Question

Find three rational numbers between $ - 1$ and $ - 2$ ?

Answer 1

Hint: In this question, we have to find three rational numbers lying between the $2$ rational numbers already given to us. Hence, we first find the equivalent rational numbers of both the numbers between which three rational numbers are to be found with a larger denominator so that the rational numbers can be found without any problem. We can find equivalent rational numbers by multiplying or dividing the numerator and denominator of the rational by the same number. Then, we find the rational numbers between the two numbers by choosing the numerator accordingly.

Complete step by step answer:
So, we are given the rational numbers $ - 1$ and $ - 2$ in the question itself. We can write these numbers in $\dfrac{p}{q}$ form having their denominators as one. So, we get, $\dfrac{{ - 1}}{1}$ and $\dfrac{{ - 2}}{1}$. So, we first find out the equivalent rational numbers of these. Hence, we multiply the numerator and denominator by the same number so that the value does not change.

So, multiplying the numerator and denominator by $10$, we get the numbers as,
$ - \dfrac{1}{1} \times \dfrac{{10}}{{10}} = - \dfrac{{10}}{{10}}$ and $ - \dfrac{2}{1} \times \dfrac{{10}}{{10}} = - \dfrac{{20}}{{10}}$
Now, we can easily find three rational numbers between $ - \dfrac{{10}}{{10}}$ and $ - \dfrac{{20}}{{10}}$ by choosing the numerators accordingly. So, we can choose the numerator of the rational numbers between $ - 10$ and $ - 20$ with denominator as $10$ so that the rational numbers lie between $ - \dfrac{{10}}{{10}}$ and $ - \dfrac{{20}}{{10}}$.

Now, we know that $ - 11$, $ - 13$ and $ - 17$ lie between $ - 10$ and $ - 20$. So, the rational numbers $\dfrac{{ - 11}}{{10}}$, $\dfrac{{ - 13}}{{10}}$ and $\dfrac{{ - 19}}{{10}}$ lie between $\dfrac{{ - 10}}{{10}}$ and $\dfrac{{ - 20}}{{10}}$. Also, these rational numbers don’t have any common factor in numerator and denominator. So, they are in their simplest forms.

Hence, three rational numbers between $ - 1$ and $ - 2$ are: $\dfrac{{ - 11}}{{10}}$, $\dfrac{{ - 13}}{{10}}$ and $\dfrac{{ - 19}}{{10}}$.

Note: Equivalent rational numbers are the rational numbers that have different numerator and denominator but are equal to the same value. Following the information and the steps mentioned in the above solution, we can solve similar questions. Care should be taken while doing calculations in order to be sure of the answer. Also, there are infinite rational numbers between any two given rational numbers. So, the answer to the given problem may vary from person to person.