Sanity of lending out made-up money
There are two ways I can think of interpreting this question.

My first thought is that a single bank doesn't lend out more money than it has, but the banking system does. So let's say someone deposits 100 dollars in Bank A and the reserve requirement is 10%. Then Bank A can lend out 90 dollars. Let's say that money is deposited in Bank B eventually. That bank can then lend out 81 dollars. This money then gets deposited in Bank C eventually, and it can lend out 72.9 dollars. This process continues, and if you take the sum of all this lending, you get that the money supply is now:

$$\sum_{n=0}^\infty 0.9^n(100) = \frac{100}{0.1} = 1000$$

So the banking system can lend out 1000 dollars without being considered unstable under this 10% reserve requirement. Notice how it kind of looks like we are double counting the money that is in the economy, and that's the point of fractional reserve banking. There isn't actually 1000 dollars, but there is credit worth that much floating around in the system.



The second way you could argue though is that individual banks in fact can lend more money than they have in their account. It creates an electronic record crediting the borrower's deposit with the sum of credit. So here, the limitation is still the reserve requirement. Say Bank A gets a deposit of 100 dollars. It can then lend out 1000 dollars at a 10% reserve requirement. Since it doesn't have 1000 in cash, it gives out credit.

I think that the real economy will use a combination of these two methods to get its lending out, but regardless, the result is the same. A simple money multiplier.



Edit: To reiterate (read: copy-pasted) from the comments below, how can a bank lend out credit that isn't backed up by paper/physical money?


FDIC insurance from the central bank, but more to the point,
The bank isn't going to back up all the credit at the same time. Say in our above scenario (100 dollars, 10% req.) the bank lends out "air money" of 5 dollars to someone, who then spends the credit purchasing from a business. The business then claims on that credit, and the bank still had some reserve (10 dollars in cash) to pay the business. Meanwhile, the bank, because it lent out so much credit, is also collecting a lotttt of interest, and borrowers pay back that interest...in cash. That replenishes the reserve. So in an equilibrium money market, hopefully the rate at which the bank gets cash in interest is the same as the rate which they pay back the "air money" with that actual cash.


Take the car example from in the comments. That is, say you "promise both your friends a car, but you only have one car and both friends know this". Well, the analogy is a little stilted, not quite parallel to what a bank is like. If you were a bank, you would only lend (not give away forever) out the car if you were promised to get it back with some interest, (maybe more gas or something). Problems only happen if both friends want the car the same time.