Question

By what rational number should be divided by \[\left( {\dfrac{{ - 2}}{9}} \right)\] to get \[\left( { - 1} \right)\]?

Answer 1

Hint: A fraction is part of a whole number. It has two parts – a numerator and a denominator.
Dividing a fraction by another fraction is the same as multiplying the fraction by the reciprocal (inverse) of the other. We get the reciprocal of a fraction by interchanging its numerator and denominator.
You can solve most division problems by following these three steps:
* Flip the divisor into a reciprocal
* Change the division sign to a multiplication symbol and multiply
* Simplify your answer if possible

Complete step-by-step answer:
Let the number be x.
So, \[x{\text{ }} \div \left( {\dfrac{{ - 2}}{9}} \right) = {\text{ }} - 1\]
To divide fractions, take the reciprocal (invert the fraction) of the divisor and multiply the dividend.
To find the reciprocal of a fraction you simply flip the numbers. The denominator becomes the numerator and vice versa.
Invert the fraction that you are dividing by
\[ \Rightarrow x \times \left( {\dfrac{9}{{ - 2}}} \right){\text{ }} = {\text{ }} - 1\]
Now try to isolate x on the LHS and multiply and divide the both sides by \[\left( {\dfrac{{ - 2}}{9}} \right){\text{ }}\]
\[ \Rightarrow x = ( - 1)\left( {\dfrac{{ - 2}}{9}} \right)\]
Simplify the fraction and we get,
To get a fraction down to its simplest form, you divide the numerator and denominator by their greatest common factor.
\[ \Rightarrow x{\text{ }} = {\text{ }}\dfrac{2}{9}\]

Note: When the product of two numbers is one, they are called reciprocals or multiplicative inverses of each other. For example, $\dfrac{2}{7}$and $\dfrac{7}{2}$ are reciprocals because $\dfrac{2}{7} \times \dfrac{7}{2} = \dfrac{{14}}{{14}} = 1$. This is the motivation for the following property of fractions.
Inverse Property for Fraction Multiplication
$\dfrac{a}{b} \times \dfrac{b}{a} = \dfrac{b}{a} \times \dfrac{a}{b} = 1$ where a and b are nonzero. The fraction $\dfrac{b}{a}$ is called the multiplicative inverse of $\dfrac{a}{b}$ (or reciprocal) and vice versa.