Let Amy's, Ben's, and Chris's ages be $ a $, $ b $, and $ c $, respectively. We have the equations \begin{align*} \tag{1} \frac{a+b+c}{3}=6 \Rightarrow a+b+c&=18 \\ \tag{2} c-4&=a\\ \tag{3} b+4&=\frac{3}{5}(a+4) \end{align*} From Equation (3), we have $ b=\frac{3}{5}(a+4)-4 $. We substitute Equation (2) into Equation (3) to eliminate $ a $, to get $ b=\frac{3}{5}(c)-4 $. Substituting this last equation and Equation (2) into Equation (1) to eliminate $ a $ and $ b $, we have \[[c-4]+[\frac{3}{5}(c)-4]+c=18\] Solving for $ c $, we find that $ c=10 $. Thus, Chris's age is $ \boxed{10} $.