Question

Verify commutativity of addition of rational numbers for each of the following pairs of rational numbers:

(iv) $\frac{2}{-7}\text{and}\frac{12}{-35}$

Answer 1

Given: $\frac{2}{-7}\text{and}\frac{12}{-35}$

Commutativity of addition: Addition of two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ is commutative if $\frac{a}{b}+\frac{c}{d}=\frac{c}{d}+\frac{a}{b}$

$\therefore \frac{2}{-7}+\frac{12}{-35}=\frac{-2}{7}+\frac{-12}{35}$
$=\frac{-2\times 5}{7\times 5}+\frac{-12}{35}$
$=\frac{-10}{35}+\frac{-12}{35}$
$=\frac{-10-12}{35}$
$=\frac{-22}{35}$

and $\frac{12}{-35}+\frac{2}{-7}=\frac{-12}{35}+\frac{-2}{7}$
$=\frac{-12}{35}+\frac{-2\times 5}{7\times 5}$
$=\frac{-12}{35}+\frac{-10}{35}$
$=\frac{-12-10}{35}$
$=\frac{-22}{35}$
Thus,$\frac{2}{-7}+\frac{12}{-35}=\frac{12}{-35}+\frac{2}{-7}$

Hence, commutativity of addition is verified.