At a college 70 out of 100 students studied Math 75 out of 100 studied English 85 out of 100 studied French and 80 out of 100 studied German. How many must have studied all 4 subjects?

Answer 1

Most of us would tend to add up the number of students taking each course and try figure out the least common inclusion. I don't know if it can be done this way. I think this problem needs an indirect approach for solution. Think about exclusions instead of inclusions. When we say 70% students take Maths, it also means 30% do not take Maths. By the same rule, 25% do not take English, 15% do not take French and 20% do not take German. If we add up these exclusions (assuming these non-takers are mutually exclusive to maximize their count) we get a figure of 90%. So there is a possibility that 90% students do not take at least one of the subjects. It may be highly unlikely on a probability perspective, but still is mathematically possible. The more important point here is that it is mathematically impossible to increase the number of non-takers with the given data. So the answer will be AT LEAST 10% of the students must have taken all four subjects.