The question you opened
Thirty miles out, sixty miles back
You drive 60 miles out at 30 mph, then the same 60 miles back at 60 mph.
What is your average speed for the whole trip?
40 mph
120 miles took 3 hours: 2 out and 1 back. 120 over 3 is 40. Average speed is total distance over total time.
You drive 60 miles out at 30 mph and the same 60 miles back at 60 mph. The average speed for the round trip is 40 mph, and almost everyone says 45 first. The 45 comes from averaging the two speeds, which feels fair because the two distances are equal. The catch is that average speed measures total distance divided by total time, and the two legs contribute very different amounts of time. Do the clock.
The outbound leg is 60 miles at 30 mph, so it takes 2 hours. The return leg is 60 miles at 60 mph, so it takes 1 hour. The whole trip is 120 miles in 3 hours. Divide and you get 40 mph. The reason the slow leg dominates is simple once you see it. You spend twice as long crawling at 30 as you do cruising at 60, so the slow speed gets twice the weight in the average.
Averaging the speeds gives them equal weight, which would only be correct if you spent equal time at each, meaning equal times rather than equal distances. The general rule for equal distances at two speeds is the harmonic mean: 2ab over a plus b. Here that is 2 times 30 times 60, over 90, which is 3600 over 90, so 40. Same answer, one line. There is a practical consequence for anyone who drives.
Speeding up on the fast part of a journey buys you very little, because you were already spending little time there. The time you can actually save sits in the slow section, the traffic, the town centre, the queue. That is where minutes live. Whenever a question mixes rates and asks for an average, put the totals on top and bottom. Total distance over total time, total work over total hours, total cost over total items. The averaging instinct is the thing to distrust.
Technique: Total distance over total time