Why does "carrying" work in addition? I've been asked this many times from my most inquisitive students. Grab a pencil and paper and follow me...
 
We'll work on a simple example. If you want to try a more complex example, just post your question to this post.
 
On your piece of paper write this,
 
  35
+47 
 
If you already know how to add this, go ahead and complete the problem. Your answer should be 82, and you should have carried a 1 over the tens place. So why does this “carrying” method work?!

Do you agree that
35 = 30 + 5, and that
47 = 40 + 7? Yes, right?

Do you agree that 5 + 7 = 12? Of course, right?!

So we can write the following,
35 + 47
= 30 + 5 + 40 + 7
= 30 + 40 + 5 + 7
= 30 + 40 + 12        (here we are substituting 5 + 7 for 12)
= 30 + 40 + 10 + 2  (by substituting 12 for 10 + 2)

Now we can add in our head WITHOUT “carrying”.
  30
+40
+10
+  2
  82
By “carrying” a 1, you are essentially splitting the 12 into a 10 and 2, and that’s why carrying works!