To subtract the fraction 2 2/5 from 7, we can first convert 7 to an improper fraction:

7 = 7/1

Now, we can find a common denominator to subtract the fractions:

7/1 - 12/5 = 35/5 - 12/5 = 23/5

Next, we can find the common denominator to subtract the remaining fractions:

23/5 - 8/15 = 69/15 - 8/15 = 61/15

Lastly, we can add the final fraction of 5 5/8 to the result:

61/15 + 69/15 = 130/15 = 8 10/15 = 8 2/3

Therefore, $7-22/5:8/15$ + 5 5/8 = 8 2/3.