Oooh, I like this question. Here's a nice way to visualize it.
Suppose you've got 100 students. 45 of them study French, and 65 study Spanish. What's the smallest possible number of students who study both French and Spanish?
Imagine that you're lining up the students in a row:
Use colored lines to represent the ones that study Spanish (red), and the ones that study French (blue):
In the scenario above, all 45 of the French students also study Spanish. The overlap is 45 students. There are 35 students left over who study neither.
This is a different scenario. Scoot the blue line over to the right a bit; now it only partially overlaps the red line. The overlap is 25 students. There are 20 students who only study French, and 15 students who study neither.
Notice that the overlap gets smaller the further you move the blue line. To find the
smallest overlap, move the blue line as far to the right as possible:
You know there are 35 students on the right (everyone who doesn't take Spanish), so the overlap must be the remainder of the 45, or 10 students.