Simplify :

$\Big(\dfrac{1}{4} - \dfrac{1}{9}\Big) ÷ \Big(\dfrac{1}{2} + \dfrac{1}{4} ÷ \dfrac{1}{3}\Big)$

$\Rightarrow \Big(\dfrac{1}{4} - \dfrac{1}{9}\Big) ÷ \Big(\dfrac{1}{2} + \dfrac{1}{4} ÷ \dfrac{1}{3}\Big)\\[1em] \Rightarrow \Big(\dfrac{1 \times 9}{4 \times 9} - \dfrac{1 \times 4}{9 \times 4}\Big) ÷ \Big(\dfrac{1}{2} + \dfrac{1}{4} \times \dfrac{3}{1}\Big)\\[1em] \Rightarrow \Big(\dfrac{9}{36} - \dfrac{4}{36}\Big) ÷ \Big(\dfrac{1}{2} + \dfrac{1 \times 3}{4 \times 1}\Big)\\[1em] \Rightarrow \Big(\dfrac{9 - 4}{36}\Big) ÷ \Big(\dfrac{1}{2} + \dfrac{3}{4}\Big)\\[1em] \Rightarrow \Big(\dfrac{5}{36}\Big) ÷ \Big(\dfrac{1}{2} + \dfrac{3}{4}\Big)\\[1em] \Rightarrow \Big(\dfrac{5}{36}\Big) ÷ \Big(\dfrac{1 \times 2}{2\times 2} + \dfrac{3}{4}\Big)\\[1em] \Rightarrow \Big(\dfrac{5}{36}\Big) ÷ \Big(\dfrac{2}{4} + \dfrac{3}{4}\Big)\\[1em] \Rightarrow \Big(\dfrac{5}{36}\Big) ÷ \Big(\dfrac{2 + 3}{4}\Big)\\[1em] \Rightarrow \Big(\dfrac{5}{36}\Big) ÷ \Big(\dfrac{5}{4}\Big)\\[1em] \Rightarrow \dfrac{5}{36} \times \dfrac{4}{5}\\[1em] \Rightarrow \dfrac{5 \times 4}{36 \times 5}\\[1em] \Rightarrow \dfrac{4}{36}\\[1em] \Rightarrow \dfrac{1}{9}.$

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