Question

Is \[0.6\] the multiplicative inverse of \[6\dfrac{2}{3}\]? Why? or why not?

Answer 1

Hint: Multiplicative inverse of a number is a number when multiplied with the original number equals to one. To find whether \[0.6\] is the multiplicative inverse of \[6\dfrac{2}{3}\], we will first write the given number in a simplified way i.e., \[\dfrac{{20}}{3}\]. Then using the definition of multiplicative inverse, we will find the multiplicative inverse of \[6\dfrac{2}{3}\]. If it is equal to \[0.6\], then \[0.6\] is the multiplicative inverse of \[6\dfrac{2}{3}\].

Complete step by step answer:
The multiplicative inverse of a number is a number when multiplied with the original number equals to one. Here, the original number must never be equal to zero. The multiplicative inverse of a number is also referred to as its reciprocal. The multiplicative inverse of a number \[X\] is represented as \[{X^{ - 1}}\] or \[\dfrac{1}{X}\].
We have to first find the multiplicative inverse of \[6\dfrac{2}{3}\] i.e., \[\dfrac{{20}}{3}\].
We know that the multiplicative inverse of a fraction \[\dfrac{a}{b}\] is given by \[\dfrac{b}{a}\], where \[a\] and \[b\] are real numbers and\[a,b \ne 0\].
Therefore, the multiplicative inverse of \[6\dfrac{2}{3}\] i.e., \[\dfrac{{20}}{3}\] will be \[\dfrac{3}{{20}}\].
On calculating we get, \[\dfrac{3}{{20}} = 0.15\] which is not equal to \[0.6\].
Therefore, \[0.6\] is not the multiplicative inverse of \[6\dfrac{2}{3}\].

Note:
The multiplicative inverse of zero does not exist. This is because zero multiplied by any number is zero and \[\dfrac{1}{0}\] is undefined. And the multiplicative inverse of \[1\] is \[1\] only because \[1 \times 1 = 1\]. Also, if \[{X^{ - 1}}\] or \[\dfrac{1}{X}\] is the multiplicative inverse of \[X\], then \[X\] is the multiplicative inverse of \[{X^{ - 1}}\] or \[\dfrac{1}{X}\]. This is due to the commutative property of multiplication.