A modification of blokhead's idea... what golf can go without regular expressions? 72 strokes:

for($_=pop;/\//,$`;$_=$x*$`-$'.'/'.$'*$x){1while++$x<$'/$`;print"1/$x
"}
[download]

I counted the line break as one character, is that kosher?
Update: I see also shmem had a similar thought, I had missed that before posting somehow.
Update again: Changed the title of this node to not be dumb. Sorry all, thanks for the feedback those who messaged me!