Question

A rectangular field is 16 m long and 10 m wide. There is a path of uniform width all around it, having an area of 120 m². Find the width of the path.

Answer 1

$$\begin{gathered} \text{Let the width of the path be}x\text{m.} \\ \therefore \text{Length of the field including the path = 16 +}x+x=16+2x \\ \text{Breadth of the field including the path = 10 +}x+x=10+2x \\ \mathrm{Now}, \\ \left(\text{Area of the field including path}\right)-\left(\mathrm{A}\text{rea of the field excluding path}\right)=\text{Area of the path} \\ \Rightarrow (16+2x)(10+2x)-(16\times 10)=120 \\ \Rightarrow 160+32x+20x+4x^2-160=120 \\ \Rightarrow 4x^2+52x-120=0 \\ \Rightarrow x^2+13x-30=0 \\ \Rightarrow x^2+(15-2)x+30=0 \\ \Rightarrow x^2+15x-2x+30=0 \\ \Rightarrow x(x+15)-2(x+15)=0 \\ \Rightarrow (x-2)(x+15)=0 \\ \Rightarrow x-2=0\text{or}x+15=0 \\ \Rightarrow x=2\text{or}x=-15 \\ \Rightarrow x=2(\because \text{Width cannot be negative}) \\ \mathrm{Thus}\text{, the width of the path is 2 m.} \end{gathered}$$