Question

A carpet is laid on the floor of a room 8 m by 5 m. There is a border of constant width all around the carpet. If the area of the border is 12 m², find its width.

Answer 1

Let the width of the border be x m.
The length and breadth of the carpet are 8 m and 5 m, respectively.
Area of the carpet = $8\times 5=40{\mathrm{m}}^2$
Length of the carpet without border = $(8-2x)$
Breadth of carpet without border = $(5-2x)$
Area of the border = 12 m²
Area of the carpet without border = $(8-2x)(5-2x)$
$$\begin{gathered} \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ 12=40-\left[(8-2x)(5-2x)\right] \\ \Rightarrow 12=40-(40-26x+4x^2) \\ \Rightarrow 12=26x-4x^2 \end{gathered}$$

$$\begin{gathered} \Rightarrow 26x-4x^2=12 \\ \Rightarrow 4x^2-26x+12=0 \\ \Rightarrow 2x^2-13x+6=0 \end{gathered}$$

$$\begin{gathered} \Rightarrow (2x-1)(x-6)=0 \\ \Rightarrow 2x-1=0\mathrm{and}x-6=0 \\ \Rightarrow x=\frac{1}{2}\mathrm{and}x=6 \end{gathered}$$

Because the border cannot be wider than the entire carpet, the width of the carpet is $\frac{1}{2}$ m, i.e., 50 cm.