A boat can go 18 miles downstream in the same time it takes to go 12miles upstream. The rate of the water is 6 mph. What is the rate of the boat?

To solve this problem, we need to understand that the speed of the boat is affected by the speed of the water. When the boat is going downstream, the speed of the water is helping it, so we add the speed of the boat and the speed of the water. When the boat is going upstream, the speed of the water is hindering it, so we subtract the speed of the water from the speed of the boat.

Let's denote the speed of the boat in still water as 'b' (in mph).

1. When the boat is going downstream (with the current), its speed is (b + 6) mph (boat speed plus water speed). It covers 18 miles at this speed. So, the time it takes is distance/speed = 18/(b + 6) hours.

2. When the boat is going upstream (against the current), its speed is (b - 6) mph (boat speed minus water speed). It covers 12 miles at this speed. So, the time it takes is distance/speed = 12/(b - 6) hours.

According to the problem, these two times are equal. So, we can set up the following equation and solve for 'b':

18/(b + 6) = 12/(b - 6)

Cross-multiplying gives us:

18*(b - 6) = 12*(b + 6)

Expanding both sides gives:

18b - 108 = 12b + 72

Subtracting 12b from both sides gives:

6b = 180

Finally, dividing both sides by 6 gives:

b = 30

So, the speed of the boat in still water is 30 mph.