A rectangular lawn has dimensions 80 m by 60 m. Inside this lawn there are two roads, each 10 m wide, running through the middle: one road is parallel to the length and the other road is parallel to the breadth, forming a cross. Find the total cost of gravelling these two roads at the rate of Rs. 3 per square metre.

Introduction / Context:
This mensuration question deals with a rectangular lawn that has two perpendicular roads crossing each other inside the lawn. The roads run along the middle of the length and the breadth, so together they form a plus sign. We must find the total area covered by both roads and then multiply this area by the given cost per square metre to obtain the total cost of gravelling.

Given Data / Assumptions:

• Outer lawn dimensions: 80 m by 60 m.
• Each road is 10 m wide.
• One road is parallel to the length (80 m) and the other is parallel to the breadth (60 m).
• Cost of gravelling = Rs. 3 per sq m.

Concept / Approach:
The two roads form a cross. If we add their rectangular areas directly, the overlapping central square will be counted twice. Therefore, the total road area equals the sum of the areas of the two rectangular roads minus the area of the overlapping square. After obtaining the road area, we multiply by the cost rate to get the total cost.

Step-by-Step Solution:
Area of road parallel to length = 80 * 10 = 800 sq m.Area of road parallel to breadth = 60 * 10 = 600 sq m.If we add these, overlap at the centre is counted twice.Overlap area is a square of side 10 m, so area = 10 * 10 = 100 sq m.Total road area = 800 + 600 - 100 = 1300 sq m.Total cost = 1300 * 3 = Rs. 3900.

Verification / Alternative check:
A quick sense check: each road alone covers 800 sq m and 600 sq m. Since there is a central overlapping square, subtracting 100 sq m once is correct. A total of 1300 sq m is slightly more than one quarter of the entire lawn area (80 * 60 = 4800 sq m), which is reasonable for two wide roads. Multiplying by Rs. 3 gives a moderate cost of Rs. 3900.

Why Other Options Are Wrong:
Rs. 3600 and Rs. 3800 arise if the overlap is ignored or subtracted incorrectly. Rs. 3700 is another incorrect result from arithmetic errors. Rs. 3000 would correspond to only 1000 sq m of road area, which underestimates the cross shaped region significantly.

Common Pitfalls:
Common mistakes include: not subtracting the overlapping 10 m by 10 m square, treating the path as only one long strip, or confusing perimeter with area while computing costs. Some learners also forget to multiply by the rate per square metre at the end. Careful handling of the overlap ensures a correct area and therefore a correct cost.

Final Answer:
Rs. 3900