No summing is necessary. Bitcoin doesn't track addresses. It tracks outputs. When a transaction spends from an address, it spends an entire output, and the excess is sent to a "change" address. If a node sees two transactions spending the same output, it will reject one of them.
Yes, no node will approve both of transactions. But the unlikely event OP is talking about is that both transactions are made at a same time. In this case, we don't know which node will receive transaction A first and which node will receive transaction B first. So some nodes have approved transaction A and some nodes have approved transaction B.
Now both transactions have the chance of being included in the next block.
A miner may mine a block including transaction A and another miner may mine a block including transaction B.
Now we have two different blocks. Both of them are valid if they are taken into account independently. So, we have two different chains.
Next blocks will determine that which of those transactions was the valid transaction.
Miners A will try to mine the next block considering the hash of block A (The block including transaction A) and miners B will try to mine the next block considering the hash of block B (The block including transaction B).
If miners A become successful, block A will be the valid block.
If miners B become successful, block B will be the valid block.