### 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: I need to find Katie's speed in miles per hour, so I'll start by writing an equation for that.
Let x be Katie's speed in miles per hour.
Then I have:

x = Katie's speed
Now I need to relate x to the other runners' speeds.
I know that Brianna runs $\frac{2}{3}$ as fast as Eugene, so I can write an equation for that too.
Let y be Brianna's speed in miles per hour.
Then I have:

y = Brianna's speed

y = $\frac{2}{3}$ * Eugene's speed
I also know that Eugene runs at 4 miles per hour, so I can plug that in:

y = $\frac{2}{3}$ * 4
I can simplify this by multiplying:

y = $\frac{8}{3}$
So Brianna runs at $\frac{8}{3}$ miles per hour.
Now I can use the same idea to relate Katie's speed to Brianna's speed.
I know that Katie runs $\frac{7}{5}$ as fast as Brianna, so I can write another equation for that:

x = $\frac{7}{5}$ * Brianna's speed
I can plug in the value of y that I found earlier:

x = $\frac{7}{5}$ * $\frac{8}{3}$
I can simplify this by multiplying the fractions:

x = $\frac{56}{15}$
This is the exact value of Katie's speed, but I can also convert it to a mixed number or a decimal if I want.
To do that, I can divide the numerator by the denominator:

x = $\frac{56}{15}$

x = 3 + $\frac{11}{15}$

x $\approx$ 3.733
So Katie runs at about 3.733 miles per hour.