If a kayaker's speed while traveling upstream is 7 miles per hour and downstream is 4 miles per hour. What is the kayaker's speed in still water? What is the speed of the current?

5.5

To find the kayaker's speed in still water and the speed of the current, we can use the concept of relative velocity.

Let's assume the kayaker's speed in still water is "x" miles per hour, and the speed of the current is "c" miles per hour.

When the kayaker is traveling upstream, the speed of the current acts against them, so their effective speed is reduced. Therefore, the kayaker's speed while traveling upstream is given by:
x - c = 7 miles per hour

Similarly, when the kayaker is traveling downstream, the speed of the current assists them, so their effective speed is increased. Therefore, the kayaker's speed while traveling downstream is given by:
x + c = 4 miles per hour

We now have two equations:

x - c = 7 (Equation 1)
x + c = 4 (Equation 2)

We can solve this system of equations using either substitution or elimination.

Let's use the elimination method here:
Adding Equation 1 and Equation 2:
(x - c) + (x + c) = 7 + 4
2x = 11
x = 11/2

So, the kayaker's speed in still water is 5.5 miles per hour.

Substituting the value of x into either Equation 1 or Equation 2, we can find the value of c:

5.5 - c = 7
-c = 7 - 5.5
-c = 1.5
c = -1.5

The speed of the current is 1.5 miles per hour. Note that the negative sign indicates that the current is flowing opposite to the direction of the kayaker's travel upstream.