With the current, a person can canoe 24 miles in 4 hours. Against the same current, the same person can canoe 18 miles in 6 hours. Find the person’s average rate in still water and the average rate of the current.

speed of canoe in still water --- x mph

speed of current --- y mph

4(x+y) = 24 ---> x + y = 6
6(x-y) = 18 ---> x - y = 3
add them:
2x = 9
x = 4.5
then y = 1.5

state conclusion

To solve this problem, we can use the concept of relative speeds. Let's denote the person's average rate in still water as "s" and the average rate of the current as "c".

When canoeing with the current, the effective speed of the person is increased by the speed of the current, so we can set up the equation:

s + c = 24/4

And when canoeing against the current, the effective speed is decreased by the speed of the current, giving us another equation:

s - c = 18/6

Simplifying both equations, we get:

s + c = 6 (Equation 1)
s - c = 3 (Equation 2)

Now, we can solve this system of equations simultaneously to find the values of "s" and "c".

Adding Equation 1 and Equation 2, we eliminate the "c" term:

(s + c) + (s - c) = 6 + 3
2s = 9
s = 9/2
s = 4.5 mph

Substituting the value of "s" into Equation 1, we can find the value of "c":

4.5 + c = 6
c = 6 - 4.5
c = 1.5 mph

So, the person's average rate in still water is 4.5 mph, and the average rate of the current is 1.5 mph.

To find the average rate in still water and the average rate of the current, we can set up a system of equations based on the given information.

Let's assume the person's average rate in still water is represented by "r" (in miles per hour), and the average rate of the current is represented by "c" (in miles per hour).

We are given two scenarios: one with the current and one against the current.

1) With the current:
In this scenario, the person can cover 24 miles in 4 hours. Since the current helps the person, their effective speed is increased.
The equation for this scenario can be written as:
(r + c) * 4 = 24

2) Against the current:
In this scenario, the person can cover 18 miles in 6 hours. Here, the current acts against the person, reducing their effective speed.
The equation for this scenario can be written as:
(r - c) * 6 = 18

We now have a system of equations:
(r + c) * 4 = 24
(r - c) * 6 = 18

Let's solve this system of equations to find the values of "r" and "c".

First, simplify the equations:
4r + 4c = 24 (equation 1)
6r - 6c = 18 (equation 2)

Now, we can solve for "r" by multiplying equation 2 by 2 to eliminate "c":
12r - 12c = 36 (equation 3)

Next, add equation 1 and equation 3 to eliminate "c":
4r + 4c + 12r - 12c = 24 + 36
16r = 60

Divide both sides of the equation by 16:
r = 3.75

Now, substitute this value of "r" back into equation 1 to solve for "c":
4(3.75) + 4c = 24
15 + 4c = 24
4c = 24 - 15
4c = 9
c = 2.25

Therefore, the person's average rate in still water (r) is 3.75 miles per hour, and the average rate of the current (c) is 2.25 miles per hour.