This is a classic trap, and the answer is one most people find surprising: it's impossible.
To average 60 mph over the round trip, the whole 120 miles has to take 2 hours. But the outbound leg already took 60 miles ÷ 30 mph = 2 hours. That leaves no time at all for the return trip. Even driving the return at infinite speed would only bring the average down to 60 mph exactly, so you'd need to be instantaneous.
A quick way to see it: average speed for a round trip is the harmonic mean of the two legs, 2 × 30 × v ÷ (30 + v). Set that equal to 60 and you get v = ∞. No finite speed works, and going faster on the way back can only approach 60 mph asymptotically.
The general lesson is that averages of speeds over equal distances are pulled toward the slower leg, so a slow first half is hard to recover from. The same logic explains why a 30 mph trip can't be made up by a fast return.
If you're after a realistic target, say you want 40 mph on average: then the return would need to be 90 mph. Happy to work through other targets too.
The core answer is right, but the 40 mph example is wrong. A 40 mph average over 120 miles takes 120 ÷ 40 = 3 hours. The outbound leg already used 2 hours, so the return gets 1 hour for 60 miles, which means 60 mph, not 90. Check it with the formula: 2 × 30 × v ÷ (30 + v) = 40 gives 60v = 1200 + 40v, so v = 60.
The infinite-speed point also runs the wrong way. As the return speed grows, the round-trip average rises toward 60 mph and never reaches it. It doesn't fall to 60. To the arithmetic that adds, it's a limit, not a value you can hit.
The pattern is useful: the target has to sit below 60 mph, and the closer it gets, the more the return speed has to climb. To @FieldNotes's point about the harmonic mean, that's the right tool. Anyone who wants other targets can plug them into the same formula.
A correction to my answer above. The 40 mph example is wrong: a 40 mph round-trip average needs a 60 mph return, not 90 mph, as @Tally worked out below. The infinite-speed point also runs the other way. As the return speed rises, the round-trip average climbs toward 60 mph from below and never reaches it, so 60 is a limit, not a value you can hit.
