Insert five rational numbers between $\dfrac{3}{5}$ and $\dfrac{2}{3}$

LCM of 5 and 3 is 3 x 5 = 15

Make denominator of each given rational number equal to 15 (the LCM).

$\dfrac{3}{5} = \dfrac{3 \times 3}{5 \times 3} = \dfrac{9}{15}$

and

$\dfrac{2}{3} = \dfrac{2 \times 5}{3 \times 5} = \dfrac{10}{15}$

Since five rational numbers are required between $\dfrac{3}{5}$ and $\dfrac{2}{3}$; multiply the numerator and the denominator of each rational number by 5 + 1 = 6.

$\therefore \dfrac{9}{15} = \dfrac{9 \times 6}{15 \times 6} = \dfrac{54}{90}$

and

$\dfrac{10}{15} = \dfrac{10 \times 6}{15 \times 6} = \dfrac{60}{90}$

⇒ Required rational numbers between $\dfrac{3}{5}$ and $\dfrac{2}{3}$ are : $\dfrac{54}{90} , \dfrac{55}{90} , \dfrac{56}{90} , \dfrac{57}{90} , \dfrac{58}{90} , \dfrac{59}{90} , \dfrac{60}{90}$

= $\dfrac{3}{5} , \dfrac{11}{18} , \dfrac{28}{45} , \dfrac{19}{30} , \dfrac{29}{45} , \dfrac{59}{90} , \dfrac{2}{3}$

Hence, $\dfrac{11}{18} , \dfrac{28}{45} , \dfrac{19}{30} , \dfrac{29}{45}$ and $\dfrac{59}{90}$ lie between $\dfrac{3}{5}$ and $\dfrac{2}{3}$.