Find two rational numbers between :

(i) 2 and 3

(ii) $\dfrac{1}{3} \text{ and } \dfrac{2}{5}$

(iii) $\dfrac{3}{4} \text{ and } 1\dfrac{1}{5}$

(iv) –2 and 1

(i) Let the first rational number between 2 and 3 be x.

$\Rightarrow x = \dfrac{1}{2}\Big(2 + 3\Big) \\[1em] \Rightarrow x = \dfrac{1}{2} \times 5 \\[1em] \Rightarrow x = \dfrac{5}{2} \\[1em]$

Let the second rational number be y.

$\Rightarrow y = \dfrac{1}{2} \Big(\dfrac{5}{2} + 3\Big) \\[1em] \Rightarrow y = \dfrac{1}{2} \Big(\dfrac{5+6}{2}\Big) \\[1em] \Rightarrow y = \dfrac{1}{2} \Big(\dfrac{11}{2}\Big) \\[1em] \Rightarrow y = \dfrac{11}{4} \\[1em]$

Hence, two rational numbers between 2 and 3 are $\dfrac{5}{2} \text{ and } \dfrac{11}{4}$.

(ii) Let the first rational number between $\dfrac{1}{3}$ and $\dfrac{2}{5}$ be x.

$\Rightarrow x = \dfrac{1}{2} \Big(\dfrac{1}{3} + \dfrac{2}{5}\Big) \\[1em] \Rightarrow x = \dfrac{1}{2} \Big(\dfrac{5 + 6}{15}\Big) \\[1em] \Rightarrow x = \dfrac{1}{2} \Big(\dfrac{11}{15} \Big) \\[1em] \Rightarrow x = \dfrac{11}{30}$

Let the second rational number be y.

$\Rightarrow y = \dfrac{1}{2} \Big(\dfrac{11}{30} + \dfrac{2}{5}\Big) \\[1em] \Rightarrow y = \dfrac{1}{2} \Big(\dfrac{11 + 12}{30}\Big) \\[1em] \Rightarrow y = \dfrac{1}{2} \Big(\dfrac{23}{30}\Big) \\[1em] \Rightarrow y = \dfrac{23}{60} \\[1em]$

Hence, two rational numbers between $\dfrac{1}{3}$ and $\dfrac{2}{5}$ are $\dfrac{11}{30} \text{ and } \dfrac{23}{60}$ .

(iii) Let the first rational number between $\dfrac{3}{4}$ and $1\dfrac{1}{5}$ be x.

$\Rightarrow x = \dfrac{1}{2}\Big(\dfrac{3}{4} + 1\dfrac{1}{5}\Big) \\[1em] \Rightarrow x = \dfrac{1}{2}\Big(\dfrac{3}{4} + \dfrac{6}{5}\Big) \\[1em] \Rightarrow x = \dfrac{1}{2}\Big(\dfrac{15 + 24}{20}\Big) \\[1em] \Rightarrow x = \dfrac{1}{2}\Big(\dfrac{39}{20} \Big) \\[1em] \Rightarrow x = \dfrac{39}{40}$

Let the second rational number be y.

$\Rightarrow y = \dfrac{1}{2}\Big(\dfrac{39}{40} + \dfrac{6}{5}\Big) \\[1em] \Rightarrow y = \dfrac{1}{2}\Big(\dfrac{39 + 48}{40}\Big) \\[1em] \Rightarrow y = \dfrac{1}{2}\Big(\dfrac{87}{40}\Big) \\[1em] \Rightarrow y = \dfrac{87}{80} \\[1em]$

Hence, two rational numbers between $\dfrac{3}{4}$ and $1\dfrac{1}{5}$ are $\dfrac{39}{40} \text{ and } \dfrac{87}{80}$ .

(iv) Let the first rational number between -2 and 1 be x.

$\Rightarrow x = \dfrac{1}{2}(-2 + 1) \\[1em] \Rightarrow x = \dfrac{1}{2} \times -1 \\[1em] \Rightarrow x = -\dfrac{1}{2} \\[1em]$

Let the second rational number be y.

$\Rightarrow y = \dfrac{1}{2}\Big[-\dfrac{1}{2} + (-2)\Big] \\[1em] \Rightarrow y = \dfrac{1}{2}\Big(\dfrac{-1-4}{2}\Big) \\[1em] \Rightarrow y = \dfrac{1}{2} \times -\dfrac{5}{2} \\[1em] \Rightarrow y = -\dfrac{5}{4}.$

Hence, two rational numbers between -2 and 1 are $-\dfrac{1}{2} \text{ and } -\dfrac{5}{4}$.