Insert three fractions between: \[\dfrac{3}{8}\] and \[\dfrac{6}{{11}}\]

Hint: Here we are asked to find three fractions that lie in between the given two fractions. For that, we first need to find the equivalent fractions for the given fraction with the same denominator. When we get fractions with the same denominator it will be easy to find the fractions in between them.

Complete step by step answer:
We are given two fractions \[\dfrac{3}{8}\] and \[\dfrac{6}{{11}}\], we aim to find three fractions in between them.
To find a fraction in between two fractions, those fractions have to have the same denominators so that it will be easy for us to find some other fractions in between them.
To make the given fractions to the fractions with the same denominator we will find the equivalent fractions.
A fraction that is obtained by multiplying or dividing the numerator and the denominator of that fraction by the same non-zero number is called an equivalent fraction.
Let us consider the first given fraction and multiplying its numerator and denominator by eleven we get
\[\dfrac{3}{8} \times \dfrac{{11}}{{11}} = \dfrac{{33}}{{88}}\]
Now let’s consider the second given fraction and multiply its numerator and denominator by eight we get
\[\dfrac{6}{{11}} \times \dfrac{8}{8} = \dfrac{{48}}{{88}}\]
Now we got the equivalent fractions of the given two fractions with the same denominators.
Now let us write the fractions in between them.
\[\dfrac{{33}}{{88}} = \dfrac{3}{8} < \dfrac{{34}}{{88}} < \dfrac{{35}}{{88}} < \dfrac{{36}}{{88}} < \dfrac{{37}}{{88}} < \dfrac{{38}}{{88}} < \dfrac{{39}}{{88}} < \dfrac{{40}}{{88}} < \dfrac{{41}}{{88}} < \dfrac{{42}}{{88}} < \dfrac{{43}}{{88}} < \dfrac{{44}}{{88}} < \dfrac{{45}}{{88}} < \dfrac{{46}}{{88}} < \dfrac{{47}}{{88}} < \dfrac{{48}}{{88}} < \dfrac{{48}}{{88}} = \dfrac{6}{{11}}\]
We want three fractions in between \[\dfrac{3}{8}\] and \[\dfrac{6}{{11}}\] so we will take \[\dfrac{3}{8} < \dfrac{{34}}{{88}} < \dfrac{{47}}{{88}} < \dfrac{{48}}{{88}} < \dfrac{6}{{11}}\].
Thus, \[\dfrac{{34}}{{88}},\dfrac{{47}}{{88}},\dfrac{{48}}{{88}}\] are the three fractions in between \[\dfrac{3}{8}\] and \[\dfrac{6}{{11}}\].

Note:
The fractions are nothing but the number of the form \[\dfrac{a}{b}\] where \[a\] is called the numerator and \[b\] is called the denominator. The condition lies in the fractions are \[a,b\] have to be whole numbers and \[b\] should not be equal to zero since any number divided by zero is indefinite. There are many types of fractions are there few of them are proper, improper, like, unlike, mixed, equivalent fractions.