Question

What is the additive inverse of $\dfrac{-p}{q}$ where $\dfrac{-p}{q}$ is a rational number.
 (A). $\dfrac{p}{q}$
 (B). $\dfrac{p}{-q}$
 (C). $\dfrac{-p}{q}$
 (D). $\dfrac{q}{p}$

Answer 1

Hint: We will take help from the definition of additive inverse. Basically, additive inverse in mathematics means a number. We can say that a is the additive inverse of b, if when a is added to b, we get an answer which equals 0. Like 4 is the additive inverse of $-4$, because $4+\left( -4 \right)=4-4=0$. We will use this concept to find the additive inverse of $\dfrac{-p}{q}$.

Complete step-by-step solution -
It is given in the question that we have to find the additive inverse of $\dfrac{-p}{q}$, where $\dfrac{-p}{q}$ is a rational number. Since it is given that $\dfrac{-p}{q}$ is a rational number which confirms that p and q are integers also $q\ne 0$. Now, we will look over the basic definition of additive inverse.
In mathematics additive inverse means when a number ‘a’ is added with another number ‘b’ and if the result is equal to zero then ‘a’ is an additive inverse of ‘b’ and ‘b’ is an additive inverse of ‘a’. For example, $\dfrac{-1}{2}$ is an additive inverse of $\dfrac{1}{2}$. We will use this concept of additive inverse to find the additive inverse of $\dfrac{-p}{q}$.
Let the additive inverse of $\dfrac{-p}{q}$ be x, then, according to definition of additive inverse, we get the following equation $\dfrac{-p}{q}+x=0$ or $x=0+\dfrac{p}{q}=\dfrac{p}{q}$. Thus, the additive inverse of $\dfrac{-p}{q}$ is $\dfrac{p}{q}$. Also we can verify it as $\dfrac{-p}{q}+\dfrac{p}{q}=0$ and thus option a) is the correct answer.

Note: Students usually confuse with additive inverse and multiplicative inverse. Basically, an additive inverse sum of two numbers is equal to zero as $\dfrac{-p}{q}+\dfrac{p}{q}=0$. But the multiplicative inverse product of two numbers is 1, that is, $a\times b=1$ like $2\times \dfrac{1}{2}=1$. Thus 2 is the multiplicative inverse of $\dfrac{1}{2}$. Therefore, it is recommended to keep these small points and definitions in mind to solve such questions successfully.