... using mental arithmetic tricks. I'm not going to teach you how to do long addition on paper, although I will assume you are comfortable with doing it. This Instructable is for those times when you don't have paper or a calculator handy.
Step 1: Borrow Ones
"From where?", I hear you cry. Essentially from nowhere- sometimes you can make an addition (or sometimes a subtraction or multiplication) easier by borrowing a one, doing the sum and putting the one back. Example:
13+29 is a bit of a pain to do in your head. If you like the previous technique you could call it thirty-twelve, but to do that precise addition is a little awkward. 13+30 is much easier, however. Borrowing ones essentially means you say "13+29 is hard- I'm going to borrow a one and add it to the 29, 13+30 is 43, then I put back the one I borrowed so the answer is 42". It sounds complicated, but with a little practise I now do all of those steps in my head much faster than just trying to add 13+29.
Borrowing ones can also work in reverse by "burying" a one you would rather ignore for the time being- for example, 13+31 can be turned into 13 + 30 = 43 and then you add the one you ignored earlier, but additions that become easier by "burying" a one are less common that those made easier by borrowing.
Pros: simple, conceptually easy
Cons: limited application, doesn't allow very large additions.