Chapter 7.4, Problem 12CE

To determine

The proportions that are equivalent to the given equation.

Answer to Problem 12CE

There are five types of expressions which are equivalent to the given expression.

  $\begin{array}{l}\frac{AB}{CD}=\frac{WX}{YZ}\\ \frac{AB}{WX}=\frac{CD}{YZ}\\ \frac{CD}{AB}=\frac{YZ}{WX}\\ \frac{AB+CD}{CD}=\frac{WX+YZ}{YZ}\\ \frac{AB+WX}{CD+YZ}=\frac{AB}{CD}\end{array}$

Explanation of Solution

Given information:

  $AB\cdot YZ=CD\cdot WX$

Property of proportions:

(1) If $\frac{a}{b}=\frac{c}{d}$ is equivalent to:

  $\begin{array}{l}ad=bc\cdots \cdots (1)\hfill \\ \frac{a}{c}=\frac{b}{d}\dots \dots (2)\hfill \\ \frac{b}{a}=\frac{d}{c}\dots \dots (3)\hfill \\ \frac{a+b}{b}=\frac{c+d}{d}\dots \dots (4)\hfill \end{array}$

(2) If $\frac{a}{b}=\frac{c}{d}=\frac{e}{f}=\dots ,\text{then}\frac{a+c+e+\dots }{b+d+f+\dots }=\frac{a}{b}=\dots \dots (5)$

Now, consider an expression, which is similar to the equation $(1),AB\cdot YZ=CD\cdot WX$

Therefore,

  $\begin{array}{l}\frac{AB}{CD}=\frac{WX}{YZ}\\ \frac{AB}{WX}=\frac{CD}{YZ}\\ \frac{CD}{AB}=\frac{YZ}{WX}\\ \frac{AB+CD}{CD}=\frac{WX+YZ}{YZ}\\ \frac{AB+WX}{CD+YZ}=\frac{AB}{CD}\end{array}$

Hence, there are five types of expressions which are equivalent to the given expression.