We put this question to Shaykh ‘Abd al-Rahmaan al-Barraak (may Allaah preserve him), and he answered as follows: 

The basic principle regarding the exchange of gold for gold and silver for silver is that equal amounts must be given, like for like. It makes no difference whether something is faulty or broken, etc. 

Banknotes nowadays have taken the place of gold and silver in that the value of things is measured in terms of them. 

But we cannot apply all the rulings on gold and silver to banknotes, because bank notes do not have any value in and of themselves, rather their value is conferred by the state which issues them. 

Hence I say that there is nothing wrong with exchanging torn banknotes for half of their value, because they have now lost their value (i.e., by becoming torn).