What the question is really asking you is:
How much saving can be achieved by "compressing" a DFA to its equivalent minimal DFA?
The answer is: the amount of saving which can achieved is unbounded. There are (at least) two ways to measure the saving:
- Number of states which are eliminated.
- Number of states in the new automaton as a fraction of the original number of states.
The number of states which are eliminated is unbounded, and the number of new states as a fraction of the number of original states can be arbitrarily close to zero.
An example is not hard to come by: pick you favorite minimal DFA, and add as many dummy states as you want. These dummy states will not be reachable from the initial state of the original DFA, so thy won't affect the language accepted by the DFA; but they will increase the number of states by an amount of your choice.
What if we disallow this example by asking all states in the DFA to be reachable? What if ask, moreover, that there be an accepting state reachable from any state? (This makes sense as stated if we're allowing the DFA to get stuck, in other words, for the transition function to be a partial function; but a similar condition can be defined for DFAs with total transition functions.) The statements above still hold true, and an example is not difficult to come up to. I challenge the readers to figure out such an example.