Journey from Somewhere to Elsewhere

Alice and Bob are both residents of Somewhere, and they are both coworkers in a company located in Elsewhere. Every weekday morning and evening, Alice and Bob commute between these two towns, which lie on adjacent exits on a stretch of Highway 773. They make their commutes at roughly the same time each day.

One day, at the water cooler, Bob complained to Alice about the other cars on Highway 773. During his commutes, Bob would see many more cars that violated the speed limit than those that adhered to it. Alice, in response, stated that in her experience, the opposite was true; she had seen more cars following the speed limit during her commutes than she had seen violating it.

In spite of this apparent contradiction, both Alice and Bob were honest in stating their own observations. How can this be so?

Alice is speeding, and Bob is not.

Assume 100 cars are spaced evenly and do the speed limit of 60 mph
100 additional cars are doing 70 mph and spaced evenly

Bob will only see the 60 mph car in front and in back
Bob will see the 70 mph cars pass him
Alice will only see the 70 mph car in front and in back
Alice will see the 60 mph cars she passes

I think that the answer is:

Alice is speeding, and Bob isn't.

My reasoning for this is:

I'll use a scenario to illustrate my point. Let's say that the speed limit on Highway 773 is 100km/h (or 60mph, depending on what system Somewhere uses), and that every hour, 12 cars use the highway. In each hour, half of these cars (so 6) drive lawfully, and 6 drive above the speed limit. Let's use the assumption that speeders drive 50% faster than they should.

Hence, it takes speeders two-thirds of the time to travel from Somewhere to Elsewhere, compared to lawful drivers. Hence, there would be, on average, $1\div\frac{2}{3}=\frac{3}{2}$, or 50% more drivers who are speeding past Bob, even though the amount of speeders and lawful drivers on the road are equivalent.

Whereas for Alice, who is speeding, in the time it takes her to get from Somewhere to Elsewhere, she will pass all the lawful drivers that started with her in Somewhere, but she will also pass the lawful drivers who passed Somewhere $\frac{1}{3}x$ before she did, where $x$ is the amount of time it takes her to drive from Somewhere to Elsewhere. Hence, she will see a disproportionate skew of lawful drivers to speeding drivers during her journey.

• This is the right train of thought, but the reasoning is incorrect. – redyoshi49q Jul 15 '17 at 1:09
• @redyoshi49q I'm sorry, could you please explain what you mean by this being the right train of thought? – Lucas - Better Coding Academy Jul 15 '17 at 1:12
• The crux of the apparent paradox is a result of Alice and Bob driving at different speeds, and while there isn't enough information to deternine how fast each person drives, there is enough to determine that one drives faster than the other (and who that is). However, your answer is not quite correct. – redyoshi49q Jul 15 '17 at 1:23
• I can give a further hint if such is desired. – redyoshi49q Jul 15 '17 at 1:26
• @redyoshi49q Ahhh :) did I get it now? – Lucas - Better Coding Academy Jul 15 '17 at 2:42