Chapter 7.2, Problem 32WE

To determine

Show that the given proportions are equivalent.

Explanation of Solution

Given information:

The given proportion is $\frac{a}{b}=\frac{c}{d}$ and $\frac{a+c}{b+d}=\frac{a-c}{b-d}$ .

Formula used: 

Two proportional are said to be equivalent if one can be obtained from the other.

Proof:

It is given $\frac{a}{b}=\frac{c}{d}$

Let $\frac{a}{b}=\frac{c}{d}=k$ then $a=bk,c=dk$

Now,

  $\begin{array}{c}\frac{a+c}{b+d}=\frac{bk+dk}{b+b}\\ =\frac{k\left(b+d\right)}{b+d}\\ =k\end{array}$

And

  $\begin{array}{c}\frac{a-c}{b-d}=\frac{bk-dk}{b-d}\\ =\frac{k\left(b-d\right)}{b-d}\\ =k\end{array}$

Hence,

  $\frac{a+c}{b+d}=\frac{a-c}{b-d}$ and it is obtained from $\frac{a}{b}=\frac{c}{d}$

Therefore, the given proportion is equivalent.