Question 1153479: A group of 2457 students were surveyed about the courses they were taking at their college with the following results:
1075 students said they were taking English.
1249 students said they were taking Science.
1108 students said they were taking Psychology.
473 students said they were taking Psychology and English.
534 students said they were taking Science and Psychology.
594 students said they were taking Science and English.
251 students said they were taking all three courses.
a) Fill in the following Venn Diagram with the cardinality of each region.
a) How many students took Science & Psychology or took Psychology & English?
b) How many students took Psychology and English, but not Science?
c) How many students took Science or didn't take Psychology?
d) How many students took Science, Psychology, or English?
e) How many students took Science or English, but not Psychology?
f) How many students took none of the courses?
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Given Facts
- Fact 1: 2457 students total.
- Fact 2: 1075 students taking English.
- Fact 3: 1249 students taking Science.
- Fact 4: 1108 students taking Psychology.
- Fact 5: 473 students taking Psychology and English.
- Fact 6: 534 students taking Science and Psychology.
- Fact 7: 594 students taking Science and English.
- Fact 8: 251 students taking all three courses.
Draw 3 overlapping circles inside of a rectangle. The circles in the diagram below are labeled E, S and P for English, Science, and Psychology respectively. The set U, shown as the surrounding rectangle, is the universal set to represent everyone surveyed. The circles are always inside set U because this is the set of everything we care about for the problem.
We have 8 distinct regions as shown with the labels (a) through (h). They are defined as such:
- Region (a) = people taking english only
- Region (b) = people taking english and science only (not taking psychology)
- Region (c) = people taking science only
- Region (d) = people taking english and psychology only (not taking science)
- Region (e) = people taking all three courses (english, science, psychology)
- Region (f) = people taking science and psychology only (not taking english)
- Region (g) = people taking psychology only
- Region (h) = people not in any of the classes mentioned (ie not in english, not in science, and not in psychology)
Use facts 5,6,7 and 8 to find that...
594 in set E and S (Fact 7), 251 in all three sets(Fact 8), 594-251 = 343 in set E and S only. Write "343" in region (b)
473 in set E and P (Fact 5), 251 in all three sets (Fact 8), 473-251 = 222 in set E and P only. Write "222" in region (d)
534 in set S and P (Fact 6), 251 in all three sets(Fact 8), 534-251 = 283 in set S and P only. Write "283" in region (f)
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Here's what we have so far
Add up the values in circle E
343+222+251 = 816
There are 816 students who are taking English in addition to one or two other classes. There are 1075 students taking English (Fact 2), so that means 1075 - 816 = 259 students are taking english only (they aren't taking science, they aren't taking psychology). Write the value 259 in region (a).
Repeat for circle S
Add up the values in circle S
343+283+251 = 877
There are 877 students taking science along with one or two other classes.
There are 1249 students in set S (Fact 3)
So we have 1249-877 = 372 students taking science only (not taking english, not taking psychology). Write this value in region (c).
Repeat for circle P
Add up values in circle P
222+283+251 = 756
There are 756 students taking psychology along with one or two other classes.
1108 in set P (Fact 4)
1108-756 = 352 students taking psychology only (not taking english, not taking science). Write this value in region (g).
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You should have the following numbers for the regions (a) through (g)
Region (a) = 259
Region (b) = 343
Region (c) = 372
Region (d) = 222
Region (e) = 251
Region (f) = 283
Region (g) = 352
as shown below
Add up those values:
259 + 343 + 372 + 222 + 251 + 283 + 352 = 2082
We have 2082 students who are taking at least one course of english, science, or psychology.
Subtract this from the total number of students surveyed (2457 from Fact 1) to get
2457 - 2082 = 375
There are 375 students who take neither of the three courses mentioned. This value goes in region (h)
Here is the fully completed Venn Diagram
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Now we can address the questions
a) How many students took Science & Psychology or took Psychology & English?
Answer: 756
Explanation: Add up the values in regions (d), (e), and (f). This represents the "Psychology & English" group (they didn't take Science), the group that took all three courses, and the "Science & Psychology" group (they didn't take English).
b) How many students took Psychology and English, but not Science?
Answer: 222
Explanation: This is region (d) which is the region in circles P and E, but outside circle S.
c) How many students took Science or didn't take Psychology?
Answer: 1883
Explanation: Add up the regions (a), (b), (c), (e), (f), and (h). Alternatively, add up regions (d) and (g), then subtract from 2457. Regions (a), (b), (c), (e), (f), and (h) represent those who either took Science ( region (b), (c), (e), (f) ) or those who did not take Psychology (region (a), (b), (c), (h) )
d) How many students took Science, Psychology, or English?
Answer: 2082
Explanation: Subtract the value in region (h) from the total number of people surveyed (2457) to get 2457-375 = 2082. This value was found earlier by adding up regions (a) through (g).
e) How many students took Science or English, but not Psychology?
Answer: 974
Explanation: The answer is found by adding up regions (a) through (c). All of which are either in circle E or circle S (or both), but not inside circle P.
f) How many students took none of the courses?
Answer: 375
Explanation: This is the value in region (h)
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