If it's just the 7 that's repeating, let
x = 0.1777...
Then
10x = 1.777...
100x = 17.777...
100x - 10x = 17.777... - 1.777...
90x = 16
x = 16/90 = 8/45
If both 1 and 7 are repeating, let
x = 0.171717...
Then
100x = 17.171717...
100x - x = 17.171717... - 0.171717...
99x = 17
x = 17/99
