Question

List five rational numbers between:
(i) $ - 1 $ and 0
(ii) $ - 2 $ and $ - 1 $
(iii) $ - \dfrac{4}{5} $ and $ - \dfrac{2}{3} $
(iv) $ \dfrac{1}{2} $ and $ \dfrac{2}{3} $

Answer 1

Hint: To find the rational numbers, we will have to multiply the rational numbers with a number. We will convert the given numbers into fractions. For Fractions, we will have to make denominators the same by finding the least common multiple. Then we can find the five rational between the given numbers.

Complete step-by-step answer:
(i) The given numbers are $ - 1 $ and 0. We will have to find the five rational numbers between these numbers. We can find them by multiplying both the numbers by $ \dfrac{6}{6} $ . Since any number multiplied by 0 will be zero and hence by considering 6 we can find five rational numbers.
 $ - 1 \times \dfrac{6}{6} = \dfrac{{ - 6}}{6} $
And
 $ 0 \times \dfrac{6}{6} = 0 $
We can write five rational number between $ - 1 $ and 0 as:
 $ - \dfrac{5}{6} $ , $ - \dfrac{4}{6} $ , $ - \dfrac{3}{6} $ , $ - \dfrac{2}{6} $ , $ - \dfrac{1}{6} $ .

(ii) The given numbers are $ - 2 $ and $ - 1 $ . We will have to find the five rational numbers between these numbers. We can find them by multiplying both the numbers by $ \dfrac{6}{6} $ . Since any number multiplied by 0 will be zero and hence by considering 6 we can find five rational numbers.
 $ - 2 \times \dfrac{6}{6} = \dfrac{{ - 12}}{6} $
And
 $ - 1 \times \dfrac{6}{6} = - \dfrac{6}{6} $
We can write five rational number between $ - 2 $ and $ - 1 $ as:
 $ - \dfrac{{11}}{6} $ , $ - \dfrac{{10}}{6} $ , $ - \dfrac{9}{6} $ , $ - \dfrac{8}{6} $ , $ - \dfrac{7}{6} $ .
(iii) The given numbers are $ - \dfrac{4}{5} $ and $ - \dfrac{2}{3} $ . We will have to find the five rational numbers between these numbers. We will have to make their denominator to the rational numbers between them. For this we will find the least common multiple of the denominators 5 and 3. This can be expressed as:
 $ \begin{array}{c}
{\rm{LCM of }}3\;{\rm{and }}5 = 3 \times 5\\
 = 15
\end{array} $
We will multiply both numerator and denominator with the number to make the denominator equal to 15.
Hence we can write the numbers like
 $ - \dfrac{4}{5} = - \dfrac{4}{5} \times \dfrac{3}{3} = - \dfrac{{12}}{{15}} $
And
 $ - \dfrac{2}{3} = - \dfrac{2}{3} \times \dfrac{5}{5} = - \dfrac{{10}}{{15}} $
We will have to find the five rational numbers between these numbers. We can find them by multiplying both the numbers by $ \dfrac{6}{6} $ .
 $ - \dfrac{{12}}{{15}} \times \dfrac{6}{6} = \dfrac{{ - 72}}{{90}} $
And
\[ - \dfrac{{10}}{{15}} \times \dfrac{6}{6} = \dfrac{{ - 60}}{{90}}\]
We can write five rational number between $ \dfrac{{ - 72}}{{90}} $ and \[\dfrac{{ - 60}}{{90}}\] as:
 $ - \dfrac{{71}}{{90}} $ , $ - \dfrac{{70}}{{90}} $ , $ - \dfrac{{69}}{{90}} $ , $ - \dfrac{{68}}{{90}} $ , $ - \dfrac{{67}}{{90}} $ .
(iv) The given numbers are $ \dfrac{1}{2} $ and $ \dfrac{2}{3} $ . We will have to find the five rational numbers between these numbers. We will have to make their denominator to the rational numbers between them. For this we will find the least common multiple of the denominators 2 and 3. This can be expressed as:
 $ \begin{array}{c}
{\rm{LCM of 2}}\;{\rm{and 3}} = 3 \times 2\\
 = 6
\end{array} $
We will multiply both numerator and denominator with the number to make the denominator equal to 6.
Hence we can write the numbers like
 $ \dfrac{1}{2} = \dfrac{1}{2} \times \dfrac{3}{3} = \dfrac{3}{6} $
And
 $ \dfrac{2}{3} = \dfrac{2}{3} \times \dfrac{2}{2} = \dfrac{4}{6} $
We will have to find the five rational numbers between these numbers. We can find them by multiplying both the numbers by $ \dfrac{6}{6} $ .
 $ \dfrac{3}{6} \times \dfrac{6}{6} = \dfrac{{18}}{{36}} $
And
\[\dfrac{4}{6} \times \dfrac{6}{6} = \dfrac{{24}}{{36}}\]

We can write five rational number between $ \dfrac{{18}}{{36}} $ and \[\dfrac{{24}}{{36}}\] as:
 $ \dfrac{{19}}{{36}} $ , $ \dfrac{{20}}{{36}} $ , $ \dfrac{{21}}{{36}} $ , $ \dfrac{{22}}{{36}} $ , $ \dfrac{{23}}{{36}} $ .

Note: In the question, we need to find five rational numbers between the given rational numbers. That’s why we are multiplying the number by 6/6 . According to the number of rationals required, we will multiply with the respective fraction.