200 students
80 in science
70 in geography
60 in art
35 in geography and science 33 have geography and art 31 have science and art 15 have geo. sci and art How many of the 200 students have none of these courses
Set up your diagram with 3 circles, S, G, A, for each subject. The middle should have a small space that includes part of all 3 circles. Start from this spot and move outward.
This very middle spot represents the students who are in all 3 subjects. As given in the problem, there are 15, so write that in the center portion. Now lets try to fill out the rest of the Geography circle.
The next portion of G we can find is the # of students who are in both Geography and Science but NOT art. You are given that 35 students are in G and S, however 15 of these students are ALSO in art. To get those only in G and S, subtract the 15 from the entire 35 taking both subjects. This gives us 20 students taking G&S but NOT A.
Next is the students taking G and A, but NOT S. It is given that 33 students are taking G and A. Again, 15 of these are ALSO in Science, therefore subtract the 15 from the entire 33 to get 18 students in G&A only.
Now we have everything but the students taking STRICTLY geography. You are given that there are 70 students total in G, and have found 20 are in G&S, 18 in G&A, and 15 in G&S&A. Subtract these students from the entire 70 to get 17 that are only taking Geography.
Now do the same procedure with the circles for Art and for Science. At the end, you can sum all of the sections you have filled out to find the total # of students enrolled in the subjects and subtract that # from 200 to find those not in any of them.
To find the number of students who have none of these courses, we need to subtract the total number of students taking at least one of these courses from the total number of students.
First, let's calculate the total number of students taking at least one of these courses by adding up the number of students taking each course:
Number of students taking science: 80
Number of students taking geography: 70
Number of students taking art: 60
However, this count includes some students who are double-counted because they are taking multiple courses. We need to subtract those students who are taking more than one course.
Number of students taking science and geography (35 students): 80 - 35 = 45
Number of students taking geography and art (33 students): 70 - 33 = 37
Number of students taking science and art (31 students): 80 - 31 = 49
Again, this count includes some students who are triple-counted because they are having all three courses. So we need to subtract the students who are taking all three courses.
Number of students taking all three courses (15 students): 45 - 15 = 30
Now, we can find the total number of students taking at least one of these courses:
80 + 70 + 60 - 45 - 37 - 49 + 30 = 109
Finally, to find the number of students who have none of these courses, we subtract the total number of students taking at least one of these courses from the total number of students.
Total number of students - Number of students taking at least one of these courses = 200 - 109 = 91
Therefore, there are 91 students who have none of these courses.