Insert four rational numbers between $\dfrac{4}{5}$ and $\dfrac{5}{6}$.

The L.C.M of 5 and 6 is 30.

$\dfrac{4}{5} = \dfrac{4 \times 6}{5 \times 6} = \dfrac{24}{30} \\[1.5em] \dfrac{5}{6} = \dfrac{5 \times 5}{6 \times 5} = \dfrac{25}{30} \\[1.5em] \text{Since } 24 \lt 25, \\[1.5em] \therefore \dfrac{4}{5} \lt \dfrac{5}{6}$

To find four rational number between $\dfrac{4}{5}$ and $\dfrac{5}{6}$ multiply the numerator and denominator of $\dfrac{24}{30}$ and $\dfrac{25}{30}$ by 4 + 1 i.e. by 5 we get, $\dfrac{120}{150}$ and $\dfrac{125}{150}$

$\text{Since}, 120 \lt 121 \lt 122 \lt 123 \lt 124 \lt 125 \\[1.5em] \Rightarrow\dfrac{120}{150} \lt \dfrac{121}{150} \lt \dfrac{122}{150} \lt \dfrac{123}{150} \lt \dfrac{124}{150} \lt \dfrac{125}{150} \\[1.5em] \Rightarrow\dfrac{4}{5} \lt \dfrac{121}{150} \lt \dfrac{61}{75} \lt \dfrac{123}{150} \lt \dfrac{62}{75} \lt \dfrac{5}{6} \\[1.5em]$

Four rational number between $\dfrac{4}{5}$ and $\dfrac{5}{6}$ are:

$\bold{\dfrac{121}{150}}, \bold{\dfrac{61}{75}}, \bold{\dfrac{123}{150}}, \bold{\dfrac{62}{75}}$