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 m^{2}. Find the width of the path.

Let the width of the path be x m

Length of the field including the path = 16 + x + x = 16 + 2x

Breadth of the field including the path = 10 + x + x = 10 + 2x

Area of field including the path – Area of field excluding the path = Area of path

(16 + 2x) (10 + 2x) – (16.10) = 120

160 + 32x + 20x + 4x^{2} – 160 = 120

4x^{2} + 52x – 120 = 0

x^{2} + 13x – 30 = 0

Using the splitting middle term – the middle term of the general equation is divided in two such values that:

Product = a.c

For the given equation a = 1 b = 13 c = – 30

= 1. – 30 = – 30

And either of their sum or difference = b

= 13

Thus the two terms are 15 and – 2

Difference = 15 – 2 = 13

Product = 15. – 2 = – 30

x^{2} + 15x – 2x – 30 = 0

x(x + 15) – 2(x + 15) = 0

(x + 15) (x – 2) = 0

x = 2 or x = – 15

x = 2 (width cannot be negative)

Thus the width of the path is 2 m

63