Question

In which of the following pairs, the numbers are not additive inverses of each other?

• A. $\frac{2}{3}$, $\frac{-2}{3}$
• B. $\frac{4}{-5}$, $\frac{-4}{-5}$
• C. $\frac{1}{2}$, $\frac{-1}{-2}$
• D. $\frac{2}{1},-2$

Answer 1

The correct option is C

$\frac{1}{2}$, $\frac{-1}{-2}$

For a rational number $\frac{a}{b}$, we have $\frac{a}{b}+(-\frac{a}{b})=(-\frac{a}{b})+\left(\frac{a}{b}\right)=0$.
We say that $(-\frac{a}{b})$ is the additive inverse of $\frac{a}{b}$ and $\frac{a}{b}$ is the additive inverse of $(-\frac{a}{b}).$

We have, $\frac{2}{3}$ + $\frac{-2}{3}=\frac{-2}{3}+\frac{2}{3}=0$
$\therefore \frac{-2}{3}\text{and}\frac{2}{3}\text{are additive inverses of each other}.$

Note that $\frac{-4}{-5}=\frac{4}{5}.$
$\Rightarrow \frac{4}{-5}+\frac{4}{5}=\frac{-4}{5}+\frac{4}{5}=0$
$\therefore \frac{4}{-5}\text{and}\frac{-4}{-5}\text{are additive inverses of each other}.$

Similarly, since $\frac{2}{1}+-2=-2+\frac{2}{1}=0$, we have $\frac{2}{1}\text{and}-2\text{are additive inverses of each other}.$

But, $\frac{-1}{-2}$ + $\frac{1}{2}=\frac{1}{2}$ + $\frac{1}{2}=1$.
$\text{Hence}\frac{2}{1}\text{and}-2\text{are not additive inverses of each other}.$