A rectangular park measuring 60 m by 40 m has two concrete roads of equal width running through the middle, one along its length and one along its breadth. The remaining area is a lawn of 2109 square metres. What is the width of each road (in metres)?

Introduction / Context:
This question involves area calculation in a rectangular park with two perpendicular roads crossing through the centre. The roads have the same width and cover some part of the park, while the remaining region is grass lawn. By expressing the road area in terms of the unknown width and subtracting it from the total area, we can find the width. Such problems are common in aptitude exams and help develop algebraic modelling skills using geometric shapes.

Given Data / Assumptions:
• The park is a rectangle with length 60 metres and breadth 40 metres.
• Two concrete roads of equal width run through the middle, one along the length and one along the breadth.
• The remaining grass lawn area is 2109 square metres.
• Let the width of each road be x metres.
• The roads cross each other, so their overlapping area must not be counted twice.

Concept / Approach:
The total area of the park is length * breadth. The area covered by the roads is the sum of the two rectangular strips minus the overlapping square at the centre, because that intersection gets counted twice if we simply add the areas. Each road has area equal to width times the full dimension it crosses. So one road has area 60x and the other has area 40x. Their overlap is a square of side x with area x². The lawn area is the total area of the park minus the road area. Setting this equal to the given lawn area gives a quadratic equation in x, which we can solve.

Step-by-Step Solution:
Step 1: Compute the total area of the park: 60 * 40 = 2400 square metres. Step 2: Let x be the width of each road in metres. Step 3: Area of the road along the length is 60x square metres. Step 4: Area of the road along the breadth is 40x square metres. Step 5: The overlapping central rectangle (actually a square) has area x² square metres. Step 6: So total road area is 60x + 40x - x² = 100x - x². Step 7: Lawn area = total area - road area = 2400 - (100x - x²) = 2400 - 100x + x². Step 8: Set this equal to the given lawn area: 2400 - 100x + x² = 2109. Step 9: Rearrange: x² - 100x + (2400 - 2109) = 0, so x² - 100x + 291 = 0. Step 10: Solve the quadratic: x² - 100x + 291 = 0 has roots x = 3 and x = 97. Step 11: Since the width cannot exceed the smaller side of 40 m, x = 3 m is the only feasible value.

Verification / Alternative check:
Using x = 3 m, the road along the length has area 60 * 3 = 180 square metres, and the road along the breadth has area 40 * 3 = 120 square metres. The overlap is a 3 m by 3 m square with area 9 square metres. So total road area is 180 + 120 - 9 = 291 square metres. The lawn area is then 2400 - 291 = 2109 square metres, which matches the given value. This confirms that a width of 3 metres is correct. The other root 97 m is impossible because it is greater than the width of the entire park.

Why Other Options Are Wrong:
2.5 m and 2.91 m both lead to lawn areas different from 2109 square metres when substituted into the area expressions.
3.5 m and 5.82 m also produce incorrect lawn areas and may exceed realistic proportions relative to the park size. Only 3 m satisfies the quadratic equation and keeps the geometry practical.

Common Pitfalls:
A common mistake is forgetting to subtract the overlapping area x², which causes double counting and leads to an incorrect equation. Another pitfall is miscomputing the quadratic roots or not checking which root is physically meaningful. Learners may also confuse which sides correspond to 60 and 40 metres when calculating road areas. Drawing a diagram and writing the expression for road area carefully helps avoid these errors.

Final Answer:
The width of each road is 3 m.