Chapter 8.1, Problem 19WE

To determine

To prove the properties of the proportion $\frac{a}{b}=\frac{c}{d}$ .

Explanation of Solution

Given information:

The given property is:

  $\frac{a-b}{c-d}=\frac{a}{c}\text{ and }c\ne d$

Proof :

The property of the proportion can be proved by:

  $\begin{array}{l}\frac{a}{b}=\frac{c}{d}\\ \frac{a}{b}-1=\frac{c}{d}-1\left[\text{Subtract 1 from each side}\right]\\ \frac{a-b}{b}=\frac{c-d}{d}\\ \frac{a-b}{c-d}=\frac{b}{d}\left[\text{Cross multiplication}\right]\left(\text{eq}.1\right)\\ \frac{a}{b}=\frac{c}{d}\\ \frac{a}{b}\cdot \frac{1}{c}=\frac{c}{d}\cdot \frac{1}{c}\left[\text{Divide both sides by }c\right]\\ \frac{a}{bc}\cdot b=\frac{1}{d}\cdot b\left[\text{Multiply both sides by }b\right]\\ \frac{a}{c}=\frac{b}{d}\left(\text{eq}\text{.2}\right)\\ \frac{a-b}{c-d}=\frac{a}{c}\left[\text{Substitute the value of }\frac{b}{d}=\frac{a}{c}\text{ from eq}\text{.2}\right]\end{array}$

Hence, the given property is proved.