To solve this problem, we need to follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and convert all the mixed numbers to improper fractions for easier computation.

6 3/14 4 5/9 - 6 3/14 1 5/9 + 2 11/14 3
= (6 14 + 3) / 14 (4 9 + 5) / 9 - (6 14 + 3) / 14 (1 9 + 5) / 9 + (2 14 + 11) / 14 3
= (84 + 3) / 14 (36 + 5) / 9 - (84 + 3) / 14 (9 + 5) / 9 + (28 + 11) / 14 3
= 87 / 14 41 / 9 - 87 / 14 14 / 9 + 39 / 14 3
= 6.214285714285714 4.555555555555555 - 6.214285714285714 1.5555555555555554 + 2.7857142857142856 3
≈ 28.285714285714285 - 9.642857142857142 + 8.357142857142858
≈ 27.000000000000004

Therefore, the answer is approximately 27.