To solve this expression, we first need to simplify each fraction:

1/6 = 4/24
1 1/24 = 25/24
2 2/7 = 16/7
6 2/3 = 20/3
3 5/9 = 32/9

So the expression becomes:

(4/24 - 25/24) * 16/7 + 20/3 : 32/9

(-21/24) * 16/7 + 20/3 ÷ 32/9

(-21/24) 16/7 + 20/3 9/32

(-7/8) * 16/7 + 60/96

-224/56 + 60/96

-4 + 5/8

Therefore, the final result is -4 5/8.