His reasoning is wrong because he was traveling for a longer time at the slower speed of 40 mph, so his average speed will be closer to 40 mph than 60 mph.
This is easier to see with an extreme case. If you travel around the world at 1 mph, and then turn around and travel around the world again at the speed of light (Einstein might have something to say about that...), which is roughly 670 million mph, would your average speed be (670,000,000+1)/2, or about 335 million mph? Of course not-it would have to be much slower because that first part of the trip takes a very long time.
Back to Brian's problem, to find average speed we need the total distance traveled and the time taken. We know the distance is 240 miles. The time taken for the first part of the trip (when he was going 60 mph) is 2 hours, since that's how long it takes to go 120 miles at 60 mph. The time taken for the second part is
(120 miles/40 mph)=3 hours. Thus the total time taken is 5 hours, and the distance is 240, so his average speed is
(240 miles/5 hours)= 48 mph, which is closer to 40 than 60, as we reasoned at the beginning. For practice, find the average speed of the light speed-capable traveler's trip.