CAT Venn Diagrams Questions

Master CAT Venn Diagrams Questions with practice questions and detailed solutions.

Question 1.

In a class of 100 students, 73 like coffee, 80 like tea and 52 like lemonade. It may be possible that some students do not like any of these three drinks. Then the difference between the maximum and minimum possible number of students who like all the three drinks is

Text Explanation:

Let n, s, d and t be the number of students who likes none of the drinks, exactly one drink, exactly 2 drinks and all three drinks, respectively.

It is given,

n + s + d + t = 100 ...... (1)

s + 2d + 3t = 73 + 80 + 52

s + 2d + 3t = 205 ...... (2)

(2)-(1), we get

d + 2t - n = 105

Maximum value t can take is 52, i.e. t = 52, d = 1 and n = 0

Minimum value t can take is 5, i.e. t = 5, d = 95 and n = 0 (This also satisfies equation (1))

Difference = 52 - 5 = 47

The answer is option C.

Question 2.

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

Video Explanation

Text Explanation:

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.

Question 3.

A club has 256 members of whom 144 can play football, 123 can play tennis, and 132 can play cricket. Moreover, 58 members can play both football and tennis, 25 can play both cricket and tennis, while 63 can play both football and cricket. If every member can play at least one game, then the number of members who can play only tennis is

Video Explanation

Question 4.

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

Video Explanation

Text Explanation:

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 $75-2*20 = 35$.

Question 5.

Text Explanation:

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

Question 6.

Text Explanation:

It has been given that among 200 students, 105 like pizza and 134 like burger.
The question asks us to find out the number of students who can be liking only burgers among the given values.
The least number of students who like only burger will be obtained when everyone who likes pizza likes burger too.
In this case, 105 students will like pizza and burger and 134-105 = 29 students will like only burger. Therefore, the number of students who like only burger cannot be less than 29.
The maximum number of students who like only burger will be obtained when we try to separate the 2 sets as much as possible.
There are 200 students in total. 105 of them like pizza. Therefore, the remaining 95 students can like only burger and 134-95 = 39 students can like both pizza and burger. As we can see, the number of students who like burger cannot exceed 95.
The number of students who like only burger should lie between 29 and 95 (both the values are included).
93 is the only value among the given options that satisfies this condition and hence, option D is the right answer.

Question 7.

Question 8.

Text Explanation:

If we carefully observe set A, then we find that $6^{2n} -35n - 1$ is divisible by 35. So, set A contains multiples of 35. However, not all the multiples of 35 are there in set A, for different values of $n$.
For $n = 1$, the value is 0, for $n = 2$, the value is 1225 which is the 35th multiple of 3.
If we observe set B, it consists of all the multiples of 35 including 0.
So, we can say that every member of set A will be in B while every member of set B will not necessarily be in set A.
Hence, option A is the correct answer.

Question 9.

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)?

Text Explanation:

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

Instructions

Instructions
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.

Question 10.

Question 11.

Question 12.

Text Explanation:

FR is greater than ER, thus z is 4,5,6.
If z=4,
FR=20, ER= 18
If z=5,
FR=21, ER= 17
If z=6,
FR=22, ER= 16
Now even if we transfer one of the volunteers opted out of the TR project thus he will now be handling FR and ER, and one opted out of the ER project will be handling TR and FR, while the remaining ones involved in all the three projects opted out of the FR project.
The worst-case scenario will be when z=4. Now 2 are out of FR, 1 from ER and 1 from TR.

FR=20-2=18 and ER=18-1=17
So we can see that even if volunteers are withdrawn, the number of volunteers in FR are more as in, total if we calculate comes out to be more. Hence option B.

Question 13.

Question 14.

Question 15.

On her walk through the park, Hamsa collected 50 coloured leaves, all either maple or oak. She sorted them by category when she got home, and found the following:
The number of red oak leaves with spots is even and positive.
The number of red oak leaves without any spot equals the number of red maple leaves without spots.
All non-red oak leaves have spots, and there are five times as many of them as there are red spotted oak leaves.
There are no spotted maple leaves that are not red.
There are exactly 6 red spotted maple leaves.
There are exactly 22 maple leaves that are neither spotted nor red.
How many oak leaves did she collect?

Instructions

Instructions
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.

Question 16.

Question 17.

Instructions

Instructions
For the following questions answer them individually

Question 18.

Text Explanation:

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

Instructions

Instructions
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.

Question 19.

Question 20.

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CAT Venn Diagrams Questions

Question 1.

In a class of 100 students, 73 like coffee, 80 like tea and 52 like lemonade. It may be possible that some students do not like any of these three drinks. Then the difference between the maximum and minimum possible number of students who like all the three drinks is

Question 2.

Question 3.

Question 4.

Question 5.

Question 6.

<div>If among 200 students, 105 like pizza and 134 like burger, then the number of students who like only burger can possibly be</div>

Question 7.

<div>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</div>

Question 8.

<div>If A = {$6^{2n} -35n - 1$}, where $n$ = 1,2,3,... and B = {35($n$-1)}, where $n$ = 1,2,3,... then which of the following is true?</div>

Question 9.

Instructions

Question 10.

<div>Based on the information given above, the minimum number of volunteers involved in both FR and TR projects, but not in the ER project is:</div>

Question 11.

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

Question 12.

Question 13.

Question 14.

Question 15.

Instructions

Question 16.

<div>What is the number of projects in which Gyani alone is involved?</div>

Question 17.

<div>What is the number of projects in which Medha alone is involved?</div>

Instructions

Instructions For the following questions answer them individually

Question 18.

Instructions

Instructions 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.

Question 19.

<div>What is the maximum percentage of people who can watch all the three channels?</div>

Question 20.

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

Free CAT Quant Questions

Instructions
For the following questions answer them individually

Instructions
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.