Question

What is the opposite and reciprocal of $2\dfrac{3}{4}$?

Answer 1

Hint: We are given in this question a mixed fraction. So, we have to first convert the mixed fraction into a proper fraction. Then, we are to find the opposite and reciprocal of the proper fraction. We know, the opposite of a number is the additive inverse of the number, that is, the number with addition to which we get zero for the given number. And reciprocal is the multiplicative inverse of the given number, that is, on multiplying by which we get $1$ as the result.

Complete step-by-step answer:
The mixed fraction we are given is, $2\dfrac{3}{4}$.
The proper fraction will be, $2\dfrac{3}{4} = \dfrac{{11}}{4}$.
So, the opposite of $\dfrac{{11}}{4}$ is the additive inverse of the number, that is, the number with addition to which we get zero for the given number.
Let the opposite number be $x$.
Therefore, by definition of opposite number,
$\dfrac{{11}}{4} + x = 0$
Now, subtracting both sides by $\dfrac{{11}}{4}$, we get,
$ \Rightarrow x = - \dfrac{{11}}{4}$
Therefore, the additive inverse is, $ - \dfrac{{11}}{4}$.
Now, the reciprocal of $\dfrac{{11}}{4}$ is the multiplicative inverse of the given number, that is, on multiplying by which we get $1$ as the result.
Let the reciprocal be $y$.
Therefore, by definition of multiplicative inverse,
$\dfrac{{11}}{4}.y = 1$
Now, multiplying both sides by $\dfrac{4}{{11}}$, we get,
$ \Rightarrow y = \dfrac{4}{{11}}$
Therefore, the reciprocal is $\dfrac{4}{{11}}$.

Note: We can find the opposite of any number by just finding the negation of that number. The negation of a number is the opposite of that number, or, as we say the additive inverse of that number. And, the reciprocal of any number is just the fractional inverse of that number. That is, to find the reciprocal or multiplicative inverse, we are just to find the fractional inverse of that number.