There are 100 freshmen at a particular college, all of whom

Question

There are 100 freshmen at a particular college, all of whom must take at least one of the three core classes: Art, Biology, and Calculus. Of these freshmen, 17 take only Biology, 10 take only Calculus, 5 take all three classes, and 20 take Art and exactly one of the other two core classes. If the number of freshmen who take only Art is 3 times the number of freshmen who take every core class except Art, how many freshmen take Art?

(A) 25

(B) 32

(C) 36

(D) 48

(E) 61

(A) 25
(B) 32
(C) 36
(D) 48 
(E) 61

SergeyOrshanskiy · Accepted Answer

During an actual exam I would not even introduce any letters.

100 - 17 - 10 - 5 - 20 = 48 students take either only Art or only Biology and Calculus.

Dividing 48 in proportion 3:1, we conclude that 48*3/4=36 students take only Art.

Thus, we get 36 + 20 + 5 = 61.
(Only Art, Art and one course, Art and two courses, respectively.)

KarishmaB · Answer

Make a venn diagram to get a clear picture. Look at the diagram: Each letter represents only one color. b represents the people who take only Art. d represents people who take only Art and Bio etc. Ques3.jpg d + f = 20 (People who take Art and one other class) b = 3e (people who take only Art is 3 times the people who take Bio and Calculus) 17 + 10 + 5 + b + d + e + f = 100 (Total people) b + b/3 = 48 b = 36 Number of freshmen who take Art = 36 + 20 + 5 = 61

holidayhero · Answer

Hello Karishma, very nice job with this question. 
I solved by using the formula below and got the same answer.

Total = (# in A + # in B + # in C) - (# enrolled in 2 courses) - 2(# enrolled in 3 courses) + (# in 0 courses)

Because of all the variables, solving the problem using the formula took me too much time. Your approach is far better! Could you describe a situation when you would be required to use the formula above or will the method you used always be appropriate?

Thanks

KarishmaB · Answer

I use venn diagrams for most sets questions. It's very easy to see the relation between what is given and what is asked when you see it in a venn diagram. The process becomes completely mechanical and quick. There are various ways to represent the formulas in sets and that can get a little messy hence I avoid them.

taleesh · Answer

Hey Karishma ,

Could you please explain ,where from b/3 is derive?

KarishmaB · Answer

The question says:
"If the number of freshmen who take only Art is 3 times the number of freshmen who take every core class except Art"

Freshmen who take every class except art are the freshmen who take Bio and Calculus only. 
So b = 3e
or e = b/3

Now, Total = 100 = 17 + 10 + 5 + b + d + e + f
100 - 32 = b + (d + f) + e
68 = b + 20 + b/3

(Note that d + f = 20 and e = b/3)

You get b = 36

SriramK · Answer

Very good Question..

Another Approach:-

3k(k- freshmen who did not opt for Arts at all) + 20(arts+one other) +5 (freshmen who took all) => 3k+25....

Only 61 among all other options is in the form of 3k+25....

Michaelkalend13 · Answer

Can someone please provide a more easier approach with detailed explanation? 
Thank you!

Michaelkalend13 · Answer

Hi Sergey! 
How did you derive this one "Dividing 48 in proportion 3:1, we conclude that 48*3/4=36 students take only Art."

Please clarify.

Bunuel · Answer

My friend, you can find detailed solutions  HERE  and  HERE . You can also find link to the theory  HERE . If the question is still unclear please ask more specific question on it.

In addition please check the links below:

Overlapping Sets Made Easy!   
   ADVANCED OVERLAPPING SETS PROBLEMS   
   How to draw a Venn Diagram  
   Data Sufficiency Questions on Overlapping Sets  
   Problem Solving Questions on Overlapping Sets

Hope it helps.

ColumbiaEMBA · Answer

Great question where the advanced overlapping set formula from the GMAT Club math book becomes helpful.

When a question has exactly 2 sets in it, we can calculate total as:

Total = A + B + C - SUM(Exactly 2 overlapping sets) - 2*(all three) + neither

Here:

Total = 100
A = 3*(BC, let's refer to BC as x from hereon) 
B = 17
C = 10
All 3 = 5
A and (B or C) = 20 (falls under exactly 2 as it's only A + B or A + C
Neither = 0 per stem

Therefore: 100 = 3x + 17 + 10 - sum(x + 20) - 2*5 + 0 --> 100 = 2x-3 --> x = 103/2 --> x = 61. Either I'm doing something wrong or the question has an error, but 61 is closest to 61.5.

Kinshook · Answer

There are 100 freshmen at a particular college, all of whom must take at least one of the three core classes: Art, Biology, and Calculus. Of these freshmen, 17 take only Biology, 10 take only Calculus, 5 take all three classes, and 20 take Art and exactly one of the other two core classes.

If the number of freshmen who take only Art is 3 times the number of freshmen who take every core class except Art, how many freshmen take Art?

Attachment:

Screenshot 2025-06-19 at 10.05.13 AM.png [ 46.78 KiB | Viewed 978 times ]

The number of freshmen who take only Art is 3 times the number of freshmen who take every core class except Art
y = 3z

There are 100 freshmen at a particular college
20-x+x+5+17+10+y+z = 100
52+4z = 100
z = 12

The number of freshmen who take art = 3z + 25 = 61

IMO E

There are 100 freshmen at a particular college, all of whom

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There are 100 freshmen at a particular college, all of whom

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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