The time required for a trip 108 miles downstream on a steamer is 3 hours less than the time required for the upstream trip. A boat whose rate is 6 miles per hour less than that of the steamer required 9 hours more for the upstream trip than for the downstream trip. Find the rate of the stream.

PLZ HELP Thank you!

let speed of steamer in still water be x mph

let the speed of the current be y mph
108/(x-y) - 108/(x+y) = 3
108x + 108y - 108x + 108y = 3(x^2 - y^2)
216y = 3(x^2 - y^2)
72y = x^2 - y^2 ---- #1

let speed of "boat" be x-6
108/(x-6 - y) - 108/(x-6 + y) = 9
108x - 648 + 108y - 108x + 648 + 108y = 9(x^2+xy-6x-xy-y^2+6y-6x-6y+36

216y = 9(x^2 - 12x - y^2 + 36)
24y = x^2 - y^2 - 12x + 36
x^2 - y^2 = 12x + 24y - 36 ---- #2

then 12x + 24y - 36 = 72y
12x - 48y = 36
x - 4y = 3
x = 4y + 3

sub into #1

x^2 - y^2 - 72y = 0
(4y+3)^2 - y^2 - 72y = 0
16y^2 + 24y + 9 - y^2 - 72y = 0
15y^2 - 48y + 9 = 0
5y^2 - 16y + 3 = 0
(5y + 1)(y - 3) = 0
y = -1/5 , not possible
or
y = 3 mph

the current has a speed of 3 mph

To solve this problem, you'll need to set up a system of equations based on the given information. Let's denote the speed of the steamer as S (in miles per hour) and the speed of the boat as B (also in miles per hour). The rate of the stream is denoted by R (in miles per hour).

First, let's consider the downstream trip. The formula to calculate time is distance divided by speed. So, the time taken for the downstream trip is 108 / (S + R) hours.

Next, let's consider the upstream trip. The time taken for the upstream trip is 108 / (S - R) hours.

We are told that the time required for the downstream trip is 3 hours less than the time required for the upstream trip. So, we can write the equation:
108 / (S + R) = 108 / (S - R) - 3

Now, we are also given that a boat, whose rate is 6 miles per hour less than that of the steamer, required 9 hours more for the upstream trip than for the downstream trip. This can be expressed as:
108 / (B - R) = 108 / (B + R) + 9

However, we need to relate B to S. We know that the boat's rate is 6 miles per hour less than that of the steamer, which means B = S - 6. We can substitute this into the equation we obtained above:
108 / ((S - 6) - R) = 108 / ((S - 6) + R) + 9

Now, we have two equations with two variables. We can solve these simultaneously to find the values of S and R.

After solving, we find that the rate of the stream (R) is 3 miles per hour.