Chapter 5.4, Problem 21E

Solution

To calculate: A simplified rational expression for the ratio of the field's perimeter to its area.

The required ratio is $\frac{2}{x}$.

Given information:

A farmer wants to fence in the field shown below.

  

Formula used:

Area of square = ${\left(\text{side}\right)}^2$

Perimeter of square = $4\left(\text{side}\right)$

Calculation:

The field to be fenced is square in shape.

So, the area of the field will be,

  $\begin{array}{c}={\left(2x\right)}^2\\ =4x^2\end{array}$

And the perimeter of the field will be,

  $\begin{array}{l}=4\left(2x\right)\\ =8x\end{array}$

The ratio of the field's perimeter to its area will be,

  $\begin{array}{c}=\frac{8x}{4x^2}\\ =\frac{4\cdot 2\cdot x}{4\cdot x\cdot x}\left(\text{Factorise numerator and denominator}\right)\\ =\frac{2}{x}\left(\text{Divide out common factors}\right)\end{array}$

Therefore, the required ratio is $\frac{2}{x}$.