How would you convert a repeating decimal like 1.27 or 0.6 into a fraction?

The examples in your question are not examples of repeating decimals. A repeating decimal is one in which the same number is repeated infinitely. An example would be ##0.666...##, also written as ##0.bar6##, or ##0.0909...##, also written as ##0.bar09##.

##1.27=1 (27)/(100)##

##0.6=6/10=3/5##