8) At a certain school, each of the 150 students takes between 1 and 3 classes. The 3 classes available are Math, Chemistry, and English. 53 students study math, 88 study chemistry, and 58 study English. If 6 students take all 3 classes, how many take exactly 2 classes?
A) 37
B) 43
C) 45
D) 60
E) 70
I am getting the answer as 55. Not among the options while the answer is 37 can someone tell me how.
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aditiniyer
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Here is a useful formula for 3 overlapping groups:aditiniyer wrote:8) At a certain school, each of the 150 students takes between 1 and 3 classes. The 3 classes available are Math, Chemistry, and English. 53 students study math, 88 study chemistry, and 58 study English. If 6 students take all 3 classes, how many take exactly 2 classes?
A) 37
B) 43
C) 45
D) 60
E) 70
T = A + B + C - (AB + AC + BC) - 2(ABC)
The big idea with overlapping group problems is to SUBTRACT THE OVERLAPS.
When we add together everyone in A, everyone in B, and everyone in C:
Those in exactly 2 of the groups (AB+AC+BC) are counted twice, so they need to be subtracted from the total ONCE.
Those in all 3 groups (ABC) are counted 3 times, so they need to be subtracted from the total TWICE.
By subtracting the overlaps, we ensure that no one is overcounted.
In the problem above:
T = 150.
Math = 53.
Chemistry = 88.
English = 58.
Let exactly 2 of the groups = x.
All 3 groups = 6.
Plugging these values into the formula, we get:
150 = 53 + 88 + 58 - x - 2*6
150 = 187 - x
x = 37.
The correct answer is A.
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Jay@ManhattanReview
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We are interested in finding out the number of students studying: "Math & Chem but not Eng" + "Math & Eng but not Chem" + "Chem & Eng but not Math"aditiniyer wrote:8) At a certain school, each of the 150 students takes between 1 and 3 classes. The 3 classes available are Math, Chemistry, and English. 53 students study math, 88 study chemistry, and 58 study English. If 6 students take all 3 classes, how many take exactly 2 classes?
A) 37
B) 43
C) 45
D) 60
E) 70
I am getting the answer as 55. Not among the options while the answer is 37 can someone tell me how.
Total number of students = #Chem + #Eng + #Math - (# of students studying exactly two subjects) - 2*(# of students studying all the three subjects)
I subtracted "(# of students studying exactly two subjects)" once since each of "Math & Chem but not Eng," "Math & Eng but not Chem," and "Chem & Eng but not Math" were counted twice in the total number of students.
But I subtracted "(# of students studying all the three subjects)" twice since "Math & Chem & Eng" were counted thrice in the total number of students.
Thus,
Total number of students = #Chem + #Eng + #Math - (# of students studying exactly two subjects) - 2*(# of students studying all the three subjects)
150 = 88 + 58 + 53 - (# of students studying exactly two subjects) - 2*6
=> # of students studying exactly two subjects = 199 - 12 -150 = 37.
Answer: A
Hope this helps!
-Jay
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Jeff@TargetTestPrep
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This is an overlapping sets question. We can use the following formula to determine how many students took exactly two classes:aditiniyer wrote:8) At a certain school, each of the 150 students takes between 1 and 3 classes. The 3 classes available are Math, Chemistry, and English. 53 students study math, 88 study chemistry, and 58 study English. If 6 students take all 3 classes, how many take exactly 2 classes?
A) 37
B) 43
C) 45
D) 60
E) 70
Total students = # Math + # Chemistry + # English - # Exactly two classes - 2(# All three classes) + # None
150 = 53 + 88 + 58 - T - 2(6) + 0
150 = 199 - T - 12
150 = 187 - T
T = 37
Answer: A
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crackverbal
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Hi Aditiniyer,
Another way of looking at this question using Venn diagram, without using any Formulae,
If you have a clear understanding of Venn-diagram then formula not necessary
So let's draw the Venn diagram,
Let "a" be the number of people who opted both Math and chemistry,
"b" be the number of people who opted both Math and English,
"c" be the number of people who opted both chemistry and English,
Given,
Neither is zero because people chose between 1 to 3 subjects.
And the number of people who chose all three subject is 6.
So,
We have to find (a+b+c),
From the diagram,
150 = 53-a-6-b + 88-a-6-c+58-b-6-c + a+b+c+6
Solving this we get,
150 = 187 -(a+b+c)
So a+b+c = 37.
So the answer is A.
Hope it is clear.
Another way of looking at this question using Venn diagram, without using any Formulae,
If you have a clear understanding of Venn-diagram then formula not necessary
So let's draw the Venn diagram,
Let "a" be the number of people who opted both Math and chemistry,
"b" be the number of people who opted both Math and English,
"c" be the number of people who opted both chemistry and English,
Given,
Neither is zero because people chose between 1 to 3 subjects.
And the number of people who chose all three subject is 6.
So,
We have to find (a+b+c),
From the diagram,
150 = 53-a-6-b + 88-a-6-c+58-b-6-c + a+b+c+6
Solving this we get,
150 = 187 -(a+b+c)
So a+b+c = 37.
So the answer is A.
Hope it is clear.
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