Question

Find the multiplicative inverse of the following:
(i) $ - 13$
(ii) $\dfrac{1}{5}$
(iii) $\dfrac{{ - 5}}{8} \times \dfrac{{ - 3}}{7}$
(iv) $ - 1 \times \dfrac{{ - 2}}{5}$
(v) $ - 1$

Answer 1

Hint: In this question, we have been asked to find multiplicative inverse of 5 numbers. At first, read about what is multiplicative inverse and then use its property to find the multiplicative inverse.

Complete step-by-step solution:
We are given five different parts and we have to find their multiplication inverse. Bute, what is multiplication inverse?
Multiplication inverse is just another name for ‘reciprocal’. A very important property of multiplication inverse is that when a number is multiplied with its reciprocal or multiplicative inverse, we get $1$. Using this property only, we find multiplicative inverse of numbers.
Let us move towards the question.
(i) $ - 13$
$ \Rightarrow $Let the multiplicative inverse of $ - 13$ be x. We will keep the product of $ - 13$ and x equal to 1. And then, we will find the value of x.
$ \Rightarrow - 13 \times x = 1$
$ \Rightarrow x = - \dfrac{1}{{13}}$
Hence, multiplicative inverse of $ - 13$ is $ - \dfrac{1}{{13}}$.
(ii) $\dfrac{1}{5}$
Let the multiplicative inverse be x, Keeping the product equal to 1,
$ \Rightarrow \dfrac{1}{5} \times x = 1$
$ \Rightarrow x = 5$
Hence, multiplicative inverse of $\dfrac{1}{5}$ is 5.
(iii) $\dfrac{{ - 5}}{8} \times \dfrac{{ - 3}}{7}$
First, we will find the product.
$ \Rightarrow \dfrac{{ - 5}}{8} \times \dfrac{{ - 3}}{7} = \dfrac{{15}}{{56}}$
Let the multiplicative inverse be x, Keeping the product equal to 1,
$ \Rightarrow \dfrac{{15}}{{56}} \times x = 1$
$ \Rightarrow x = \dfrac{{56}}{{15}}$
Hence, multiplicative inverse of $\dfrac{{ - 5}}{8} \times \dfrac{{ - 3}}{7}$ is $\dfrac{{56}}{{15}}$.
(iv) $ - 1 \times \dfrac{{ - 2}}{5} = \dfrac{2}{5}$
Let the multiplicative inverse be x, keeping the product equal to 1,
$ \Rightarrow \dfrac{2}{5} \times x = 1$
$ \Rightarrow x = \dfrac{5}{2}$
Hence, the multiplicative inverse of $ - 1 \times \dfrac{{ - 2}}{5}$ is $\dfrac{5}{2}$.
(v) $ - 1$
Let the multiplicative inverse be x, Keeping the product equal to 1,
$ \Rightarrow - 1 \times x = 1$
$ \Rightarrow x = - 1$
Hence, multiplicative inverse of $ - 1$ is $ - 1$.

Option v is the correct answer.

Note: Instead of keeping the product of the number and the multiplicative inverse equal to 1, you can just find the reciprocal of the number also. In the reciprocal, we just interchange the numerator and the denominator with each other. For example: the reciprocal of $\dfrac{1}{5}$ is $5$$\left( {\dfrac{1}{5} \nearrow \dfrac{5}{1}} \right)$.