Convert a repeated decimal to a fraction
Perl 6 (93 101 100 80 68 66 bytes)

$/=split ".",get;say ($0+($1+$2/(9 x$2.comb||1))/10**$1.comb).nude


The size was increased to handle nothing, instead of just failing. Mouq proposed to use $/, so it's now being used, and the code is 20 bytes shorter. Ayiko proposed replacing / with , so the code is even shorter (by 12 bytes). Then Mouq proposed replacing chars with comb (in numeric context, they are identical, because list of characters after conversion to number is number of characters).

Sample output:

$ perl6 script.p6
5.3.87
889 165
$ perl6 script.p6
2.0.0
2 1
$ perl6 script.p6
0..3
1 3
$ perl6 script.p6
0.0.3
1 30
$ perl6 script.p6
0.0.0
0 1
$ perl6 script.p6
0.1.6
1 6
$ perl6 script.p6
0.01041.6
1 96
$ perl6 script.p6
0.2.283950617
37 162
$ perl6 script.p6
123.456.789
41111111 333000