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655-705 (Hard)|   Overlapping Sets|         
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12 Days of Christmas GMAT Competition 2025 - Day 4: At a certain 18 Dec 2025, 09:00
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At a certain university, each of two advanced courses, Biology and Chemistry, has more than 25 students enrolled. Is there at least one student enrolled in both courses?

(1) Every student who is not enrolled in Biology is also not enrolled in Chemistry.

(2) Every student enrolled in Chemistry is also enrolled in Biology.

I31-08

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D
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12 Days of Christmas GMAT Competition 2025 - Day 4: At a certain 18 Dec 2025, 10:00
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GMAT Club Official Explanation:



At a certain university, each of two advanced courses, Biology and Chemistry, has more than 25 students enrolled. Is there at least one student enrolled in both courses?

(1) Every student who is not enrolled in Biology is also not enrolled in Chemistry.

This means anyone in Chemistry must also be in Biology because if a student were in Chemistry but not in Biology, then that student would be a counterexample to the statement. Since such a student cannot exist, all students in Chemistry must also be in Biology. So every Chemistry student is also in Biology. Since Chemistry has more than 25 students, at least those 25+ students are definitely in both courses. Sufficient.

(2) Every student enrolled in Chemistry is also enrolled in Biology.

This says the same thing: anyone in Chemistry must also be in Biology because if someone belonged to Chemistry without belonging to Biology, the condition would be violated. Therefore every Chemistry student must already be a Biology student. So again, every Chemistry student is in both courses. Since Chemistry has more than 25 students, there are definitely students in both. Sufficient.

Answer: D.
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12 Days of Christmas GMAT Competition 2025 - Day 4: At a certain 18 Dec 2025, 09:21
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I. If a student is in Chemistry, then they must be in Biology. Since Chemistry has more than 25 students, all of them are also in Biology, so there is overlap.
Sufficient

II. Every Chemistry student is enrolled in Biology. Chemistry has more than 25 students, so those students are also in Biology. Overlap exists.
Sufficient

Answer: D
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12 Days of Christmas GMAT Competition 2025 - Day 4: At a certain 18 Dec 2025, 09:39
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Total atleast 25 enrolled in each
(1) => Whoever is not in Biology is not in Chemistry also, meaning if somebody is in chemistry then they are also in biology
(2) => Every student in chemistry is also in biology. chemistry is a subset of biology.
Both alone sufficient.
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12 Days of Christmas GMAT Competition 2025 - Day 4: At a certain 18 Dec 2025, 09:41
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Number of studentsBiology~ BiologyTotal
Chemistryx>0?>25
~ Chemistry
Total>25

(1) Every student who is not enrolled in Biology is also not enrolled in Chemistry

Number of studentsBiology~BiologyTotal
Chemistryx>250>25
~ Chemistry
Total>25

Since there is no student who is enrolled in Chemistry but not in Biology.
Number of students enrolled in both Chemistry & Biology > 25.
Sufficient

(2) Every student enrolled in Chemistry is also enrolled in Biology.

Number of studentsBiology~ BiologyTotal
Chemistryx>250>25
~ Chemistry
Total>25

Since there is no student who is enrolled in Chemistry but not in Biology
Number of students enrolled in both Chemistry & Biology > 25
Sufficient

IMO D
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Let x be overlapping people for Bio and chem, is x>=1??
b>=25
c>=25
1)
not in b is also not in c, this does not say anything about people enrolled in c and b
we just get not enrolled outside the the overlapping set value as not equal to zero.
Not sufficient
2)we know there are more than 25 students in bio and chem ;if enrolled in chem also means enrolled in bio we can say there is at least 1 enrolled in both suffient.
so B
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12 Days of Christmas GMAT Competition 2025 - Day 4: At a certain 18 Dec 2025, 09:54
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At a certain university, each of two advanced courses, Biology and Chemistry, has more than 25 students enrolled. Is there at least one student enrolled in both courses?
(1) Every student who is not enrolled in Biology is also not enrolled in Chemistry.

(2) Every student enrolled in Chemistry is also enrolled in Biology.

two advanced courses B & C > 25 students enrolled...

target is there >=1 student in both courses


---B---NB---
C--- --- ---
NC-- --- ---

#1
every student who is not enrolled in B is also not enrolled in Chemistry
this means there will be an overlap of students and there must be at least student in both
sufficient
#2
Every student in C is also enrolled in B
the statement is self sufficient to say that students enrolled in both courses
sufficient

OPTION D is correct
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12 Days of Christmas GMAT Competition 2025 - Day 4: At a certain 18 Dec 2025, 09:56
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(1) If every student NOT in B is also NOT in C, there must be students taking both courses, since we know neither B nor C are empty

(2) If every student in C is also in B, and C is not empty, there bust be people taking both courses (there must be more than 25)
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12 Days of Christmas GMAT Competition 2025 - Day 4: At a certain 18 Dec 2025, 10:03
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(1) every student who is not enrolled in biology is also not enrolled in chemistry.
This is a contrapositive statement of, “every student who is enrolled in chemistry is also enrolled in biology.”
Sufficient
(2) every student enrolled in chemistry is also enrolled in biology. This directly means atleast 1 student is enrolled in both courses.
Sufficient

D
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Let Biology, Chemistry and both be B,C and b

Is b>= 1?

S1
Ok, so if total is say T = 100
And B = 60
Then 40 are not enrolled in B and C both. That means C overlaps with B
This suggests b>= 1
Sufficient

S2
This means C = b, and since C> = 25, b >= 25
Sufficient

Answer D

Bunuel
At a certain university, each of two advanced courses, Biology and Chemistry, has more than 25 students enrolled. Is there at least one student enrolled in both courses?

(1) Every student who is not enrolled in Biology is also not enrolled in Chemistry.

(2) Every student enrolled in Chemistry is also enrolled in Biology.

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12 Days of Christmas GMAT Competition 2025 - Day 4: At a certain 18 Dec 2025, 10:09
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QUESTION ASKS THAT IS THERE ATLEAST ANY STUDNT ENROLLED IN BOTH? YES/NO

STATEMENT 1 SUGGESNT THAT NO ONE WHO IS NOT IN BIO IS ALSONOT IN CHEM
MEANING THTA THERE IS NO SOLE BIO STUDENT EXISTS WHICH ASNWERS THE QUES

STATEMENT2 SAYS ALL CHEM STUDENT IS ALSO BIO STUDENT WHICH ASNWERS THE QUES

HENCE BOTHE ARE GREAT INDIVIDUALLY
Bunuel
At a certain university, each of two advanced courses, Biology and Chemistry, has more than 25 students enrolled. Is there at least one student enrolled in both courses?

(1) Every student who is not enrolled in Biology is also not enrolled in Chemistry.

(2) Every student enrolled in Chemistry is also enrolled in Biology.

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Bunuel
At a certain university, each of two advanced courses, Biology and Chemistry, has more than 25 students enrolled. Is there at least one student enrolled in both courses?

(1) Every student who is not enrolled in Biology is also not enrolled in Chemistry.

(2) Every student enrolled in Chemistry is also enrolled in Biology.

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Given:
• Biology has more than 25 students
• Chemistry has more than 25 students
• Is at least one student enrolled in both?

STATEMENT 1: If we flip: If a student is enrolled in Chemistry, then the student must be enrolled in Biology.
Since Chemistry has more than 25 students, all of whom are also in Biology, there must be at least one student in both. - SUFFICIENT.
STATEMENT 2: This is the same condition we derived above. - SUFFICIENT

Hence, OPTION D.
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12 Days of Christmas GMAT Competition 2025 - Day 4: At a certain 18 Dec 2025, 10:33
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Bunuel
At a certain university, each of two advanced courses, Biology and Chemistry, has more than 25 students enrolled. Is there at least one student enrolled in both courses?

(1) Every student who is not enrolled in Biology is also not enrolled in Chemistry.

(2) Every student enrolled in Chemistry is also enrolled in Biology.

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(1) SUFFICIENT: "Not bio -> Not chem" means that chem is a subset of bio. Since chem > 25, both > 25. This means that both will intersect

(2) SUFFICIENT: "Chem -> Bio" means that chem is a subset of bio. Thus both will intersect and is sufficient

D: Each statement ALONE is sufficient
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Statement 1 alone is sufficient and statement 2 alone also is sufficient.
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From statement 1, we can infer that only chemistry portion is zero. Therefore, if more than 25 students study chemistry, then we can say that more than one study both the subjects. Sufficient

From statement 2, it also conveys that the only chemistry portion is zero. Similar to statement 2, we can answer the question as yes. Sufficient

Therefore, Option D
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1) We know that if someone is not enrolled in Biology, neither in Chemistry. We know there's someone enrolled in Chemistry and also there's people enrolled in Biology, but doesn't give us any information of the people enrolled, only about the not enrolled people. NOT SUFFICIENT

2) We know anyone enrolled in Chemistry is also enrolled in Biology, As we know there's at least 25 students enrolled in Chemistry and at least 25 enrolled in Biology, there's no class empty, therefore, there's people enrolled in both courses. SUFFICIENT
Bunuel
At a certain university, each of two advanced courses, Biology and Chemistry, has more than 25 students enrolled. Is there at least one student enrolled in both courses?

(1) Every student who is not enrolled in Biology is also not enrolled in Chemistry.

(2) Every student enrolled in Chemistry is also enrolled in Biology.

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Bunuel
At a certain university, each of two advanced courses, Biology and Chemistry, has more than 25 students enrolled. Is there at least one student enrolled in both courses?

(1) Every student who is not enrolled in Biology is also not enrolled in Chemistry.

(2) Every student enrolled in Chemistry is also enrolled in Biology.

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1. Not in Biology = Not in Chemistry, (But not in chemistry is not necessarily equal to not in biology) i.e. Chemistry is subset of Biology.
2. All in chemistry = All in Bio, i.e Chemistry is subset of biology.

Clearly each statement is self sufficient.
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12 Days of Christmas GMAT Competition 2025 - Day 4: At a certain 18 Dec 2025, 11:27
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Bunuel
At a certain university, each of two advanced courses, Biology and Chemistry, has more than 25 students enrolled. Is there at least one student enrolled in both courses?

(1) Every student who is not enrolled in Biology is also not enrolled in Chemistry.

(2) Every student enrolled in Chemistry is also enrolled in Biology.

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From the stem we know that Biology > 25 students, Chemistry > 25 students, and that this is a yes or no question about whether at least 1 student is in either class.

Statement 1: Has a double negative, so let's just get rid of both mentions of "not" in the sentence and see what it says.

Every student who is enrolled in biology is also enrolled in chemistry.

We know each class has at least 25 students, every student in one class is enrolled in the other, so there's definitely more than 1 student enrolled in both classes. Sufficient.

Statement 2: This states the same thing as Statement 1 once we removed the double negative, so the same logic applies to this. Sufficient.

Therefore answer is D.
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each bio & chem has 25 students each
1. if 25 students are in chem , then they must have opted for bio. so YES, SUFFICIENT
2. if every student in chem is also in bio, then YES...SUFFICIENT

Ans D
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Bunuel
At a certain university, each of two advanced courses, Biology and Chemistry, has more than 25 students enrolled. Is there at least one student enrolled in both courses?

(1) Every student who is not enrolled in Biology is also not enrolled in Chemistry.

(2) Every student enrolled in Chemistry is also enrolled in Biology.

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Let’s assume a two set Venn diagram.

Where a = biology only.

c = chemistry only,

b = both Biology and Chemistry.

d = neither.

Biology = (a+b) , Chemistry = (c+d) .

Given that : (a+b) >25 and (c+d) >25.

Statement 1:

Every student who is not enrolled in Biology is also not enrolled in Chemistry.

So, (c+d) = d

c = 0.

We know that: (b+c) >25

If c = 0, then b > 25.

Sufficient.

Statement 2:

Every student enrolled in chemistry is also enrolled in biology.

c+b = b

c=0

b+c > 25, with c=0

b>25

Sufficient.

Option D
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