Suppose the money supply is currently $500 billion and the Fed wishes to increase it by $100 billion. Given a required reserve rate of .25, what...

In the situation described here, the Federal Reserve should inject $25 billion into the economy, presumably by buying that much worth of government securities from banks.  Because of the money multiplier, the $25 billion injection will increase the money supply by $100 billion.

When the Fed injects money into the economy, the money supply does not simply increase by the amount of money that it has injected.  Instead, there is a multiplier effect.  This effect exists because the banks will loan out as much of the money that they receive from the Fed as possible.  When the bank loans the money, that money gets deposited in another bank and loaned out again.  Thus, much more money is created than the Fed originally injected.

The size of the multiplier effect depends on the required reserve ratio (RRR) because the RRR prevents banks from loaning out all of the money they receive as deposits.  The multiplier is found by using the equation multiplier = 1/RRR.  When the RRR is .25, the multiplier is 4.

If the multiplier is 4, the Fed needs to inject one-fourth of the amount by which it wishes the money supply to rise.  This is because the multiplier will cause the money supply to increase by four times the amount injected.  In this case, the Fed wants to increase the money supply by $100 billion.  One-fourth of $100 billion is $25 billion.  Therefore, the Fed should inject $25 billion of new money into the banking system in order to increase the money supply by $100 billion.