Given,

$\frac{P}{Q}=\frac{2}{3}$

$\frac{Q}{R}=\frac{4}{5}$

$\frac{R}{S}=\frac{6}{7}$

$\frac{P}{Q}\times\frac{Q}{R}\times\frac{R}{S}=\frac{P}{S}$

$\frac{P}{S}=\frac{2}{3}\times\frac{4}{5}\times\frac{6}{7}$

$\frac{P}{S}=\frac{16}{35}$

Hence,Option$(C)16 : 35$.