It doesn't matter what the actual distance is from London to New York, so for the solution it's easiest to use 3,600 miles, also Time = Distance ÷ Speed:
The outward journey took 3600 ÷ 225 = 16 hours.
The return journey took 3600 ÷ 180 = 20 hours.
Therefore the total time was 36 hours, over a total distance of 7,200 miles. Using Speed = Distance ÷ Time we can see that the average speed is 7200 / 36 = 200 mph. QED.
To show that it doesn't matter what distance we use, we can call the distance D. We then have the outward journey taking D ÷ 225, and the return journey taking D ÷ 180. So the average speed is 2D / (D ÷ 225 + D ÷ 180). The D will cancel and simplification will leave the answer as 81000 ÷ 405 = 200 mph, as required.