Your first impulse might be to give the answer as 500 mph, the aircraft’s own airspeed and the arithmetic mean of 450 and 550. But first impulses can be wrong. Let the distance between the cities be d and the required speed, v. The time it takes the aircraft to complete the outbound stretch (flying against the headwind) is d/450, whereas the time of the return stretch is d/550. The total time is, therefore, d/450 + d/550. Setting this equal to the time it would take to complete the round trip at a constant ground speed v, we have