### Instruction: Eugene, Brianna, and Katie are going on a run. Eugene runs at a rate of 4 miles per hour. If Brianna runs $\frac{2}{3}$ as fast as Eugene, and Katie runs $\frac{7}{5}$ as fast as Brianna, how fast does Katie run?### Response: To solve this problem, I need to find the relationship between the speeds of the three runners.
I can use the given information to write equations that relate their speeds.
Let E be Eugene's speed, B be Brianna's speed, and K be Katie's speed.
The problem says that Brianna runs $\frac{2}{3}$ as fast as Eugene, so I can write:

\[B = \frac{2}{3}E\]
The problem also says that Katie runs $\frac{7}{5}$ as fast as Brianna, so I can write:

\[K = \frac{7}{5}B\]
I can use these equations to find Katie's speed in terms of Eugene's speed, by substituting B with $\frac{2}{3}E$ in the second equation:

\[K = \frac{7}{5}(\frac{2}{3}E) = \frac{14}{15}E\]
Now I have an expression for Katie's speed in terms of Eugene's speed, but I need to find a numerical value.
The problem gives me Eugene's speed: 4 miles per hour.
So I can plug that in for E:

\[K = \frac{14}{15}(4) = \frac{56}{15}\]