Question

how to simplify 0.111111111 repeating as a fraction

Answer 1

Depending on how many times it repeats, and I used 9 1's from your question, the answer would be 1/9.

Answer 2

Any numerical repeating pattern can be rationalized...let x=0.111 then:

10x=1.111 and upon taking differences:

10x-x=1.111-0.111

9x=1  then divide both sides by 9

x=1/9

(This method works for converting any repeating numerical pattern to a rational number)