Problem B: Dead Fraction
Mike is frantically scrambling to finish his thesis at the last minute. He
needs to assemble all his research notes into vaguely coherent form in the
next 3 days. Unfortunately, he notices that he had been extremely sloppy in
his calculations. Whenever he needed to perform arithmetic, he just plugged
it
into a calculator and scribbled down as much of the answer as he felt was
relevant. Whenever a repeating fraction was displayed, Mike simply
reccorded the first few digits followed by "...". For
instance, instead of "1/3" he might have written down "0.3333...".
Unfortunately, his results require exact fractions! He doesn't have time to
redo every calculation, so he needs you to write a program (and FAST!) to
automatically deduce the original fractions.
To make this tenable, he assumes that the original fraction is always the
simplest one that produces the given sequence of digits; by simplest, he means the the
one with smallest denominator. Also, he assumes that he did not neglect
to write down important digits; no digit from the repeating portion of the
decimal expansion was left unrecorded (even if this repeating
portion was all zeroes).
There are several test cases. For each test case there is one line of input
of the form "0.dddd..." where dddd is a string of 1 to 9 digits, not all zero.
A line containing 0 follows the last case. For each case, output the original
fraction.
Note that an exact decimal fraction has two repeating expansions
(e.g. 1/5 = 0.2000... = 0.19999...).
Sample Input
0.2...
0.20...
0.474612399...
0
Output for Sample Input
2/9
1/5
1186531/2500000
D. Kisman