Insert a rational number between $\dfrac{2}{9}$ and $\dfrac{3}{8}$ arrange in descending order.

The L.C.M of 9 and 8 is 72.

$\dfrac{2}{9} = \dfrac{2 \times 8}{9 \times 8} = \dfrac{16}{72} \\[0.5em] \dfrac{3}{8} = \dfrac{3 \times 9}{8 \times 9} = \dfrac{27}{72} \\[0.5em] \text{Since } 16 \lt 27, \dfrac{2}{9} \lt \dfrac{3}{8}$

A rational number between $\dfrac{2}{9}$ and $\dfrac{3}{8}$

$= \dfrac{\dfrac{2}{9} + \dfrac{3}{8}}{2} \\[0.5em] = \dfrac{\dfrac{16 + 27}{72}}{2} \\[0.5em] = \bold{\dfrac{43}{144}} \\[0.5em]$

Numbers in descending order are:

$\bold{\dfrac{3}{8}}, \bold{\dfrac{43}{144}}, \bold{\dfrac{2}{9}}$