Let n?stand for the number of students who belong to both clubs. The 15?members of the math club can be broken down into two groups: those who are in both clubs (there are n?students in this category) and those who are in the math club only (there are 15-n?students in this category).

The 12?members of the chess club can also be broken down into two groups: nstudents who are in both clubs and 12-n?students who are in the chess club only.

Since a total of 13?students belong to only one of the two clubs, you know that?. Solving this equation gives n=7, so 7?students belong to both clubs.