Question

The multiplicative inverse of $ - \dfrac{2}{3}$ is:
(A) $\dfrac{2}{3}$
(B) $ - \dfrac{3}{2}$
(C) $\dfrac{3}{2}$
(D) none of these

Answer 1

Hint: In the given question, we have to find the multiplicative inverse of the given number. To find the multiplicative inverse of a number, we must know what the term multiplicative inverse actually means. The multiplicative inverse of x is a number which when multiplied to x gives unity as a result. So, to find the multiplicative inverse of the given number, we first equate the product of the number and its multiplicative inverse as one. Then, we solve the mathematical equation obtained using the transposition method to find the value of the variable.

Complete step-by-step solution:
So, the number given to us is $ - \dfrac{2}{3}$.
Let us assume the multiplicative inverse of the number to be x. Then, the product of $ - \dfrac{2}{3}$ and x should be equal to one. So, we get,
$ \Rightarrow x \times \left( { - \dfrac{2}{3}} \right) = 1$
Now, we use the transposition method to find the value of variable x in the equation above.
So, multiplying both sides of the equation by $\dfrac{3}{2}$, we get,
$ \Rightarrow x \times \left( { - \dfrac{2}{3}} \right) \times \dfrac{3}{2} = 1 \times \dfrac{3}{2}$
Cancelling the common factors in numerator and denominator, we get,
$ \Rightarrow x \times \left( { - 1} \right) = \dfrac{3}{2}$
Multiplying both sides of the equation by $ - 1$, we get,
$ \Rightarrow x \times \left( { - 1} \right) \times \left( { - 1} \right) = \dfrac{3}{2} \times \left( { - 1} \right)$
Now, we know that multiplication of two negative signs yields a positive sign. So, simplifying the calculations, we get,
$ \Rightarrow x = - \dfrac{3}{2}$
So, we obtain the value of x as $ - \dfrac{3}{2}$. Hence, the multiplicative inverse of $ - \dfrac{2}{3}$ is $ - \dfrac{3}{2}$.
Therefore, option (B) is the correct answer.

Note: We can see that the multiplicative inverse of the number is the reciprocal of the original number. This is true for every real number. We can use this trick directly to find the multiplicative inverse of a number as this saves a lot of time and calculations.