How do you find \[90 \div \left( { - \dfrac{2}{3}} \right)\]?

Hint: Here, we need to evaluate the given expression. The division of a number by a fraction is equal to the product of the number and the reciprocal of the fraction. First, we will find the reciprocal of the fraction. Then, we will multiply the given number by the reciprocal of the fraction to find the required value.

Complete step-by-step solution:
First, we will find the reciprocal of the given divisor, that is the fraction \[ - \dfrac{2}{3}\].
Thus, we get
Reciprocal of \[ - \dfrac{2}{3} = - \dfrac{3}{2}\]
The division of a number by a fraction is equal to the product of the number and the reciprocal of the fraction. This can be written as \[a \div \dfrac{b}{c} = a \times \dfrac{c}{b}\], where \[b\] and \[c\] are not equal to 0.
The division of 90 by the fraction \[ - \dfrac{2}{3}\] is equal to the product of the number 90 and the reciprocal of the fraction \[ - \dfrac{2}{3}\], that is \[ - \dfrac{3}{2}\].
Therefore, we get the expression
\[90 \div \left( { - \dfrac{2}{3}} \right) = 90 \times \left( { - \dfrac{3}{2}} \right)\]
Rewriting the expression, we get
\[ \Rightarrow 90 \div \left( { - \dfrac{2}{3}} \right) = - \dfrac{{90 \times 3}}{2}\]
A number is divisible by 2 if it has the digit 0, 2, 4, 6, or 8 at the unit’s place.
The number 90 has the digit 0 at the unit’s place.
This means that 90 is divisible by 2.
Dividing 90 by 2 in the expression, we get
\[ \Rightarrow 90 \div \left( { - \dfrac{2}{3}} \right) = - \dfrac{{45 \times 3}}{1}\]
Multiply 45 by 3 in the expression, we get
\[ \Rightarrow 90 \div \left( { - \dfrac{2}{3}} \right) = - \dfrac{{135}}{1}\]
Simplifying the expression, we get
\[\therefore 90\div \left( -\dfrac{2}{3} \right)=-135\]

Therefore, we get the value of the given expression \[90 \div \left( { - \dfrac{2}{3}} \right)\] as \[ - 135\].

Note:
We used the term ‘reciprocal’ in the solution. The reciprocal of a number \[a\] is given by \[\dfrac{1}{a}\]. It is the number by which a number \[a\] must be multiplied to get 1 as the result. It is also called the multiplicative inverse of a number. There are three main types of fractions i.e. proper fractions, improper fractions, and mixed fractions. A proper fraction is a fraction having the numerator less or lowers in degree, than the denominator. The value of proper fraction after simplification is always less than 1. An improper Fraction is a fraction where the numerator is greater than or equals to the denominator, then it is known as an improper fraction. A mixed Fraction is the combination of a natural number and fraction.