Find six rational numbers between $\dfrac{1}{2}$ and $\dfrac{2}{3}$.

Writing the given numbers with same denominator 6 (L.C.M. of 2 and 3), we get:

$\dfrac{1}{2} = \dfrac{3}{6} \\[0.5em] \dfrac{2}{3} = \dfrac{4}{6}$

Since we want to find six rational numbers between the given numbers, multiplying the numerator and denominator of the above numbers by 6 + 1 i.e. by 7, we get $\dfrac{21}{42}$ and $\dfrac{28}{42}$, which are equivalent to the given numbers.

$\text{As } 21 \lt 22 \lt 23 \lt 24 \lt 25 \lt 26 \lt 27 \lt 28, \\[0.7em] \dfrac{21}{42} \lt \dfrac{22}{42} \lt \dfrac{23}{42} \lt \dfrac{24}{42} \lt \dfrac{25}{42} \lt \dfrac{26}{42} \lt \dfrac{27}{42} \lt \dfrac{28}{42} \\[0.7em] \Rightarrow \dfrac{1}{2} \lt \dfrac{11}{21} \lt \dfrac{23}{42} \lt \dfrac{4}{7} \lt \dfrac{25}{42} \lt \dfrac{13}{21} \lt \dfrac{9}{14} \lt \dfrac{2}{3}$

Therefore, six rational numbers between $\dfrac{1}{2}$ and $\dfrac{2}{3}$ are:

$\bold{\dfrac{11}{21}}, \bold{\dfrac{23}{42}}, \bold{\dfrac{4}{7}}, \bold{\dfrac{25}{42}}, \bold{\dfrac{13}{21}}, \bold{\dfrac{9}{14}}$