5. Find five rational numbers between.

(i)  \frac{2}{3}  and  \frac{4}{5}              (ii)  \frac{-3}{2} and  \frac{5}{3}             (iii)  \frac{1}{4} and  \frac{1}{2}

Answers (1)

(i)     For finding rational numbers between 2 numbers one method is to find means between the numbers repeatedly.

         Another method is:- For    \frac{2}{3}  and  \frac{4}{5} 

             \frac{2}{3}  can be written as  \frac{10}{15}                            \left ( \frac{2}{3}\times \frac{5}{5} = \frac{10}{15}\right )

          and  \frac{4}{5} can be written as  \frac{12}{15}                              \left ( \frac{4}{5}\times \frac{3}{3} = \frac{12}{15}\right )

         Thus numbers between \frac{10}{15} and \frac{12}{15} are the required numbers.

         Now since we require 5 numbers in between, thus we multiply numerator and denomenator both by 4.

         It becomes numbers between \frac{40}{60} and \frac{48}{60}.

         Thus numbers are \frac{41}{60}, \frac{42}{60}, \frac{43}{60}, \frac{44}{60}, \frac{45}{60}.


(ii)      Similarly for \frac{-3}{2}and \frac{5}{3}

          Required numbers fall between \frac{-9}{6} and \frac{10}{6}                 \left \{ \left ( \frac{-3}{2}\times \frac{3}{3} \right )= \frac{-9}{6} \right \}

          Thus numbers are \frac{-8}{6}, \frac{-7}{6}, \frac{-6}{6}, \frac{-5}{6}, \frac{-4}{6}

(iii)      For   \frac{1}{4}  and  \frac{1}{2}

           Required numbers lie between \frac{1}{4} and \frac{2}{4}   or we can say between \frac{8}{32} and \frac{16}{32}

         Thus numbers are \frac{9}{32}, \frac{10}{32}, \frac{11}{32}, \frac{12}{32}, \frac{13}{32}