namely, how many times does 1 and 5/9 go into 3 1/2
let's first convert the mixed fractions to "improper", then divide then.
[tex]\bf \stackrel{mixed}{3\frac{1}{2}}\implies \cfrac{3\cdot 2+1}{2}\implies \stackrel{improper}{\cfrac{7}{2}}
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\stackrel{mixed}{1\frac{5}{9}}\implies \cfrac{1\cdot 9+5}{9}\implies \stackrel{improper}{\cfrac{14}{9}}\\\\
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\cfrac{\quad\frac{7}{2} \quad }{\frac{14}{9}}\implies \cfrac{7}{2}\cdot \cfrac{9}{14}\implies \cfrac{7\cdot 9}{2\cdot 14}\implies \cfrac{63}{28}\implies \cfrac{9}{4}\implies 2\frac{1}{4}[/tex]
