Insert a rational number between $\dfrac{5}{9}$ and $\dfrac{7}{13}$, and arrange in ascending order.

The L.C.M. of 9 and 13 is 117.

$\dfrac{5}{9} = \dfrac{5 \times 13}{9 \times 13} = \dfrac{65}{117} \\[1.5em] \dfrac{7}{13} = \dfrac{7 \times 9}{13 \times 9} = \dfrac{63}{117} \\[1.5em] \text{Since } 9 \lt 13, \dfrac{1}{13} \lt \dfrac{1}{9}$

A rational number between $\dfrac{5}{9}$ and $\dfrac{7}{13}$

$= \dfrac{\dfrac{5}{9} + \dfrac{7}{13}}{2} \\[1.5em] = \dfrac{\dfrac{65 + 63}{117}}{2} \\[1.5em] = \dfrac{128}{117 × 2} \\[1.5em] = \bold{\dfrac{64}{117}} \\[1.5em]$

$\therefore\dfrac{7}{13} \lt \dfrac{64}{117} \lt \dfrac{5}{9}$

Hence, numbers in ascending order are:

$\bold{\dfrac{7}{13}}, \bold{\dfrac{64}{117}}, \bold{\dfrac{5}{9}}$