To solve these problems, we need to convert all mixed numbers to improper fractions and then perform the operations.

4 4/9 - 1 5/6:
Convert to improper fractions: (49 + 4)/9 - (16 + 5)/6
= (36 + 4)/9 - (6 + 5)/6
= 40/9 - 11/6

To subtract fractions with different denominators, we need to find a common denominator. In this case, it would be 18.

= (402)/(92) - (113)/(63)
= 80/18 - 33/18
= (80 - 33)/18
= 47/18

5 11/12 + 2 3/8:
Convert to improper fractions: (512 + 11)/12 + (28 + 3)/8
= (60 + 11)/12 + (16 + 3)/8
= 71/12 + 19/8

Find a common denominator (24):
= (712)/(122) + (193)/(83)
= 142/24 + 57/24
= (142 + 57)/24
= 199/24
= 8 7/24

1 7/60 + 3/20:
Convert to improper fractions: (1*60 + 7)/60 + 3/20
= (60 + 7)/60 + 3/20
= 67/60 + 3/20

Find a common denominator (60):
= (671)/60 + (33)/20
= 67/60 + 9/20

Find a common denominator (60):
= (673)/(603) + 9/20
= 201/180 + 54/180
= (201 + 54)/180
= 255/180
= 1 5/9

7 35/68 - (1 1/68 + 5 3/22):
Convert to improper fractions: (768 + 35)/68 - ((168 + 1)/68 + (5*22 + 3)/22)
= (476 + 35)/68 - ((68 + 1)/68 + 113/22)
= 511/68 - (69/68 + 113/22)

Find a common denominator (68):
= 511/68 - ((6922)/(6822) + 1133/(223))
= 511/68 - (1518/1496 + 339/66)
= 511/68 - (1518/1496 + 339/66)

Find a common denominator (1496):
= (51122)/(6822) - 1518/1496 + (3394)/(664)
= 11242/1496 - 1518/1496 + 1356/264
= (11242 - 1518 + 1356)/1496
= 1080/1496
= 540/748
= 135/187

Therefore, the solutions are:

4 4/9 - 1 5/6 = 47/185 11/12 + 2 3/8 = 8 7/241 7/60 + 3/20 = 1 5/97 35/68 - (1 1/68 + 5 3/22) = 135/187