yellowho wrote:Of the 300 graduate students in the comparative literature department at College X, 170 study Russian literature and 190 study English literature. If at least 40 students study neither Russian nor English literature, the number of students studying both Russian and English literature could be any number from


When dealing with max and min of sets. what is the best approach? My approach was to find [min. both] max neither because it is the only data point we have here to move around. and to [max both] min. neither. THe result here is a little surprising since maxing neither actually produces the largest both and min. neither produces the least. This is a little counterintuitive. Can somebody explain why this is true?

People with experiece, is it easier to deal max/min overlapping sets with equation?

Here is the formula for overlapping groups:

Total = Group 1 + Group 2 - Both + Neither

The big idea with overlapping groups is to subtract the overlap. In the problem above, there is an overlap between the two groups (those who study Russian and those who study English). When we count the total number who study Russian and the total number who study English, the overlap -- the students who study both languages -- gets counted twice. So we need to subtract the number of students who study both languages (the overlap) so that they don't get double-counted.

In the problem above:
Total = 300
Group 1 = Russian = 170
Group 2 = English = 190
B = both
N = neither

Thus, the equation becomes:

300 = 170 + 190 - B + N
-60 = -B + N
B = N + 60.

To minimize B, we need to minimize N. Since at least 40 students study neither subject, the smallest possible value is of N is N=40:
B = 40 + 60
B = 100.

The maximum value of B is determined by the number in the smaller group. Since only 170 students study Russian, no more than 170 students can study both Russian and English. Thus, the maximum value of B is 170.

Thus, B is between 100 and 170, inclusive.