a) To divide 5 2/3 by 2 1/4, we need to convert both mixed numbers to improper fractions:

5 2/3 = (3x5+2)/3 = 17/3
2 1/4 = (4x2+1)/4 = 9/4

Now we can divide by multiplying by the reciprocal of the divisor:

17/3 ÷ 9/4 = 17/3 x 4/9 = 68/27

So the answer is 68/27.

b) To solve 3 1/2 + 1/2 ÷ 4 1/2, we need to follow the order of operations (PEMDAS) and simplify the division first:

1/2 ÷ 4 1/2 = 1/2 ÷ (9/2) = 1/2 x 2/9 = 1/9

Now we can add the whole numbers and fractions separately:

3 + 1/9 = 28/9

So the answer is 28/9 or 3 1/9.