The other tutor changed to improper fractions.  It works but it isn't 
necessary.  You can just leave them as mixed fractions.  This is the 
way we did it way back in the 50's:

 

We'd write it like this:

    3&1/3 
   -2&2/5
    -----
  
Then since the denominators are not the same we
have to get a LCD for 3 and 5.  Since 3 and 5 have 
no common factors (other than 1) we find the LCD by 
multiplying them, getting 15. So we write this to the 
right of what we just wrote, leaving the numerators
 blank for a moment, where the empty
boxes are:
                      

    3&1/3 =  3&↏/15   
   -2&2/5 = -2&↏/15
    ---------------

3 goes into 15 5 times  and we multiply by the numerator 1 and get 5
for the first numerator,

5 goes into 15 3 times  and we multiply by the numerator 2 and get 6
for the lower numerator,

                        

    3&1/3 =  3&5/15
   -2&2/5 = -2&6/15
    ---------------

Sometimes we are able to do the subtaction when we get to this step,
and always when we are adding.  But we can't here because we don't 
want to get a negative answer by taking a larger number from a smaller
one and 5/15 is smaller than 6/15 so we can't subtract them as they
are.  So we extend once more on the right, this time reducing the whole
number 3 down to 2, borrowing 1 from it.  But we consider that 1 which
we borrowed from the 3 as the fraction 15/15. Since we have to
pay back what we borrowed, we add it to 5/15 making it 20/15.  
[Shortcut: add the numerator and denominator of 5/15 getting 5+15 or 20 and put that over 15]

So we have:  


    3&1/3 =  3&5/15 =  2&20/15
   -2&2/5 = -2&6/15 = -2& 6/15
    --------------------------
                       0&14/15 or just 14/15 since the whole part 
came out 2-2 or 0.

That's the answer, 14/15.  That's the way your grandfather and grandmother
learned to do it.  They didn't change them to improper fractions. I hope it's
the way you do it.

Edwin