Find six rational numbers between $3$ and $4$ . Find five rational numbers between $\dfrac{3}{5}$ and $\dfrac{4}{5}$ .

Hint: To insert rational numbers between any two rational numbers, we make the denominators of the two rational numbers the same. This way we can easily insert any number of rational numbers between any two rational numbers.

Complete step-by-step solution:
Firstly, we need to insert six rational numbers between $3$ and $4$ .
Now, to make the denominators same in the given two rational numbers, we multiply and divide the numbers with the same number, such as the denominators of both the numbers become the same, i.e.
$3 = 3 \times \dfrac{7}{7} = \dfrac{{21}}{7}$ and $4 = 4 \times \dfrac{7}{7} = \dfrac{{28}}{7}$
So, here, we multiply and divide both the numbers by $7$
Thus, $3$ and $4$ becomes $\dfrac{{21}}{7}$ and $\dfrac{{28}}{7}$ respectively.
Now, we can easily insert six rational numbers between $\dfrac{{21}}{7}$ and $\dfrac{{28}}{7}$ , such as,
$\dfrac{{22}}{7}$ , $\dfrac{{23}}{7}$ , $\dfrac{{24}}{7}$ , $\dfrac{{25}}{7}$ , $\dfrac{{26}}{7}$ , $\dfrac{{27}}{7}$
∴ six rational numbers between $3$ and $4$ are $\dfrac{{22}}{7}$ , $\dfrac{{23}}{7}$ , $\dfrac{{24}}{7}$ , $\dfrac{{25}}{7}$ , $\dfrac{{26}}{7}$ , $\dfrac{{27}}{7}$ .
Now, secondly, we need to insert five rational numbers between $\dfrac{3}{5}$ and $\dfrac{4}{5}$ .
Again, to make the denominators same in the given two rational numbers, we multiply and divide the numbers with the same number, such as the denominators of both the numbers become the same, i.e.
$\;\dfrac{3}{5} = \dfrac{3}{5} \times \dfrac{6}{6} = \dfrac{{18}}{{30}}$ and $\dfrac{4}{5} = \dfrac{4}{5} \times \dfrac{6}{6} = \dfrac{{24}}{{30}}$
So, here, we multiply and divide both the numbers by $6$
Thus, $\dfrac{3}{5}$ and $\dfrac{4}{5}$ becomes $\dfrac{{18}}{{30}}$ and $\dfrac{{24}}{{30}}$ respectively.
Now, we can easily insert five rational numbers between $\dfrac{{18}}{{30}}$ and $\dfrac{{24}}{{30}}$ , such as,
$\dfrac{{19}}{{30}}$ , $\dfrac{{20}}{{30}}$ , $\dfrac{{21}}{{30}}$ , $\dfrac{{22}}{{30}}$ , $\dfrac{{23}}{{30}}$

Five rational numbers between $\dfrac{3}{5}$ and $\dfrac{4}{5}$ are $\dfrac{{19}}{{30}}$ , $\dfrac{{20}}{{30}}$ , $\dfrac{{21}}{{30}}$ , $\dfrac{{22}}{{30}}$ , $\dfrac{{23}}{{30}}$ .

Note: A rational number is in the form of $\dfrac{p}{q}$ , where $p$ and $q$ are integers and $q \ne 0$ . We multiply and divide a rational number with another number because it does not make any change in the existing number. These rational numbers are also called equivalent rational numbers.