A boat can go 20 miles against a current in the same times that it can go 60 miles with the current. The current is 4 miles per hour. Find the speed of the boat with no current.

since time = distance/speed,

20/(x-4) = 60/(x+4)
x=6

Let's denote the speed of the boat in still water as "b" (in miles per hour).

When the boat goes against the current, its effective speed is reduced by the speed of the current, so the speed is equal to (b - 4) miles per hour.

When the boat goes with the current, its effective speed is increased by the speed of the current, so the speed is equal to (b + 4) miles per hour.

From the information given, we can create the following equations:

20 miles / (b - 4) hours = 60 miles / (b + 4) hours

To simplify the equation, we can cross multiply:

20(b + 4) = 60(b - 4)

Now let's solve for "b":

20b + 80 = 60b - 240

Subtract 20b from both sides:

80 = 40b - 240

Add 240 to both sides:

320 = 40b

Divide both sides by 40:

b = 8

Therefore, the speed of the boat in still water is 8 miles per hour.

To find the speed of the boat with no current, we can use a basic formula that relates distance, time, and speed:

Speed = Distance / Time

Let's assign variables to the unknowns in the problem:
Let B be the speed of the boat with no current.
Let C be the speed of the current.

Given that a boat can go 20 miles against the current in the same time it can go 60 miles with the current, we can set up two equations:

Equation 1: 20 / (B - C) = Time
Equation 2: 60 / (B + C) = Time

Since we know that the current is 4 miles per hour, we can substitute C with 4 in both equations:

Equation 1: 20 / (B - 4) = Time
Equation 2: 60 / (B + 4) = Time

Now, we can set these two equations equal to each other to eliminate the variable "Time":

20 / (B - 4) = 60 / (B + 4)

Next, we can cross-multiply to get rid of the fractions:

20(B + 4) = 60(B - 4)

Distribute:
20B + 80 = 60B - 240

Now, let's combine like terms by moving all the B terms to one side:

20B - 60B = -240 - 80

-40B = -320

Divide both sides by -40:

B = -320 / -40

Simplifying the equation gives us:

B = 8

So, the speed of the boat with no current is 8 miles per hour.