jmc

Since Brianna runs $\frac{2}{3}$ as fast as Eugene and he runs at a rate of 4 miles per hour, then Brianna runs at a rate of $\frac{2}{3}\cdot 4=\frac{(2)(4)}{3}=\frac{8}{3}$ miles per hour. Since Katie runs $\frac{7}{5}$ as fast as Brianna, Katie runs at a rate of $\frac{7}{5}\cdot \frac{8}{3}=\frac{(7)(8)}{(5)(3)}=\boxed{\frac{56}{15}}$ miles per hour.