How does the fractional reserve banking system work? I would like to understand better exactly how this works. I have heard a lot recently (particularly since the 2007-08 crash and Occupy movement) about money creation in particular (usually with a very negative bias). However, these descriptions often only go so far into it and I am curious whether it is really so bad once one has the full picture.

I'd like to go through a simple example and point out where I am unsure what is actually supposed to happen.

So, we have one Bank, with perhaps two customers. Alice and Bob. Alice has deposits at the Bank worth £500. Bob has £100 at this bank. A total of £600 of deposits, which the bank is liable make available to the customers at any time. The Bank has £5000 worth of its own capital. Totaling £5600 cash held (I realize that the banks' capital and the deposits are not the same - capital is an asset and deposits are a liability, for one - but this sum represents the full reserve amount, yes? The money which would be used if needed to serve depositors requests for withdrawal). We will say that the capital ratio is 10%. As I understand it, this makes the total lendable capital at the bank £5000 + £450 (Alice) + £90 (Bob) = £5540. I am pretty sure I am correct in my understanding so far.

So, Bob owes Alice a lot of money. He needs to pay her £1000. So, he takes out a loan at 5% interest. The Bank decreases its lendable capital by the amount of the loan, so it can now only loan a further £4540, however it still technically possesses the full £5540 as the £1000 loan is "new" money. Naturally, Alice now deposits Bob's £1000 payment in her account. The banks total deposits now sit at £1600, the amount it is holding is now £6600. Due to the new £1000 deposit on Alice's account, £900 is added to the lendable capital. Making it £5440. Is this correct? So far - besides the amount of lendable capital against which "new" money may be created - no deductions have been made. In addition, the £1000 loan is considered an asset that the bank owns.

So, now, the loan has eventually accumulated £80 interest and Bob manages to complete his payments to the bank. Bob pays the bank £1080 in total. Here is where I am most confused. What happens to this amount? I have heard people talk about "money destruction", implying that the bank quite literally throws away the £1000 originally lent to Bob. But somehow I find that hard to believe, and I am further confused because people mention removing it from circulation - presumably simply not making any use of it - as being the mechanism by which it is "destroyed". However, I do not believe it can be effectively removed completely from circulation, as the bank would surely either invest it or use it to make more loans, no? If indeed the money is not really destroyed, the bank has effectively just made £1080 out of nothng, right?

Perhaps I have gone wrong somewhere in my accounting. I have a Software Engineering background so a lot of this is very new/alien to me. I am just very interested in understanding it. I feel like if something is hard to believe (such as many of the claims made about money creation) it usually is for good reason, so I just wanted to understand the full picture for myself.
The bank hasn't made £1080 out of nothing. Bob paid that money into the bank. It wasn't in the bank earlier, and it is in the bank now. Alice's account has been increased by £1000. And the bank has £80 extra that it earnt by lending to Bob.

It's helpful to distinguish the money that the bank holds, from the amount of money in the whole system, and from the amount of money the bank owns.

£1080 has left the rest of the system, and entered the bank.

The bank's own capital starts at £5000 and finishes at £5080 (ignoring any interest it's paid to Alice, and its operating costs).

Alice's account starts at £500, and finishes at £1500.

Bob's account starts and ends at £100.