Puzzle - 9 Continued ....

Two bicyclists start twenty miles apart and head toward each other, each going at a steady rate of 10 m.p.h. At the same time, a fly that travels at a steady 15 m.p.h. starts from the front wheel of the southbound bicycle and flies to the front wheel of the northbound one, then turns around and flies to the front wheel of the southbound one again, and continues in this manner till he is crushed between the two front wheels.  Question: what total distance did the fly cover?

There are two ways to solve this problem. A mathematician would take lot of time in answering it as he would first find the distance travelled by the fly each time. It is in fact a converging series; he would then sum it up; this will take time.  When the question was put to von Neumann, he solved it in an instant, and thereby disappointed the questioner: "Oh, you must have heard the trick before!" "What trick", asked von Neumann, "all I did was sum the infinite series." I could not figure out the converging series so I asked my mother-in-law Prof. K.D. Singh, a mathematician, to solve this problem as a mathematician would solve. This is the answer that she sent

Suppose the fly travels x miles before it meets the other cyclist; the other cyclist must cover 20-x miles till the meeting point. The time taken by both is same hence x/15=(20-x)/10.  This gives x=12miles. Fly travels 12 miles and cyclist 8 miles before they meet. The other cyclist must also have traveled 8 miles by this time. At this time, the distance between the fly and the other cyclist is 20-16=4, which is 1/5th of the first distance. So this time fly would travel proportional distance i.e. 12/5 miles and so on. The series will be 12+ 12/5 +12/25+ 12/125+... which is a geometric series and you can sum it up.

Whoo!!! Only a genius can do it in an instant; at least I am not one of them but  the other method to find out the answer does not require any mathematics;  any one can find it out. Can you guess?