Step 1 :Let's denote the speed of the plane in still air as 'p' (in mph) and the speed of the wind as 'w' (in mph).
Step 2 :When the man is flying against the wind (from Fargo to Bismarck), his effective speed is (p - w) mph. Given that the distance is 180 miles and the time taken is 2 hours, we can set up the equation: \(180 = 2 * (p - w)\)
Step 3 :When the man is flying with the wind (from Bismarck to Fargo), his effective speed is (p + w) mph. Given that the distance is again 180 miles and the time taken is 1 hour 12 minutes (or 1.2 hours), we can set up the equation: \(180 = 1.2 * (p + w)\)
Step 4 :We can solve these two equations simultaneously to find the values of 'p' and 'w'.
Step 5 :Final Answer: The speed of the plane in still air is \(\boxed{120}\) mph and the speed of the wind is \(\boxed{30}\) mph.