One hundred people were surveyed. Of these, $87$ indicated they liked Mozart and $70$ indicated they liked Bach. What is the minimum number of people surveyed who could have said they liked both Mozart and Bach?
The minimum number of people who like both Mozart and Bach is achieved when the number of people who like Mozart but not Bach is maximized. There are $100-70=30$ people who do not like Bach. If all these people like Mozart, then the number of people who like Bach and Mozart is $87-30=\boxed{57}$.