a)

$3\frac{1}{2}\cdot \frac{5}{7}=\frac{{\cancel{7}}^1}{2}\cdot \frac{5}{{\cancel{7}}_1}=\frac{1}{2}\cdot \frac{5}{1}=\frac{1\cdot 5}{2\cdot 1}=\frac{5}{2}=2\frac{1}{2}$

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b)

$4\frac{2}{7}\cdot 28=\frac{30}{{\cancel{7}}_1}\cdot \frac{{\cancel{28}}^4}{1}=\frac{30}{1}\cdot \frac{4}{1}=30\cdot 4=120$

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c)

$2\frac{1}{2}\cdot 3\frac{1}{3}=\frac{5}{{\cancel{2}}_1}\cdot \frac{{\cancel{10}}^5}{3}=\frac{5}{1}\cdot \frac{5}{3}=\frac{5\cdot 5}{1\cdot 3}=\frac{25}{3}=8\frac{1}{3}$

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d)

$81:2\frac{1}{4}=\frac{81}{1}:\frac{9}{4}=\frac{{\cancel{81}}^9}{1}\cdot \frac{4}{{\cancel{9}}_1}=\frac{9}{1}\cdot \frac{4}{1}=9\cdot 4=36$

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e)

$5\frac{2}{3}:34=\frac{17}{3}:\frac{34}{1}=\frac{{\cancel{17}}^1}{3}\cdot \frac{1}{{\cancel{34}}_2}=\frac{1}{3}\cdot \frac{1}{2}=\frac{1\cdot 1}{3\cdot 2}=\frac{1}{6}$

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f)

$15\frac{2}{5}:2\frac{1}{5}=\frac{77}{5}:\frac{11}{5}=\frac{{\bcancel{77}}^7}{{\cancel{5}}_1}\cdot \frac{{\cancel{5}}^1}{{\bcancel{11}}^1}=\frac{7\cdot 1}{1\cdot 1}=\frac{7}{1}=7$