What confuses me is that if I apply simple mathematics, there is no way for the system to work. You can't pay back $101 (principal + 1% interest) if there are only $100 in existence.
Lets work with your VERY SIMPLIFIED example of only 100 units available and one person has all of them.
There are only 3 people in the world, Bobitza, Lisa, and Danny. There are only 100 units of currency in the world, and somehow Bobitza has managed to acquire all of it. Bobitza has deposited this currency in a bank owned by Lisa.
Bobiza has no fishing skills at all, but loves to eat fish and owns a shop that sells fishing equipment. Danny is an amazing fisherman, but his fishing equipment is now broken and since he has none of the currency, he is unable to purchase any of the fishing equipment from Bobitza. This means that neither Danny nor Bobitza will be able to eat and will therefore starve.
Danny goes to Lisa's bank and informs her that he'd like to borrow 100 units of currency so he can purchase some fishing equipment. He will be paying her back 110 units over the next 11 weeks (10 units per week). Lisa knows of Danny's fishing skills and judges him to have a low risk of default. She also knows (for some reason) that Bobitza won't be withdrawing any units from the bank this week, and won't be withdrawing more than 10 units per week in any of the following weeks. She lends him the 100 units out of the bank.
Danny takes his 100 units of currency to Bobitza's fishing equipment business and purchases all the fishing equipment he needs. Bobitza takes this 100 units and deposits them in Lisa'a bank. He's a bit confused, because he thought that he already owned all the units in existence, but he is glad that he now owns twice as much as he did before. Lisa breath's a sigh of relief when Bobitza deposits the 100 units because she now has some currency on hand in case Bobitza decideds to withdraw some. Bobitza's account at the bank indicates he has 200 units on deposit and Lisa has 100 units in reserve so she is engaging in fractional reserve lending.
Danny then proceeds to catch fish to eat and sells the extra fish to Bobitza who loves fish so much. Each week, Bobitza withdraws 10 units of currency from Lisa's bank to pay Danny for his fish. Danny then heads straight to Lisa's bank and makes a 10 unit payment on the loan which Lisa adds back to the bank's reserves to cover any additional currency Bobitza might decide to withdraw. Lisa feels more and more comfortable about her reserve situation each week as the number of units reflected in Bobitza's account drops by another 10 units bringing it closer to the actual amount she has in reserve. Meanwhile her reserve remains at 100 units since Danny keeps making his loan payments of 10 units.
So, in the tenth week, Bobitza has payed Danny 100 units of currency for the fish bringing his balance back to 100 units and Danny has paid Lisa back 100 units of currency so the bank still has 100 in reserve. At this point Lisa is no longer engaging in fractional reserve banking since the 100 units she has fully account for the 100 units that Bobitza's account indicates.
Now the 11th week comes along and Danny sells another 10 currency units of fish to Bobitza which he uses to pay off his final loan payment to Lisa. Bobitza's account now indicates that he has 90 units. Danny no longer owes anything, and Lisa's bank has 90 units in reserve for Bobitza as well as 10 units of profit from the loan.
Does this explain how it works, or am I missing something?