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Find the five rational numbers between
(i) $\dfrac{3}{5}$ and $\dfrac{2}{3}$
(ii) $-2$ and $ - 1\dfrac{1}{2}$

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Last updated date: 23rd Feb 2024
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IVSAT 2024
Answer
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Hint: Here we have to find the five rational numbers between the given two numbers. A rational number is defined as a number that can be expressed in the form of a fraction, where the denominator is not equal to zero. For that, we will find the average between the given numbers, which will be the rational numbers between the given numbers. We will then again find the average of the obtained number and one of the given numbers to find the next rational number. We will follow the same process until we get five rational numbers between the given numbers.

Complete step by step solution:
We know that rational numbers are any number that can be expressed in the form of $\dfrac{p}{q}$, where $q$ cannot be zero.
(i) $\dfrac{3}{5}$ and $\dfrac{2}{3}$
1) First rational number between $\dfrac{3}{5}$ and $\dfrac{2}{3}$ can be obtained by finding average between them, which is
$\dfrac{\dfrac{3}{5}+\dfrac{2}{3}}{2}=\dfrac{\dfrac{9+10}{15}}{2}=\dfrac{19}{30}$
Now, we have three numbers i.e. $\dfrac{3}{5}$ , $\dfrac{{19}}{{30}}$ and $\dfrac{2}{3}$ and thus, the rest rational numbers can be obtained by calculating average between $\dfrac{3}{5}$ and $\dfrac{{19}}{{30}}$ , and between $\dfrac{{19}}{{30}}$ and $\dfrac{2}{3}$.
2) Second rational numbers between $\dfrac{3}{5}$ and $\dfrac{{19}}{{30}}$can be obtained by finding the average between them.
$ \dfrac{{\dfrac{3}{5} + \dfrac{{19}}{{30}}}}{2} = \dfrac{{\dfrac{{18 + 19}}{{30}}}}{2} = \dfrac{{37}}{{60}}$
3) Third rational numbers between $\dfrac{{19}}{{30}}$ and $\dfrac{2}{3}$ can be obtained by finding the average between them.
$ \dfrac{{\dfrac{{19}}{{30}} + \dfrac{2}{3}}}{2} = \dfrac{{\dfrac{{19 + 20}}{{30}}}}{2} = \dfrac{{39}}{{60}}$
4) Now, the fourth rational number between $\dfrac{3}{5}$ and $\dfrac{{37}}{{60}}$can be obtained by finding the average between them.
$ \dfrac{{\dfrac{3}{5} + \dfrac{{37}}{{60}}}}{2} = \dfrac{{\dfrac{{36 + 37}}{{60}}}}{2} = \dfrac{{73}}{{120}}$
5) Fifth rational numbers between $\dfrac{37}{60}$ and $\dfrac{2}{3}$ can be obtained by finding the average between them.
$ \dfrac{{\dfrac{{37}}{{60}} + \dfrac{2}{3}}}{2} = \dfrac{{\dfrac{{37 + 40}}{{60}}}}{2} = \dfrac{{77}}{{120}}$
Thus, the 5 rational number between $\dfrac{3}{5}$ and $\dfrac{2}{3}$ are $\dfrac{{19}}{{30}}$, $\dfrac{{37}}{{60}}$, $\dfrac{{39}}{{60}}$, $\dfrac{{73}}{{120}}$ and $\dfrac{{77}}{{120}}$.

(ii) $-2$ and $ - 1\dfrac{1}{2}$
Now, we will first convert the mixed fraction into the simple fraction.
$ - 1\dfrac{1}{2} = - \dfrac{3}{2}$
1) First rational number between $-2$ and $- \dfrac{3}{2}$ can be obtained by finding average between them, which is
$ \dfrac{-2-\dfrac{3}{2}}{2}=\dfrac{\dfrac{-4-3}{2}}{2}=-\dfrac{7}{4}$
Now, we have three numbers i.e. $-2$ , $ - \dfrac{7}{4}$ and $ - \dfrac{3}{2}$ and thus, the rest rational numbers can be obtained by calculating average between $-2$ and $ - \dfrac{7}{4}$ , and between $ - \dfrac{7}{4}$ and $ - \dfrac{3}{2}$.
2) Second rational numbers between $-2$ and $ - \dfrac{7}{4}$ can be obtained by finding the average between them.
$ \dfrac{{ - 2 - \dfrac{7}{4}}}{2} = \dfrac{{\dfrac{{ - 8 - 7}}{4}}}{2} = \dfrac{{\dfrac{{ - 15}}{4}}}{2} = - \dfrac{{15}}{8}$
3) The third rational number between $ - \dfrac{7}{4}$ and $-\dfrac{3}{2}$ can be obtained by finding the average between them.
$ \dfrac{{ - \dfrac{7}{4} - \dfrac{3}{2}}}{2} = \dfrac{{\dfrac{{ - 7 - 6}}{4}}}{2} = - \dfrac{{13}}{8}$
4) Similarly fourth rational numbers between $-2$ and $ - \dfrac{{15}}{8}$ can be obtained by finding the average between them.
$ \dfrac{{ - 2 - \dfrac{{15}}{8}}}{2} = \dfrac{{\dfrac{{ - 16 - 15}}{8}}}{2} = - \dfrac{{31}}{{16}}$
5) Now, the fifth rational number between $ - \dfrac{{15}}{8}$ and $ - \dfrac{3}{2}$ can be obtained by finding the average between them.
$\dfrac{-\dfrac{15}{8}-\dfrac{3}{2}}{2}=\dfrac{\dfrac{-15-12}{8}}{2}=\dfrac{\dfrac{-27}{8}}{2}=\dfrac{-27}{16}$

Thus, the 5 rational number between $-2$ and $-1\dfrac{1}{2}$ are $-\dfrac{7}{4}$, $ - \dfrac{{15}}{8}$, $ - \dfrac{{13}}{8}$, $ - \dfrac{{31}}{{16}}$ and $\dfrac{{ - 27}}{{16}}$.

Note:
We can say that the given numbers are rational numbers because they all can be expressed in fraction form and the denominator is not equal to zero. Here we have used the average method to find the rational number between the given numbers. But we can also find the required rational number between the given numbers by just taking random numbers between the given numbers, which can be expressed in the form of $\dfrac{p}{q}$, where $q$ cannot be zero. A number which cannot be expressed in fraction form is called an irrational number.