Insert five rational numbers between \[\dfrac{3}{5}\] and \[\dfrac{2}{3}\].

Hint:
Here, we have to find five rational numbers between the given numbers. We can find out the mean value for the two given rational numbers. The mean value should be the required rational number. So, in order to find more rational numbers, repeat the same process with the old and the new obtained rational numbers. A rational number can be defined as any number which can be represented as the fraction of two integers.

Formula used:
We will use the formula of Mean value :\[\overline x = \dfrac{{{x_1} + {x_2}}}{2}\] where \[{x_1}\] and \[{x_2}\] are the given numbers.

Complete step by step solution:
Let \[{x_1} = \dfrac{3}{5}\] and \[{x_2} = \dfrac{2}{3}\]
Now using the formula of mean value \[\overline x = \dfrac{{{x_1} + {x_2}}}{2}\] we can find the average between \[\dfrac{3}{5}\] and \[\dfrac{2}{3}\].
\[ \Rightarrow \overline {{x_1}} = \dfrac{{\dfrac{3}{5} + \dfrac{2}{3}}}{2}\]
The denominators of the given rational numbers are \[5\] and \[3\] respectively.
The L.C.M of the denominators \[5\] and \[3\] is \[15\].
Now, we rewrite the given rational numbers into forms in which both of them have the same denominator.
\[ \Rightarrow \overline {{x_1}} = \dfrac{{\dfrac{{9 + 10}}{{15}}}}{2} = \dfrac{{19}}{{30}}\]
\[ \Rightarrow \overline {{x_2}} = \dfrac{{\dfrac{3}{5} + \dfrac{{19}}{{30}}}}{2}\]
The denominators of the given rational numbers are \[5\] and \[30\] respectively.
The L.C.M of the denominators \[5\] and \[30\] is \[30\].
Now, we rewrite the given rational numbers into forms in which both of them have the same denominator.
\[ \Rightarrow \overline {{x_2}} = \dfrac{{\dfrac{{18 + 19}}{{30}}}}{2} = \dfrac{{37}}{{60}}\]
\[ \Rightarrow \overline {{x_3}} = \dfrac{{\dfrac{2}{3} + \dfrac{{19}}{{30}}}}{2}\]
The denominators of the given rational numbers are \[3\] and \[30\] respectively.
The L.C.M of the denominators \[3\] and \[30\] is \[30\].
Now, we rewrite the given rational numbers into forms in which both of them have the same denominator.
\[ \Rightarrow \overline {{x_3}} = \dfrac{{\dfrac{{20 + 19}}{{30}}}}{2} = \dfrac{{39}}{{60}}\]
\[ \Rightarrow \overline {{x_4}} = \dfrac{{\dfrac{{37}}{{60}} + \dfrac{{19}}{{30}}}}{2}\]
The denominators of the given rational numbers are \[60\] and \[30\] respectively.
The L.C.M of the denominators \[60\] and \[30\] is \[60\].
Now, we rewrite the given rational numbers into forms in which both of them have the same denominator.
\[ \Rightarrow \overline {{x_4}} = \dfrac{{\dfrac{{37 + 38}}{{60}}}}{2} = \dfrac{{75}}{{120}}\]
\[ \Rightarrow \overline {{x_5}} = \dfrac{{\dfrac{{39}}{{60}} + \dfrac{{19}}{{30}}}}{2}\]
The denominators of the given rational numbers are \[60\] and \[30\] respectively.
The L.C.M of the denominators \[60\] and \[30\] is \[60\].
Now, we rewrite the given rational numbers into forms in which both of them have the same denominator.
\[ \Rightarrow \overline {{x_5}} = \dfrac{{\dfrac{{39 + 38}}{{60}}}}{2} = \dfrac{{77}}{{120}}\]

Therefore, the five rational numbers between \[\dfrac{3}{5}\] and \[\dfrac{2}{3}\] are \[\dfrac{{19}}{{30}},\dfrac{{37}}{{60}},\dfrac{{39}}{{60}},\dfrac{{75}}{{120}}\] and \[\dfrac{{77}}{{120}}\]

Note:
Here we have found out 5 rational numbers. We can say that the number we found is a rational number because the denominator is not equal to zero. If the denominator of a fraction is zero then they are termed as infinite numbers. We could have found the answer using a number line and placing the given numbers on the number line. And then observe which numbers come in between \[\dfrac{3}{5}\] and \[\dfrac{2}{3}\].