Question

What is the opposite and reciprocal of $\dfrac{6}{5}$?

Answer 1

Hint: In order to find the opposite, we need to find with which value we need to add the given value to get zero. And to find the reciprocal, we need to find with which value we should multiply the given number to get one.

Complete solution:
Let us consider the given number $\dfrac{6}{5}$.
We are asked to find the opposite and the reciprocal of the given value.
First let us find the opposite of the given value.
We know that the sum of a number and its opposite is zero.
So, to find the opposite of the given number, we will find the number which can be added to the given number to provide zero.
Let us suppose that x is the opposite of $\dfrac{6}{5}$ .
Then we will get \[\dfrac{6}{5} + x = 0.\]
Now, if we transpose $\dfrac{6}{5}$ from the LHS to the RHS, we will get \[x = 0 - \dfrac{6}{5} = - \dfrac{6}{5}\].
Therefore, the opposite of $\dfrac{6}{5}$ is $ - \dfrac{6}{5}$.
Now, let us find the reciprocal of the given number.
We know that the product of a number and its reciprocal is equal to one.
Let us suppose that a is the reciprocal of $\dfrac{6}{5}$.
Then, we will get \[\dfrac{6}{5} \times a = 1.\]
Now, if we transpose $\dfrac{6}{5}$ from the LHS to the RHS, we will get \[a = \dfrac{5}{6}\].
Therefore, the reciprocal of $\dfrac{6}{5}$ is $\dfrac{5}{6}$.

Hence the opposite and the reciprocal of $\dfrac{6}{5}$ are $ - \dfrac{6}{5}$ and $\dfrac{5}{6}$ respectively.

Note:
We should know that the terminology opposite stands for the additive inverse. And we can simply multiply the given value with $ - 1$ to get the opposite or additive inverse. We should also know that the terminology reciprocal stands for the multiplicative inverse of the number. So, we just need to interchange the numerator and the denominator.