Find five rational numbers between:

$\dfrac{3}{5} \text{ and } \dfrac{2}{3}$

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

$[\text{ LCM of } 5 \text{ and } 3 = 15] \\[1em] \Rightarrow \dfrac{3}{5} = \dfrac{3 \times 3}{5 \times 3} = \dfrac{9}{15} \\[1em] \Rightarrow \dfrac{2}{3} = \dfrac{2 \times 5}{3 \times 5} = \dfrac{10}{15}$

To insert 5 rational numbers, multiply both the numerator and the denominator of each rational number by 5 + 1 i.e. 6.

$\Rightarrow \dfrac{9}{15} = \dfrac{9 \times 6}{15 \times 6} = \dfrac{54}{90} \\[1em] \Rightarrow \dfrac{10}{15} = \dfrac{10 \times 6}{15 \times 6} = \dfrac{60}{90}$

As 54 < 55 < 56 < 57 < 58 < 59 < 60

$\therefore \dfrac{54}{90} \lt \dfrac{55}{90} \lt \dfrac{56}{90} \lt \dfrac{57}{90} \lt \dfrac{58}{90} \lt \dfrac{59}{90} \lt \dfrac{60}{90}$

Hence, five rational numbers between $\mathbf{\dfrac{3}{5} \text{ and } \dfrac{2}{3}}$ are $\mathbf{\dfrac{55}{90}, \dfrac{56}{90}, \dfrac{57}{90}, \dfrac{58}{90} \text{ and } \dfrac{59}{90}}$.