You're referring to a simplified description of the effect of a 10% reserve requirement on the money supply. If you deposit $10 into a transaction account, the bank may lend $9 of that (not $90), and only if you assume each dollar lent is deposited into a transaction account with the same 10% reserve requirement do you get the net effect on the entire money supply (mostly through other banks) of +$100.

In practice this does not happen. And in any case, no one lends money they don't have, as you put it, unless you have strange ideas about what constitutes money.