. Option Rs. 340: This implies an area of 425 sq m, which is too small for the given path width and rectangle size. Option Rs. 320: This corresponds to 400 sq m of path, which is clearly less than the actual border area. Option Rs. 540: This corresponds to an area of 675 sq m, again not matching the area difference of 850 sq m.

Common Pitfalls:
Some students misinterpret the width as being added outside the rectangle instead of inside, leading to wrong inner dimensions. Others subtract w only once from each side instead of 2w, forgetting that the path runs on both ends of each dimension. Another frequent error is to compute the area of the inner rectangle incorrectly or to multiply by 80 instead of 0.80 when converting paise to rupees. Careful reading of units and consistent application of the 2w adjustment prevent most of these mistakes.

Final Answer:
The total cost of gravelling the path is Rs. 680.