This is meant to sound very difficult (and might be, if you majored in journalism), but summing infinite geometric series is easy enough to do in your head if you're facile with fractions: (first_term) / (1 - step_multiplier).
Observe that, of the combined velocity of the fly and the bike approaching him, the fly always makes up 60%. That makes math easier since it eliminates division and time from the problem entirely.
On the first trip, the fly travels 12 miles and the bike approaching him travels 8 miles. There are now 4 (20 - 2 * 8) miles between the bikes.
On the second trip, the fly travels .6 * 4 = 2.4 miles and the bike approaching him travels 1.4 * 4 = 1.6 miles. There are now 0.8 miles between the bikes.
The part where people who are really good with math distinguish themselves from people who are not is realizing quickly that the problem they are looking at, with flies and bikes, quickly decomposes into "sum the series that starts 12, 2.4, etc".
12 / (1 - 0.2) = 12 * 5 / 4 = 15 miles total fly travel.
You can do it a little more formally if you want to verify the intuition that each step takes 1/5th the time (covers 1/5th the distance) of the previous step. (My intuition says "In the time that it takes the fly to go 5 units, the two bikes will chew up 4 units of that distance, so he is only left with 1 unit to travel the next time.")
I don't think it's about difficulty per se but the number of computations required. If 25 miles were covered by the fly and a bicycle, the fly covers 15 making it 60% (first calculation). Thus for 20 miles, the fly covers 12, bicycle 8 (second calculation). 20-16 or 4 miles are remaining of which the fly covers 2.4 (third computation). 2.4 is one-fifth of 12 (fifth computation). 12 / 0.8 => 12 * 5/4 = 15 (sixth computation). This route will always be slower because it requires more computations.
This is meant to sound very difficult (and might be, if you majored in journalism), but summing infinite geometric series is easy enough to do in your head if you're facile with fractions: (first_term) / (1 - step_multiplier).
Observe that, of the combined velocity of the fly and the bike approaching him, the fly always makes up 60%. That makes math easier since it eliminates division and time from the problem entirely.
On the first trip, the fly travels 12 miles and the bike approaching him travels 8 miles. There are now 4 (20 - 2 * 8) miles between the bikes.
On the second trip, the fly travels .6 * 4 = 2.4 miles and the bike approaching him travels 1.4 * 4 = 1.6 miles. There are now 0.8 miles between the bikes.
The part where people who are really good with math distinguish themselves from people who are not is realizing quickly that the problem they are looking at, with flies and bikes, quickly decomposes into "sum the series that starts 12, 2.4, etc".
12 / (1 - 0.2) = 12 * 5 / 4 = 15 miles total fly travel.
You can do it a little more formally if you want to verify the intuition that each step takes 1/5th the time (covers 1/5th the distance) of the previous step. (My intuition says "In the time that it takes the fly to go 5 units, the two bikes will chew up 4 units of that distance, so he is only left with 1 unit to travel the next time.")