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

2 1/3 = 7/3
3 3/8 = 27/8
4 4/5 = 24/5
2 4/9 = 22/9

Now we can substitute these values into the equation:

(7/3 - 27/8) * 24/5 + 22/9

To find the common denominators, we need to find the least common multiple of 3, 8, 5, and 9, which is 360.

(87 - 327) 72/360 + 40/360
(56 - 81) 72/360 + 40/360
-25 * 72/360 + 40/360
-180/360 + 40/360
-140/360

Finally, we simplify the improper fraction into a mixed number:

-140 ÷ 36 = -3 with a remainder of 20
So, -140/360 = -3 20/360
20/360 reduces to 1/18

Therefore, the answer is:
-3 1/18