The speed with the wind was

 = 300 miles per hour.


It is the sum of the plane speed at no wind  "u"  and speed of the wind  "v":

u + v = 300.    (1)



The speed against the wind was

 = 240 miles per hour.


It is the difference of the plane speed at no wind  "u"  and speed of the wind  "v":

u - v = 240.    (2)


Now add equations (1) and (2). You will get

2u = 300 + 240 = 540,   which implies  u =  = 270 mph.



Now substitute it into eq(1) to get

v = 300 - 270 = 30.


Answer.  The plain speed at no wind is 270 mph;  the wind speed is 30 mph.