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[tex]4\frac{3}{8}+5\frac{4}{9}=4\frac{3\cdot9}{8\cdot9}+5\frac{4\cdot8}{9\cdot8}=4\frac{27}{72}+5\frac{32}{72}=9\frac{59}{72}\\[/tex]
[tex]4\frac{3}{8} + 5\frac{4}{9} = \frac{35}{8} + \frac{49}{9} = \frac{315}{72} + \frac{392}{72} = \frac{707}{72} = \boxed{9\frac{59}{72}}[/tex]