Let's assume that the speed of the plane in still air is x mph and the speed of the wind is y mph.
When the plane is flying with the wind, its effective speed is increased by the speed of the wind. Therefore, the speed of the plane with the wind is x + y mph.
When the plane is flying against the wind, its effective speed is decreased by the speed of the wind. Therefore, the speed of the plane against the wind is x - y mph.
Given that the plane can travel 500 miles per hour with the wind and 400 miles per hour against the wind, we can write the following equations:
x + y = 500 (equation 1)
x - y = 400 (equation 2)
Adding equation 1 and equation 2, we get:
2x = 900
Dividing both sides of the equation by 2, we find:
x = 450
Substituting the value of x into equation 1, we can solve for y:
450 + y = 500
y = 50
Therefore, the speed of the wind is 50 mph.