Question

Insert \[{\text{5}}\] rational numbers between
(i) \[{\mathbf{3}}{\text{ }}{\mathbf{and}}{\text{ }}{\mathbf{5}}\]
(ii) \[ - {\mathbf{3}}{\text{ }}{\mathbf{and}}{\text{ }} - {\mathbf{5}}\]
(iii) $\dfrac{4}{{11}}and\dfrac{4}{{11}}$

Answer 1

Hint:To solve, find the average between given numbers and write it in the form
of $\dfrac{p}{q}$ where $p$ and $q$are integers.

Complete step-by-step solution:
We have to find \[{\text{5}}\]rational numbers between \[{\text{ - 3 and - 5}}\]
(i) First rational number between \[{\text{ - 3 and - 5}}\]can be calculated by finding average between them,
which is
$\Rightarrow \dfrac{{ - 3 - 5}}{2} = \dfrac{{ - 8}}{2}$
We can write it \[\dfrac{{ - 8}}{2}{\text{ = }} - {\text{4}}\]but rational no. are always in form of $\dfrac{p}{q}$
Now we have \[{\text{3}}\]no. i.e., \[{\text{ - 3}},\dfrac{{{\text{ - 8}}}}{2},{\text{5}}\]
So other remaining rational no can be calculated by finding average between
\[{\text{ - 3 \& }}\dfrac{{{\text{ - 8}}}}{2}{\text{ and }}\dfrac{{{\text{ - 8}}}}{2}{\text{ \& - 5}}\]
(ii) Second rational number between \[{\text{ - 3 \& }}\dfrac{{{\text{ - 8}}}}{2}{\text{ }}\]can be calculated by finding average
between them
$\Rightarrow \dfrac{{ - 3 - \dfrac{8}{2}}}{2}$
$ \Rightarrow \dfrac{{\dfrac{{ - 6 - 8}}{2}}}{2}$
$ \Rightarrow \dfrac{{ - 7}}{2}$
Now we have $ - 3,\dfrac{{ - 8}}{2},\dfrac{{ - 7}}{2},5$
(iii) Third rational number between \[ - \dfrac{{\text{8}}}{2}{\text{ and - 5 }}\]can be calculated by finding average
between them
$\dfrac{{ - \dfrac{8}{2} - 5}}{2}$
$ \Rightarrow \dfrac{{\dfrac{{ - 8 - 10}}{2}}}{2}$
$ \Rightarrow - \dfrac{9}{2}$
Now we have $ - 3, - \dfrac{7}{2}, - \dfrac{8}{2}, - \dfrac{9}{2}, - 5$
(iv) Similarly, fourth rational number between \[{\text{ - 3 \& - }}\dfrac{{\text{7}}}{2}{\text{ }}\]can be calculated by finding average
between them
$\Rightarrow\dfrac{{ - 3 - \dfrac{7}{2}}}{2}$
$ \Rightarrow \dfrac{{\dfrac{{ - 6 - 7}}{2}}}{2}$
$ \Rightarrow - \dfrac{{13}}{4}$
Now we have $ - 3, - \dfrac{7}{2}, - \dfrac{8}{2}, - \dfrac{{13}}{4}, - \dfrac{9}{2}, - 5$
(v) Fifth rational number between \[ - \dfrac{7}{2}{\text{ and - 5 }}\]can be calculated by finding average
between them
$\Rightarrow \dfrac{{ - \dfrac{7}{2} - 5}}{2}$
$ \Rightarrow \dfrac{{\dfrac{{ - 7 - 10}}{2}}}{2}$
$ \Rightarrow - \dfrac{{17}}{4}$
Now we have $ - 3, - \dfrac{7}{2}, - \dfrac{8}{2}, - \dfrac{9}{2}, - \dfrac{{13}}{4}, - \dfrac{{17}}{4}, - 5$
So five rational no. between \[{\text{ - 3 and - 5}}\] is $ - \dfrac{7}{2}, - \dfrac{8}{2}, - \dfrac{9}{2}, - \dfrac{{13}}{4}, - \dfrac{{17}}{4}$

Note:Always remember that Rational number is any number that can express in form of $\dfrac{p}{q}$,where $q$cannot be zero and there are inlimit rational no. between two rational numbers.