If a plane can travel 500 miles per hour with the wind and 400 miles per hour against the​ wind, find the speed of the wind and the speed of the plane in still air.

Let's assume the speed of the plane in still air is P mph and the speed of the wind is W mph.

When the plane is flying with the wind, its effective speed is P + W mph.
When the plane is flying against the wind, its effective speed is P - W mph.

According to the given information:
P + W = 500 ----(Equation 1)
P - W = 400 ----(Equation 2)

Adding Equation 1 and Equation 2, we have:
2P = 900
P = 450

Substituting the value of P in Equation 1:
450 + W = 500
W = 500 - 450
W = 50

Therefore, the speed of the plane in still air is 450 mph and the speed of the wind is 50 mph.