To solve this problem, we need to follow the order of operations (PEMDAS):

Perform the division first:
5 4/9 ÷ 1/9 = 49/9 ÷ 1/9 = 49/9 * 9/1 = 49

Perform the multiplication:
1 3/8 16 = 11/8 16 = 176/8 = 22

Use the power rule:
(2 1/2)^2 = (5/2)^2 = 25/4
(1/2)^2 = 1/4

Divide the squared terms:
(2 1/2)^2 ÷ (1/2)^2 = 25/4 ÷ 1/4 = 25

Finally, we have:
22 + 25 = 47

Therefore, the final answer is 47.