Simplify :

$\dfrac{1}{3}\Big(2\dfrac{1}{2} + 3\dfrac{1}{3}\Big) ÷ \dfrac{2}{9}\Big(3\dfrac{1}{8} - 1\dfrac{1}{12}\Big)$

$\Rightarrow \dfrac{1}{3}\Big(2\dfrac{1}{2} + 3\dfrac{1}{3}\Big) ÷ \dfrac{2}{9}\Big(3\dfrac{1}{8} - 1\dfrac{1}{12}\Big)\\[1em] \Rightarrow \dfrac{1}{3}\Big(\dfrac{5}{2} + \dfrac{10}{3}\Big) ÷ \dfrac{2}{9}\Big(\dfrac{25}{8} - \dfrac{13}{12}\Big)\\[1em] \Rightarrow \dfrac{1}{3}\Big(\dfrac{5 \times 3}{2 \times 3} + \dfrac{10 \times 2}{3 \times 2}\Big) ÷ \dfrac{2}{9}\Big(\dfrac{25 \times 3}{8 \times 3} - \dfrac{13 \times 2}{12 \times 2}\Big)\\[1em] \Rightarrow \dfrac{1}{3}\Big(\dfrac{15}{6} + \dfrac{20}{6}\Big) ÷ \dfrac{2}{9}\Big(\dfrac{75}{24} - \dfrac{26}{24}\Big)\\[1em] \Rightarrow \dfrac{1}{3}\Big(\dfrac{15 + 20}{6}\Big) ÷ \dfrac{2}{9}\Big(\dfrac{75 - 26}{24}\Big)\\[1em] \Rightarrow \dfrac{1}{3}\Big(\dfrac{35}{6}\Big) ÷ \dfrac{2}{9}\Big(\dfrac{49}{24}\Big)\\[1em] \Rightarrow \dfrac{35 \times 1}{6 \times 3} ÷ \dfrac{49 \times 2}{24 \times 9}\\[1em] \Rightarrow \dfrac{35}{18} ÷ \dfrac{98}{216}\\[1em] \Rightarrow \dfrac{35}{18} \times \dfrac{216}{98}\\[1em] \Rightarrow \dfrac{35}{98} \times \dfrac{216}{18} \\[1em] \Rightarrow \dfrac{5}{14} \times 12 \\[1em] \Rightarrow \dfrac{60}{14}\\[1em] \Rightarrow \dfrac{30}{7} \\[1em] \Rightarrow 4\dfrac{2}{7}$

Hence, $\dfrac{1}{3}\Big(2\dfrac{1}{2} + 3\dfrac{1}{3}\Big) ÷ \dfrac{2}{9}\Big(3\dfrac{1}{8} - 1\dfrac{1}{12}\Big) = 4\dfrac{2}{7}$.