Here’s how I would do it. First, did you really mean 3 5/5? I think you might have meant 3⅗ for the second number. That means we have 7⅘-3⅗. We can write this as 7+⅘ minus (3+⅗).
First we subtract the whole number parts: 7-3=4, so the whole number part is 4, but we also have to subtract the fractions: ⅘-⅗. This says 4 fifths take away 3 fifths. We know that 4 minus 3 of the same thing leaves one of that thing. In this case, the “thing” is a fifth, so one fifth is the result: ⅕. Now we put the whole number result with it and we have the answer 4+⅕, which we write 4⅕.
[If it was supposed to be 3⁵⁄₅, that would be 3+⁵⁄₅=3+1=4 because 5 fifths make 1 whole, so we would subtract 4 from 7⅘ leaving 3⅘, because we subtract the whole numbers first, so that’s 7-4=3. The fraction has nothing to take away from it, so we’re left with what we started with: ⅘. However, in mixed numbers, the top part of the fraction is supposed to be smaller than the bottom part.]