Fractional Reserve Banking

Suppose there are 100$ in the economy i.e. (Bank of Canada) printed 100$ and randomly distributed to the Canadian population (say 10 people). Without banks, there will only be 100$.

Now , there are some banks and all people have deposited all 100$ to a bank named Bank A. Bank A puts 10$ in its reserve and loans out 90$.

Bank A:

Assets
Reserves: 10$
Loans: 90.0$
Liabilities
Deposits : 100$

Now borrowers who took money from Bank A buy something and the sellers deposit money to bank B. Bank B, puts 10% of the deposits in the reserve and loans out the rest:

Bank B:
Assets
Reserves: 9$
Loans: 81$
Liabilities
Deposits : 90$


Now, borrowers from Bank B do the same and indirectly transfer money to Bank C:
Bank C:
Assets
Reserves: 8.10$
Loans: 72.90$
Liabilities
Deposits : 81$

If this keep on going, in the end total money will be 1000$ as:
100$ (from bank A) + 90 (from bank B) + 81 (from bank C) + ... = 1000$

Because each bank is using a reserve ratio of 10% (must keep 10% of the deposits in the highly liquid state - reserves, and 90% loans out) so this is a geometric series:
100 + 0.9*100 + 0.9^2*100+....

Sum = 100* (1-r^inf)/(1-r) = 1000$ where r= 0.9

If there is no Central bank and all of sudden everyone loses confidence in banks and starts taking out money, bank runs would occur! 900$ were created out of air.