A poll of 100 students was taken at a no-residential college to find out how they got to campus. The results were as follows: 28 said car pools;31 said buses;and 42 said they drove to school in their own car alone. In addition, 9 used both car pools and buses;10 said both car pools and using their own car; and 6 used buses and sometimes used their own car. Of the 100 respondents, only 4 used all three methods to get to school. Can you show me how to set up the problem and answer how many students used none of the three methods to get to school?

Question

Steve · Answer

This is perfect for a Venn diagram. Draw three intersecting circles, where there is a non-empty space where all three circles intersect.

Start from the center and work your way out.

The center area contains 4.
9 used carpool and bus. Of those, 4 used all three methods, leaving 5 inside the carpool-bus intersection, but outside the center.
10 used carpool and own-car. Of those, 4 used all three methods, leaving 6 inside the carpool-self intersection, but outside the center.
That makes 4+5+6=15 who used carpool and some other method. So, that leaves 13 who used ONLY carpool.

Do similar figuring for only bus and only self-car.

Add up the numbers in all the sections of the diagram, and subtract that from the Universe of 100 students. Those will be the ones who used none of the methods.