Let's break this down step by step:

Step 1: Calculate 5 3/10 - 3 4/5
Convert these mixed numbers to improper fractions:
5 3/10 = (510 + 3) / 10 = 53/10
3 4/5 = (35 + 4) / 5 = 19/5
Therefore, 5 3/10 - 3 4/5 = 53/10 - 19/5.

To subtract fractions, we need to find a common denominator, which is 10 in this case. So the equivalent fractions we need are:
53/10 - 19/5 = (53/10) - (38/10) = 15/10 = 1 1/2

Step 2: Calculate (1 1/2 13/38)
Convert 1 1/2 to an improper fraction:
1 1/2 = (12 + 1) / 2 = 3/2
Therefore, (1 1/2 13/38) = (3/2) (13/38) = 39/76

Step 3: Calculate (1 1/2 13/38) 5/24
Now multiply 39/76 by 5/24:
(39/76) (5/24) = (395) / (76*24) = 195 / 1824
Reduce the fraction if needed.

Step 4: Calculate 2 1/12 4/15
Convert 2 1/12 to an improper fraction:
2 1/12 = (212 + 1) / 12 = 25/12
Therefore, (2 1/12 4/15) = (25/12) (4/15) = 100/180
Reduce the fraction if needed.

Step 5: Add the results from Step 3 and Step 4
Add 195/1824 and 100/180 to get the final answer.