Je 2006-05-05, Navy1 <medical23FISHkidoo@sc.rr.com> skribis:

Typo?  I presume you meant 60/9 and 60/10, respectively.

This is a good approximation if the speeds are close enough to each
other, but unfortunately it doesn't give the exact answer unless the
speeds are actually equal.  As the speed difference grows, the accuracy
deteriorates, and in drastic cases this kind of calculation gives
results that don't make any sense.  Let's say a snail starts to cross a
railway bridge, and it's going to get to get to the midpoint in one hour
exactly.  At the same time, a speeding train enters the bridge, reaching
the midpoint in a few seconds.  Now it won't take half an hour for the
train and the snail to meet.

Your method, performed with exact fractions, would yield

(60/9 + 60/10) / 2 / 5 = (20/3 + 6) / 10 = 38/30 = 19/15,

which is close indeed to Vickie's correct answer, 24/19, but not the
same.  The difference is 1/285 mile, which would show up in the third
decimal.

Come to think of it, your calculation would yield the exact answer if
the faster biker suddenly slowed down to 9 mph at the midpoint.
Imagining the same in the case of the train, completely ignoring
physics, would yield a similar result: if the train rushed to the
midpoint at full speed, then suddenly slowed down literally to a snail's
pace, it'd take half an hour for the two to meet.  "Train is train and
snail is snail, and in half an hour the twain shall meet."

Actually, large part of maths research, which I do for a living,
consists in making all kinds of educated guesses at what might or might
not work.  The other part is winnowing through the guesses and
separating the wheat from the chaff.

Regards,
Taneli