Question

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

Answer 1

Hint: In this problem, first we need to take the average of the given rational numbers. Further take the average of the new obtained rational number with the minimum of the given rational numbers and so on.

Complete step-by-step answer:
The average of the given rational numbers \[\dfrac{2}{5}\] and \[\dfrac{3}{5}\] is obtained as shown below.
\[
  \,\,\,\,\,\,\,\dfrac{{\dfrac{2}{5} + \dfrac{3}{5}}}{2} \\
   \Rightarrow \dfrac{{\dfrac{{2 + 3}}{5}}}{2} \\
   \Rightarrow \dfrac{{\dfrac{5}{5}}}{2} \\
   \Rightarrow \dfrac{1}{2} \\
 \]
Now, further take the average of \[\dfrac{1}{2} \] and\[\dfrac{2}{5} \].

\[
  \,\,\,\,\,\,\,\dfrac{{\dfrac{1}{2} + \dfrac{2}{5}}}{2} \\
   \Rightarrow \dfrac{{\dfrac{{5 + 4}}{{10}}}}{2} \\
   \Rightarrow \dfrac{9}{{20}} \\
  \]
Similarly, take the average of \[\dfrac{9}{20} \] and\[\dfrac{2}{5} \].

\[
  \,\,\,\,\,\,\,\dfrac{{\dfrac{9}{{20}} + \dfrac{2}{5}}}{2} \\
   \Rightarrow \dfrac{{\dfrac{{9 + 8}}{{20}}}}{2} \\
   \Rightarrow \dfrac{{17}}{{40}} \\
   \]
Take the average of \[\dfrac{17}{40} \] and\[\dfrac{2}{5} \].

\[
  \,\,\,\,\,\,\,\dfrac{{\dfrac{{17}}{{40}} + \dfrac{2}{5}}}{2} \\
   \Rightarrow \dfrac{{\dfrac{{17 + 16}}{{40}}}}{2} \\
   \Rightarrow \dfrac{{33}}{{80}} \\
    \]
Take the average of \[\dfrac{33}{80} \] and\[\dfrac{2}{5} \].

\[
  \,\,\,\,\,\,\,\dfrac{{\dfrac{{33}}{{80}} + \dfrac{2}{5}}}{2} \\
   \Rightarrow \dfrac{{\dfrac{{33 + 32}}{{80}}}}{2} \\
   \Rightarrow \dfrac{{65}}{{160}} \\
   \]
Thus, the five rational numbers between \[\dfrac{2}{5} \] and \[\dfrac{3}{5} \] are\[\dfrac{1}{2},\dfrac{9}{{20}},\dfrac{{17}}{{40}},\dfrac{{33}}{{80}}\,\,{\text{and}}\,\,\dfrac{{65}}{{160}}\].

Note: The average of the two numbers always lies in between the numbers. There are infinite rational numbers between any two numbers.