Question

Find any ten rational numbers between $ - \dfrac{5}{6}$ and $\dfrac{5}{8}$.

Answer 1

Hint: In this question, we have to find ten 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 ten 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 solution:
So, we are given the rational numbers $ - \dfrac{5}{6}$ and $\dfrac{5}{8}$ in the questions itself.
So, we first find out the equivalent rational numbers of these such that the denominators of both the rational numbers are the same. Hence, we multiply the numerator and denominator by the same number so that the value does not change.
So, multiplying and dividing the rational number $ - \dfrac{5}{6}$ by $8$, we get,
$ - \dfrac{5}{6} \times \dfrac{8}{8} = - \dfrac{{40}}{{48}}$
Now, multiplying and dividing the rational number $\dfrac{5}{8}$ by $6$, we get,
$\dfrac{5}{8} \times \dfrac{6}{6} = \dfrac{{30}}{{48}}$
Now, we can easily find three rational numbers between $ - \dfrac{{40}}{{48}}$ and $\dfrac{{30}}{{48}}$ by choosing the numerators accordingly.
So, we can choose the numerator of the rational numbers between $ - 40$ and $30$ with denominator as $48$ so that the rational numbers lie between $ - \dfrac{{40}}{{48}}$ and $\dfrac{{30}}{{48}}$.
So, we have to choose ten rational numbers between $ - 40$ and $30$.
Now, we know that $ - 37$, $ - 31$, $ - 29$, $ - 23$, $ - 19$, $ - 17$, $ - 13$, $1$, $3$, and $5$ lie between $ - 40$ and $30$.
So, the rational numbers $\dfrac{{ - 37}}{{48}}$, $\dfrac{{ - 31}}{{48}}$, $\dfrac{{ - 29}}{{48}}$, $\dfrac{{ - 23}}{{48}}$, $\dfrac{{ - 19}}{{48}}$, $\dfrac{{ - 17}}{{48}}$, $\dfrac{{ - 13}}{{48}}$, $\dfrac{1}{{48}}$, $\dfrac{3}{{48}}$, and $\dfrac{5}{{48}}$ lie between $ - \dfrac{{40}}{{48}}$ and $\dfrac{{30}}{{48}}$.
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 $ - \dfrac{5}{6}$ and $\dfrac{5}{8}$ are: $\dfrac{{ - 37}}{{48}}$, $\dfrac{{ - 31}}{{48}}$, $\dfrac{{ - 29}}{{48}}$, $\dfrac{{ - 23}}{{48}}$, $\dfrac{{ - 19}}{{48}}$, $\dfrac{{ - 17}}{{48}}$, $\dfrac{{ - 13}}{{48}}$, $\dfrac{1}{{48}}$, $\dfrac{3}{{48}}$, and $\dfrac{5}{{48}}$.

Note: Equivalent rational numbers are the rational numbers that have different numerator and denominator but are equal to the same value. 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. If possible, we must select numerators that don’t have any common factor with the denominator as it saves the time for cancelling common factors in numerator and denominator.