A boat is rowed down a river a distance of 16 miles in 2 hours and it is rowed upstream the same distance in 8 hours. Let r represent the rate of the boat in still water in miles per hour, and let c represent the rate of the stream in miles per hour. Write the equation. What is the still water speed and rate of the stream?

still water speed of boat --- x mph

speed of river current ---- y mph

2(x+y) = 16
x+y = 8

8(x-y) = 16
x-y = 2

add them
2x = 10
x = 5, then y = 3

state the conclusions.

To solve this problem, we can use the formula:

Distance = Rate × Time

For the boat's downstream trip, the distance is 16 miles and the time is 2 hours. Since the boat is moving with the current, we can write the equation as:

16 = (r + c) × 2

For the boat's upstream trip, the distance is still 16 miles, but the time is now 8 hours. Since the boat is moving against the current, we can write the equation as:

16 = (r - c) × 8

Now, we have two equations:

16 = 2(r + c) ----------(1)
16 = 8(r - c) ----------(2)

We can solve these equations simultaneously to find the values of r (the still water speed of the boat) and c (the rate of the stream).

Let's simplify equation (1) by dividing both sides by 2:

8 = r + c

Now, let's simplify equation (2) by dividing both sides by 8:

2 = r - c

We now have a system of equations:

8 = r + c ----------(3)
2 = r - c ----------(4)

By adding equations (3) and (4), we can eliminate the c term:

8 + 2 = r + c + r - c

10 = 2r

Dividing both sides by 2:

5 = r

We have found that the still water speed of the boat is 5 miles per hour.

Now, let's substitute r = 5 into equation (3):

8 = 5 + c

Subtracting 5 from both sides:

3 = c

Therefore, the rate of the stream is 3 miles per hour.

To summarize, the still water speed of the boat is 5 miles per hour, and the rate of the stream is 3 miles per hour.