$a)$

$\frac{5}{8}:10=\frac{{\sout{5}}^1}{8}\cdot \frac{1}{{\sout{10}}^2}=\frac{1}{8}\cdot \frac{1}{2}=\frac{1}{16}$

$\frac{3}{4}:3=\frac{{\sout{3}}^1}{4}\cdot \frac{1}{{\sout{3}}^1}=\frac{1}{4}$

$1\frac{1}{3}:6=\frac{{\sout{4}}^2}{3}\cdot \frac{1}{{\sout{6}}^3}=\frac{2}{3}\cdot \frac{1}{3}=\frac{2}{9}$

$2\frac{1}{5}:22=\frac{{\sout{11}}^1}{5}\cdot \frac{1}{{\sout{22}}^2}=\frac{1}{5}\cdot \frac{1}{2}=\frac{1}{10}$

$b)$

$2:\frac{2}{5}={\sout{2}}^1\cdot \frac{5}{{\sout{2}}^1}=5$

$6:\frac{9}{10}={\sout{6}}^2\cdot \frac{10}{{\sout{9}}^3}=2\cdot \frac{10}{3}=\frac{20}{3}=6\frac{2}{3}$

$3:1\frac{1}{2}=3:\frac{3}{2}={\sout{3}}^1\cdot \frac{2}{{\sout{3}}^1}=2$

$10:3\frac{1}{3}=10:\frac{10}{3}={\sout{10}}^1\cdot \frac{3}{{\sout{10}}^1}=\frac{3}{1}=3$

$c)$

$\frac{1}{2}:\frac{3}{4}=\frac{1}{{\sout{2}}^1}\cdot \frac{{\sout{4}}^2}{3}=\frac{2}{3}$

$\frac{4}{5}:\frac{6}{7}=\frac{{\sout{4}}^2}{5}\cdot \frac{7}{{\sout{6}}^3}=\frac{2}{5}\cdot \frac{7}{3}=\frac{14}{15}$

$\frac{5}{8}:\frac{3}{10}=\frac{5}{{\sout{8}}^4}\cdot \frac{{\sout{10}}^5}{3}=\frac{5}{4}\cdot \frac{5}{3}=\frac{25}{12}=2\frac{1}{12}$

$\frac{4}{9}:\frac{5}{6}=\frac{4}{{\sout{9}}^3}\cdot \frac{{\sout{6}}^2}{5}=\frac{4}{3}\cdot \frac{2}{5}=\frac{8}{15}$

d)

$\frac{5}{7}:1\frac{2}{5}=\frac{5}{7}:\frac{7}{5}=\frac{5}{7}\cdot \frac{5}{7}=\frac{25}{49}$

$2\frac{2}{5}:\frac{4}{9}=\frac{{\sout{12}}^3}{5}\cdot \frac{9}{{\sout{4}}^1}=\frac{3}{5}\cdot \frac{9}{1}=\frac{27}{5}=5\frac{2}{5}$

$\frac{3}{4}:1\frac{1}{8}=\frac{3}{4}:\frac{9}{8}=\frac{{\sout{3}}^1}{{\sout{4}}^1}\cdot \frac{{\sout{8}}^2}{{\sout{9}}^3}=\frac{2}{3}$

$4\frac{2}{3}:\frac{7}{8}=\frac{{\sout{14}}^2}{3}\cdot \frac{8}{{\sout{7}}^1}=\frac{2}{3}\cdot \frac{8}{1}=\frac{16}{3}=5\frac{1}{3}$

$e)$  

$1\frac{1}{2}:1\frac{1}{5}=\frac{3}{2}:\frac{6}{5}=\frac{{\sout{3}}^1}{2}\cdot \frac{5}{{\sout{6}}^2}=\frac{1}{2}\cdot \frac{5}{2}=\frac{5}{4}=1\frac{1}{4}$

$1\frac{2}{3}:1\frac{1}{9}=\frac{5}{3}:\frac{10}{9}=\frac{{\sout{5}}^1}{{\sout{3}}^1}\cdot \frac{{\sout{9}}^3}{{\sout{10}}^2}=\frac{3}{2}=1\frac{1}{2}$

$2\frac{3}{4}:2\frac{1}{5}=\frac{11}{4}:\frac{11}{5}=\frac{{\sout{11}}^1}{4}\cdot \frac{5}{{\sout{11}}^1}=\frac{5}{4}=1\frac{1}{4}$

$3\frac{1}{8}:3\frac{3}{4}=\frac{25}{8}:\frac{15}{4}=\frac{{\sout{25}}^5}{{\sout{8}}^2}\cdot \frac{{\sout{4}}^1}{{\sout{15}}^3}=\frac{5}{2}\cdot \frac{1}{3}=\frac{5}{6}$