Find three rational numbers between:

4 and $4\dfrac{2}{3}$

Write both the numbers with a common denominator, then multiply the numerator and the denominator of each by one more than the number of rational numbers required.

$\Rightarrow 4 = \dfrac{12}{3} \\[1em] \Rightarrow 4\dfrac{2}{3} = \dfrac{14}{3}$

To insert 3 rational numbers, multiply both the numerator and the denominator of each rational number by 3 + 1 i.e. 4.

$\Rightarrow \dfrac{12}{3} = \dfrac{12 \times 4}{3 \times 4} = \dfrac{48}{12} \\[1em] \Rightarrow \dfrac{14}{3} = \dfrac{14 \times 4}{3 \times 4} = \dfrac{56}{12}$

As 48 < 49 < 50 < 51 < 56,

$\therefore \dfrac{48}{12} \lt \dfrac{49}{12} \lt \dfrac{50}{12} \lt \dfrac{51}{12} \lt \dfrac{56}{12}$

Hence, three rational numbers between 4 and $\mathbf{4\dfrac{2}{3}}$ are $\mathbf{\dfrac{49}{12}, \dfrac{50}{12} \text{ and } \dfrac{51}{12}}$.