Question

Find \[5\] rational numbers between \[ - 1\] and \[0\].

Answer 1

Hint:In the given problem, we have to find \[5\] rational numbers between \[ - 1\] and \[0\]. First of all, we will find one rational number between \[ - 1\] and \[0\], by taking the average of both the given numbers. Then, we will again take the average of the resultant number with the other two numbers one by one. We will repeat this process to find \[5\] rational numbers between \[ - 1\] and \[0\].

Formula Used:
To find the average of two numbers \[a\] and \[b\], we use the formula ,
Average \[ = \dfrac{{a + b}}{2}\]

Complete step by step answer:
Consider the given question, we have to find \[5\] rational number between \[ - 1\] and \[0\]
First we find one rational number between\[ - 1\] and \[0\] i.e,
\[\dfrac{{( - 1) + 0}}{2} = \dfrac{{ - 1}}{2}\]
Hence, we have \[ - 1 < \dfrac{{ - 1}}{2} < 0\]
Now, we find one rational number between \[ - 1\] and \[\dfrac{{ - 1}}{2}\].i.e. \[\dfrac{{\left( { - \dfrac{1}{2}} \right) + ( - 1)}}{2}\],
on solving by taking LCM we have
\[\dfrac{{\left( { - \dfrac{1}{2}} \right) + ( - 1)}}{2} = \dfrac{{\dfrac{{ - 1 - 2}}{2}}}{2} \\
\Rightarrow \dfrac{{\left( { - \dfrac{1}{2}} \right) + ( - 1)}}{2} = \dfrac{{ - 3}}{4}\]
Hence we have \[ - 1 < \dfrac{{ - 3}}{4} < \dfrac{{ - 1}}{2} < 0\].

Now we find a rational number between \[ - \dfrac{1}{2}\] and \[0\]
i.e. \[\dfrac{{ - \dfrac{1}{2} + 0}}{2} = \dfrac{{ - 1}}{4}\]
hence we have \[ - 1 < \dfrac{{ - 3}}{4} < \dfrac{{ - 1}}{2} < \dfrac{{ - 1}}{4} < 0\].
Again we find a rational number between \[ - 1\] and \[\dfrac{{ - 3}}{4}\] i.e. \[\dfrac{{\dfrac{{ - 3}}{4} + ( - 1)}}{2}\], on solving we have
\[\dfrac{{\dfrac{{ - 3}}{4} + ( - 1)}}{2} = \dfrac{{\dfrac{{ - 3 - 4}}{4}}}{2} \\
\Rightarrow\dfrac{{\dfrac{{ - 3}}{4} + ( - 1)}}{2}= \dfrac{{ - 7}}{8}\]
Hence we have \[ - 1 < \dfrac{{ - 7}}{8} < \dfrac{{ - 3}}{4} < \dfrac{{ - 1}}{2} < \dfrac{{ - 1}}{4} < 0\].
Now we find the last rational between \[\dfrac{{ - 1}}{4}\]and \[0\] i.e. \[\dfrac{{ - \dfrac{1}{4} + 0}}{2} = \dfrac{{ - 1}}{8}\]
Hence we have ,
\[ - 1 < \dfrac{{ - 7}}{8} < \dfrac{{ - 3}}{4} < \dfrac{{ - 1}}{2} < \dfrac{{ - 1}}{4} < \dfrac{{ - 1}}{8} < 0\].

Hence \[5\] rational number between\[ - 1\] and \[0\] are \[\dfrac{{ - 7}}{8}\], \[\dfrac{{ - 3}}{4}\], \[\dfrac{{ - 1}}{2}\], \[\dfrac{{ - 1}}{4}\] and \[\dfrac{{ - 1}}{8}\].

Note:A Rational number is of the form \[\dfrac{a}{b}\], where \[a\] and \[b\] are integers and denominators \[b \ne 0\]. To find the average, we simply add the numbers and divide the sum by \[2\]. For example, the average of \[\dfrac{1}{2}\] and \[\dfrac{1}{4}\] is \[\dfrac{{\dfrac{1}{2} + \dfrac{1}{4}}}{2}\]. To add the numbers in fraction, we will take the LCM of the denominator and multiply the numerator with the quotient of LCM and denominator. For example, \[\dfrac{1}{2} + \dfrac{1}{4} = \dfrac{{2 + 1}}{4} = \dfrac{3}{4}\].