CAT Venn Diagrams Questions & Solutions

A sample of real CAT Venn Diagrams past-year questions, each with a worked solution and the correct answer highlighted. AthenaPrep has 26 Venn Diagrams questions in all — sign up free to practise them timed.

1. Q1.CAT 2025

   In a class of 150 students, 75 students chose physics, 111 students chose mathematics and 40 students chose chemistry. All students chose at least one of the three subjects and at least one student chose all three subjects. The number of students who chose both physics and chemistry is equal to the number of students who chose both chemistry and mathematics, and this is half the number of students who chose both physics and mathematics. The maximum possible number of students who chose physics but not mathematics, is

   • 30

   • 35

   • 40

   • 55

   Show solution

   Based on the information provided, we can make the following diagram;

   

   We know that the sum of all entries in the Venn diagram should be $150$ . Therefore,

   $111+ 75-2k + 40 - (k-x) - k + 0 = 150$

   $76 = 4k-x$ , where $x\geq 1$

   We must maximise the number of students who chose Physics but not Mathematics. This number is equal to $75-2k$ . To maximise this we must minimise $k$ . 

   We have $4k = 76+x$ or $k = 19 + \dfrac{x}{4}$

   Since $x$ and $k$ are both integers, $x$ should be divisible by $4$ . To minimise $k$ , we will take the minimum value of $x$ , which is $4$ .

   Therefore, $k = 19 + \dfrac{4}{4} = 20$ .

   Thus, the maximum possible number of students who chose physics but not mathematics, is $75-2*20 = 35$ .

2. Q2.CAT 2020

   Students in a college have to choose at least two subjects from chemistry, mathematics and physics. The number of students choosing all three subjects is 18, choosing mathematics as one of their subjects is 23 and choosing physics as one of their subjects is 25. The smallest possible number of students who could choose chemistry as one of their subjects is

   • 22

   • 21

   • 20

   • 19

   Show solution

   Now 23 students choose maths as one of their subject.

   This means (MPC)+ (MC) + (PC)=23 where MPC denotes students who choose all the three subjects maths, physics and chemistry and so on.

   So MC + PM =5 Similarly we have PC+ MP =7 

   We have to find the smallest number of students choosing chemistry

   For that in the first equation let PM=5 and MC=0. In the second equation this PC=2

   Hence minimum number of students choosing chemistry will be (18+2)=20 Since 18 students chose all the three subjects.

3. Q3.CAT 2018

   Each of 74 students in a class studies at least one of the three subjects H, E and P. Ten students study all three subjects, while twenty study H and E, but not P. Every student who studies P also studies H or E or both. If the number of students studying H equals that studying E, then the number of students studying H is

   Answer: 52

   Show solution

   Let us draw a Venn diagram using the information present in the question. 

   

   It is given that the number of students studying H equals that studying E. 

   Let 'x' be the total number of students who studied H, and H and P but mot E.We can also say that the same will be the number of students who studied E, and E and P but not H.Therefore,

   x + 20 + 10 + x = 74 

   x = 22

   Hence, the number of students studying H = 22 + 10+ 20 = 52

4. Q4.CAT 2018

   For two sets A and B, let AΔB denote the set of elements which belong to A or B but not both. If P = {1,2,3,4}, Q = {2,3,5,6}, R = {1,3,7,8,9}, S = {2,4,9,10}, then the number of elements in (PΔQ)Δ(RΔS) is

   Answer: 7

   Show solution

    P = {1,2,3,4} and  Q = {2,3,5,6,}
   PΔQ = {1, 4, 5, 6}
   R = {1,3,7,8,9} and S = {2,4,9,10}
   RΔS = {1, 2, 3, 4, 7, 8, 10}
   (PΔQ)Δ(RΔS) = {2, 3, 5, 6, 7, 8, 10}
   Thus, there are 7 elements in (PΔQ)Δ(RΔS) .
   hence, 7 is the correct answer.

5. Q5.CAT 2006

   A survey was conducted of 100 people to find out whether they had read recent issues of Golmal, a monthly magazine. The summarized information regarding readership in 3 months is given below:

   Only September: 18;

   September but not August: 23;

   September and July: 8;

   September:28;

   July: 48;

   July and August: 10;

   none of the three months: 24

   What is the number of surveyed people who have read exactly two consecutive issues (out of the three)?

   • 7

   • 9

   • 12

   • 14

   • 17

   Show solution

   

   Let the areas be labelled as shown in the diagram above.

   The number of people corresponding to "none of the three months" is 24. So, H is 24.

   Only September is 18. So, G = 18
   

   September but not August is 23. So, G + D = 23.

   Hence, D = 23 - 18 = 5.

   We know that September and July is 8. So, D + E = 8

   This implies E = 3.

   September = 28. So, D + E + F + G = 28.

   So, F = 28 - 5 - 3 - 18 = 2.
   

   July and August = 10.

   So, B + E = 10.

   E = 3. So, B = 7.

   July = 48.

   So, A + B + D + E = 48

   A = 48 - 7 - 5 - 3 = 33.

   There are 100 people in total. So, C = 100 - A - B - D - E - F - G - H = 100 - 33 - 7 - 5 - 3 - 2 - 18 - 24 = 8
   

   So, number of people who read exactly two consecutive issues = (July & August) + (August & September) = B + F = 7 + 2 = 9

6. Passage

   Help Distress (HD) is an NGO involved in providing assistance to people suffering from natural disasters. Currently, it has 37 volunteers. They are involved in three projects: Tsunami Relief (TR) in Tamil Nadu, Flood Relief (FR) in Maharashtra, and Earthquake Relief (ER) in Gujarat. Each volunteer working with Help Distress has to be involved in at least one relief work project.

   • A Maximum number of volunteers are involved in the FR project. Among them, the number of volunteers involved in FR project alone is equal to the volunteers having additional involvement in the ER project.

   • The number of volunteers involved in the ER project alone is double the number of volunteers involved in all the three projects.

   • 17 volunteers are involved in the TR project.

   • The number of volunteers involved in the TR project alone is one less than the number ofvolunteers involved in ER project alone.

   • Ten volunteers involved in the TR project are also involved in at least one more project.

   Q6.CAT 2005

   Which of the following additional information would enable to find the exact number of volunteers involved in various projects?

   • Twenty volunteers are involved in FR.

   • Four volunteers are involved in all the three projects.

   • Twenty three volunteers are involved in exactly one project.

   • No need for any additional information.

   Show solution

   

   We can get the information mentioned in options B and C using the data given in the passage.

   But, we need the information in option A to find the exact number of volunteers in various projects.

   Hence, option A is the answer.

7. Passage

   DIRECTIONS for the following two questions: Answer the questions on the basis of the information given below.

   New Age Consultants have three consultants Gyani, Medha and Buddhi. The sum of the number of projects handled by Gyani and Buddhi individually is equal to the number of projects in which Medha is involved. All three consultants are involved together in 6 projects. Gyani works with Medha in 14 projects. Buddhi has 2 projects with Medha but without Gyani, and 3 projects with Gyani but without Medha. The total number of projects for New Age Consultants is one less than twice the number of projects in which more than one consultant is involved.

   Q7.CAT 2003

   What is the number of projects in which Gyani alone is involved?

   • Uniquely equal to zero.

   • Uniquely equal to 1.

   • Uniquely equal to 4.

   • Cannot be determined uniquely.

   Show solution

   
   The total number of projects = 2(3+6+8+2) - 1 = 38 - 1 = 37
   So, 19 + 2(x+y) - 16 = 37
   => x+y = 17
   The number of projects in which Medha alone is involved is 17-16 = 1
   But the number of projects in which Gyani alone is involved cannot be uniquely determined
8. Q8.CAT 2002

   Shyam visited Ram during his brief vacation. In the mornings they both would go for yoga. In the evenings they would play tennis. To have more fun, they indulge only in one activity per day, i.e. either they went for yoga or played tennis each day. There were days when they were lazy and stayed home all day long. There were 24 mornings when they did nothing, 14 evenings when they stayed at home, and a total of 22 days when they did yoga or played tennis. For how many days Shyam stayed with Ram?

   • 32

   • 24

   • 30

   • None of these

   Show solution

   Let the number of total days=N
   They played tennis for=N-14  days
   They did yoga for =N-24 days

   

   And the question says that total days when they did yoga or played tennis are 22

   which means

   N-14 + N-24 = 22

   2N - 38 = 22

   2N = 60

   N = 30

   Hence total days they stayed together were 30

9. Q9.CAT 1999

   In a survey of political preference, 78% of those asked were in favor of at least one of the proposals: I, II and III. 50% of those asked favored proposal I, 30% favored proposal II, and 20% favored proposal III. If 5% of those asked favored all three of the proposals, what percentage of those asked favored more than one of the 3 proposals.

   • 10

   • 12

   • 17

   • 22

   Show solution

   Let the distribution of votes for each of the proposal be as given below.
   From the information given, we know that 
   
   a+b+c+d+e+f+g = 78 --- (1)
   a+b+e+f = 50  ---- (2)
   b+c+f+g = 30 ----  (3)
   e+f+g+d = 20 ---- (4) and 
   f = 5 --- (5)
   
   We need to find b+e+g+f = ?
   In the above equations, (2)+(3)+(4) - (1) implies
   
   (a+b+e+f)+(b+c+f+g)+(e+f+g+d) - (a+b+c+d+e+f+g) = 50+30+20-78 = 22
   Or, b+e+g+2f=22.
   As, f = 5, it implies that b+e+g+f=17

    

   

10. Passage

    A survey of 200 people in a community who watched at least one of the three channels — BBC, CNN and DD — showed that 80% of the people watched DD, 22% watched BBC, and 15% watched CNN.
    Q10.CAT 1997

    If 5% of people watched DD and CNN, 10% watched DD and BBC, then what percentage of people watched BBC and CNN only?

    • 2%

    • 5%

    • 8.5%

    • Cannot be determined

    Show solution

    Applying AUBUC formula

    Let x be the number who watch BBC and CNN and y be the number who watch all three channels.

    100 = 80+22+15-(10+5+x)+y

    x-y = 2

    Hence only 2% people watch BBC and CNN only.

11. Q11.CAT 1996

    In a locality, two-thirds of the people have cable TV, one-fifth have VCR, and one-tenth have both. What is the fraction of people having atleast one among cable -TV and VCR?

    • $\frac{19}{30}$

    • $\frac{2}{3}$

    • $\frac{17}{30}$

    • $\frac{23}{30}$

    Show solution

    

    Let the distribution of people having cable TV and VCR be as given in the diagram above.
    Hence, $a+c =\frac{2}{3}$
    $b+c=\frac{1}{5}$
    and $c=\frac{1}{10}$

    We need to find $a+b+c=?$
    This equals $(a+b)+(b+c)-c = \frac{2}{3}+\frac{1}{5}-\frac{1}{10}$
    Which equals $\frac{20+6-3}{30}=\frac{23}{30}$

12. Q12.CAT 1991

    There are 3 clubs A, B & C in a town with 40, 50 & 60 members respectively. While 10 people are members of all 3 clubs, 70 are members in only one club. How many belong to exactly two clubs?

    • 20

    • 25

    • 50

    • 70

    Show solution

    We know that x + y + z = T and x + 2y + 3z = R, where
    x = number of members belonging to exactly 1 set = 70
    y = number of members belonging to exactly 2 sets
    z = number of members belonging to exactly 3 sets = 10
    T = Total number of members
    R = Repeated total of all the members = (40+50+60) = 150
    Thus we have two equations and two unknowns. Solving this we get y = 25

    So, 25 people belong to exactly 2 clubs.

CAT Venn Diagrams previous year questions with solutions — AthenaPrep