To solve this calculation, we first need to convert all mixed numbers into improper fractions.

4 5/7 = (47 + 5)/7 = 33/7
1 11/14 = (114 + 11)/14 = 25/14
4 2/3 = (43 + 2)/3 = 14/3
3 2/9 = (39 + 2)/9 = 29/9
1 5/6 = (16 + 5)/6 = 11/6
2 3/4 = (24 + 3)/4 = 11/4

Now we substitute the improper fractions back into the expression and simplify.

((33/7 - 25/14) 14/3 + (29/9 - 11/6) 18/25) / 11/4
( ( (198/14 - 25/14) 14/3) + ( 261/9 - 55/9) 18/25) ) / 11/4
( ( (173/14) 14/3) + ( 206/9) 18/25) ) / 11/4
( (4042/42) + (3717/9 * 18/25) ) / 11/4
( (4042/42) + (55746/225) ) / 11/4
( 96.2381 + 247.98 ) / 2.75
344.2181 / 2.75
125.169

Therefore, (4 5/7 - 1 11/14) 4 2/3 + (3 2/9 - 1 5/6) 18/25 divided by 2 3/4 equals 125.169