$\frac{a+1}{7x}\ast \frac{2x}{a+1}=\frac{1}{7}\ast \frac{2}{1}=\frac{2}{7}$

$\frac{2m}{m-n}:\frac{3mn}{m-n}=\frac{2m}{m-n}\ast \frac{m-n}{3mn}=\frac{2}{1}\ast \frac{1}{3n}=\frac{2}{3n}$

$\frac{4p}{p-3}\ast \frac{p-3}{2p^2}=\frac{2}{1}\ast \frac{1}{p}=\frac{2}{p}$

$\frac{x+y}{8a}:\frac{x+y}{16a^{2}b}=\frac{x+y}{8a}\ast \frac{16a^{2}b}{x+y}=\frac{1}{1}\ast \frac{2ab}{1}=2ab$

$\frac{2x+2y}{3}\ast \frac{6}{x+y}=\frac{2(x+y)}{3}\ast \frac{6}{x+y}=\frac{2}{1}\ast \frac{2}{1}=4$

$\frac{4a}{a^{2}b}:\frac{5ab}{3a-3b}=\frac{4a}{a^{2}b}\ast \frac{3a-3b}{5ab}=\frac{4a}{a^{2}b}\ast \frac{3(a-b)}{5ab}=\frac{4}{a^{2}b}\ast \frac{3(a-b)}{5b}=\frac{12(a-b)}{5a^2{b}^2}$

$\frac{m-3n}{6m}\ast \frac{3mn}{4m-12n}=\frac{m-3n}{6m}\ast \frac{3mn}{4(m-3n)}=\frac{1}{2}\ast \frac{n}{4}=\frac{n}{8}$

$\frac{2p-4q}{3p^2}:\frac{3p-6q}{4pq}=\frac{2p-4q}{3p^2}\ast \frac{4pq}{3p-6q}=\frac{2(p-2q)}{3p^2}\ast \frac{4pq}{3(p-2q)}=\frac{2}{3p}\ast \frac{4q}{3}=\frac{8q}{9p}$

$\frac{ax-ay}{cd}\ast \frac{cx+cy}{x-y}=\frac{a(x-y)}{cd}\ast \frac{c(x+y)}{x-y}=\frac{a}{d}\ast \frac{x+y}{1}=\frac{a(x+y)}{d}$

$\frac{mk}{am-an}:\frac{ka-k}{2m-2n}=\frac{mk}{am-an}\ast \frac{2m-2n}{ka-k}=\frac{mk}{a(m-n)}\ast \frac{2(m-n)}{k(a-1)}=\frac{m}{a}\ast \frac{2}{a-1}=\frac{2m}{a(a-1)}$