The cost of fencing a square field @ ₹ 20 per metre is ₹ 10,080. How much will it cost to lay a three metre wide pavement along the fencing inside the field @ ₹ 50 per sq. metre?

### Concept & Inner Path Area To find the cost of a pavement inside a square field, we first determine the side length of the outer square using the total fencing cost. Then, we find the area of the pavement by subtracting the inner square's area from the outer square's area. $$ \text{Perimeter} = \frac{\text{Total Cost of Fencing}}{\text{Cost per Metre}} $$ $$ \text{Path Area} = \text{Area of Outer Square} - \text{Area of Inner Square} $$ ### Step-by-Step Solution * Find the perimeter of the square field: $10080 / 20 = 504$ metres. * Find the side of the outer square field ($a$): $504 / 4 = 126$ metres. * A 3-metre wide pavement is laid *inside* the field. The side of the inner square ($b$) will be $126 - 2(3) = 120$ metres. * Calculate the area of the pavement: $\text{Outer Area} - \text{Inner Area} = 126^2 - 120^2$. * Use the algebraic identity $a^2 - b^2 = (a-b)(a+b)$ for easier calculation: $(126 - 120)(126 + 120) = 6 \times 246 = 1476$ sq. metres. * Total cost of laying the pavement = $1476 \times 50 = 73,800$. ### Exam Strategy & Shortcut Instead of squaring 126 and 120 manually, the difference of squares shortcut $(a-b)(a+b)$ is essential here. Multiplying $6 \times 246 \times 50$ can be further simplified by grouping: $246 \times (6 \times 50) = 246 \times 300 = 73800$, saving valuable calculation time. ### Common Pitfall A frequent error is subtracting the path width only once instead of twice to find the inner dimension (i.e., calculating $126 - 3$ instead of $126 - 6$). ### Final Answer Therefore, the correct answer is **₹ 73,800**.