venn diagram word problems with 3 circles is one of the topics in both school level math and quantitative aptitude. People who study quantitative aptitude to get prepared for competitive exams are stumbling to solve set theory word problems.

The reason for their stumbling is, they do not know the basic stuff to solve venn diagram word problems with 3 circles.

To understand, "How to solve venn diagram word problems with 3 circles?", we have to know the following basic stuff.

**U** --------> union **(or)**

n --------> intersection **(and)**

**ADDITION THEOREMS ON SETS**

**n(AUB) = n(A) + n(B) - n(AnB)**

**n(AUBUC) = n(A)+n(B)+n(C)-n(AnB)-n(BnC)-n(AnC)+n(AnBnC)**

Let us come to know about the following terms in details.

**n(AuB)** = Total number of elements related to any of the two events A & B.

**n(AuBuC)** = Total number of elements related to any of the three events A, B & C.

**n(A)** = Total number of elements related to A.

**n(B)** = Total number of elements related to B.

**n(C)** = Total number of elements related to C.

**For three events A, B & C, we have **

**n(A) - [n(AnB) + n(AnC) - n(AnBnC)]** =Total number of elements related to A only.

**n(B) - [n(AnB) + n(BnC) - n(AnBnC)]** =Total number of elements related to B only.

**n(C) - [n(BnC) + n(AnC) + n(AnBnC)]** =Total number of elements related to C only.

**n(AnB)** = Total number of elements related to both A & B

**n(AnB) - n(AnBnC)** = Total number of elements related to both (A & B) only.

**n(BnC)** = Total number of elements related to both B & C

**n(BnC) - n(AnBnC)** = Total number of elements related to both (B & C) only.

**n(AnC)** = Total number of elements related to both A & C

**n(AnC) - n(AnBnC)** = Total number of elements related to both (A & C) only.

**For two events A & B, we have **

**n(A) - n(AnB) ** = Total number of elements related to A only.

**n(B) - n(AnB) ** = Total number of elements related to B only.

**Let us consider the following example, to have better understanding of the above stuff explained using venn diagram.**

**Example:**

In a group of students, 65 play foot ball, 45 play hockey, 42 play cricket, 20 play foot ball and hockey, 25 play foot ball and cricket, 15 play hockey and cricket and 8 play all the three games.

Let F, H and C represent the set of students who play foot ball, hockey and cricket respectively.

**Venn diagram related to the above situation:**

**From the venn diagram, we can have the following details.**

No. of students who play foot ball = 65

No. of students who play foot ball only = 28

No. of students who play hockey = 45

No. of students who play hockey only = 18

No. of students who play cricket = 42

No. of students who play cricket only = 10

No. of students who play both foot ball & hockey = 20

No. of students who play both (foot ball & hockey) only = 12

No. of students who play both hockey & cricket = 15

No. of students who play both (hockey & cricket) only = 7

No. of students who play both foot ball and cricket = 25

No. of students who play both (foot ball and cricket) only = 17

No. of students who play all the three games = 8

Let us go through the following example problems to know "How to solve venn diagram word problems with 3 circles?"

**Problem 1 :**

**In a survey of university students, 64 had taken mathematics course, 94 had taken chemistry course, 58 had taken physics course, 28 had taken mathematics and physics, 26 had taken mathematics and chemistry, 22 had taken **** chemistry and physics course, and 14 had taken all the three courses.** Find how many had taken one course only.

**Solution :**

**Step 1 :**

Let M, C, P represent sets of students who had taken mathematics, chemistry and physics respectively

**Step 2 :**

From the given information, we have

n(M) = 64 , n(C) = 94, n(P) = 58,

n(MnP) = 28, n(MnC) = 26, n(CnP) = 22

n(MnCnP) = 14

**Step 3 :**

From the basic stuff, we have

No. of students who had taken only **Math**

= n(M) - [n(MnP) + n(MnC) - n(MnCnP)]

= 64 - [28+26-14]

= 64 - 40

= 24

**Step 4 :**

No. of students who had taken only **Chemistry**

= n(C) - [n(MnC) + n(CnP) - n(MnCnP)]

= 94 - [26+22-14]

= 94 - 34

= 60

**Step 5 :**

No. of students who had taken only **Physics**

= n(P) - [n(MnP) + n(CnP) - n(MnCnP)]

= 58 - [28+22-14]

= 58 - 36

= 22

**Step 6 :**

Total no. of students who had taken **only one** course

= 24 + 60 + 22

= 106

**Hence, the total number of students who had taken only one course is 106 **

**Alternative Method (Using venn diagram)**

**Step 1 :**

Venn diagram related to the information given in the question:

**Step 2 :**

From the venn diagram above, we have

No. of students who had taken only math = 24

No. of students who had taken only chemistry = 60

No. of students who had taken only physics = 22

**Step 3 :**

Total no. of students who had taken only one course

= 24 + 60 + 22

= 106

**Hence, the total number of students who had taken only one course is 106**

Let us see the next problem on "venn diagram word problems with 3 circles".

**Problem 2 :**

**In a group of students, 65 play foot ball, 45 play hockey, 42 play cricket, 20 play foot ball and hockey, 25 play foot ball and cricket, 15 play hockey and cricket and 8 play all the three games. Find the total number of students in the group.**(Assume that each student in the group plays at least one game.)

**Solution :**

**Step 1 :**

Let F, H and C represent the set of students who play foot ball, hockey and cricket respectively.

**Step 2 :**

From the given information, we have

n(F) = 65 , n(H) = 45, n(C) = 42,

n(FnH) = 20, n(FnC) = 25, n(HnC) = 15

n(FnHnC) = 8

**Step 3 :**

From the basic stuff, we have

Total number of students in the group = n(FuHuC)

= n(F) + n(H) + n(C) - n(FnH) - n(FnC) - n(HnC) + n(FnHnC)

= 65 + 45 + 42 -20 - 25 - 15 + 8

= 100

**Hence, the total number of students in the group is 100**

**Alternative Method (Using venn diagram)**

**Step 1 :**

Venn diagram related to the information given in the question:

**Step 2 :**

Total number of students in the group

= 28 + 12 + 18 + 7 + 10 + 17 + 8

= 100

**Hence, the total number of students in the group is 100**

Let us see the next problem on "venn diagram word problems with 3 circles".

**Problem 3 :**

In a class of 60 students, 40 students like math, 36 like science, 24 like both the subjects. Find the number of students who like

(i) Math only, (ii) Science only (iii) Either Math or Science (iv) Neither Math nor science

**Solution :**

**Step 1 :**

Let M and S represent the set of students who like math and science respectively.

**Step 2 :**

From the information given in the question, we have

n(M) = 40, n(S) = 36, n(MnS) = 24

**Step 3 :**

**Answer (i) :** No. of students who like math only

= n(M) - n(MnS)

= 40 - 24

= **16**

**Step 4 :**

**Answer (ii) :** No. of students who like science only

= n(S) - n(MnS)

= 36 - 24

= **12**

**Step 5 :**

**Answer (iii) :** No. of students who like either math or science

= n(M or S)

= n(MuS)

= n(M) + n(S) - n(MnS)

= 40 + 36 - 24

= **52**

**Step 6 :**

**Answer (iv) :**

Total no. students who like any of the two subjects = n(MuS) = 52

No. of students who like neither math nor science

= 60 - 52

= **8**

**Let us see the next problem on "venn diagram word problems with 3 circles".**

**Problem 4 :**

At a certain conference of 100 people there are 29 Indian women and 23 Indian men. Out of these Indian people 4 are doctors and 24 are either men or doctors. There are no foreign doctors. Find the number of women doctors attending the conference.

**Solution :**

**Step 1 : **

Let M and D represent the set of Indian men and Doctors respectively.

**Step 2 : **

From the information given in the question, we have

n(M) = 23, n(D) = 4, n(MuD) = 24,

**Step 3 : **

From the basic stuff, we have

n(MuD) = n(M) + n(D) - n(MnD)

24 = 23 + 4 - n(MnD)

n(MnD) = 3

n(Indian Men and Doctors) = 3

**Step 4 : **

So, out of the **4** Indian doctors, there are **3** men.

And the remaining **1** is Indian women doctor.

**Hence, the number women doctors attending the conference is 1**

**Let us see the next problem on "venn diagram word problems with 3 circles".**

**Problem 5 :**

In a college, 60 students enrolled in chemistry,40 in physics, 30 in biology, 15 in chemistry and physics,10 in physics and biology, 5 in biology and chemistry. No one enrolled in all the three. Find how many are enrolled in at least one of the subjects.

**Solution :**

Let A,B and C are the sets enrolled in the subjects Chemistry,Physics and Biology respectively.

Number of students enrolled in Chemistry n (A) = 60

Number of students enrolled in Physics n (B) = 40

Number of students enrolled in Biology n (C) = 30

No.of students enrolled in Chemistry and Physics n (A ∩ B) = 15

No.of students enrolled in Physics and Biology n (B ∩ C) = 10

No.of students enrolled in Biology and Chemistry n (C ∩ A) = 5

No one enrolled in all the three, So n (A ∩ B ∩ C) = 0

Number of students enrolled

in at least one of the subjects = 40+15+10+0+5+20+10

= 100

Let us see the next problem on "venn diagram word problems with 3 circles".

**Problem 6:**

In a town 85 % of the people speak Tamil,40 % speak English and 20 % speak Hindi. Also 32% speak English and Tamil, 13 % speak Tamil and Hindi and 10 % speak English and Hindi, find the percentage of people who can speak all the three languages.

**Solution:**

Let A,B and C are the people who speak Tamil, English and Hindi respectively.

Number of people who speak Tamil n (A) = 85

Number of people who speak English n (B) = 40

Number of people who speak Hindi n (C) = 20

Number of people who speak English and Tamil n (A ∩ B) = 32

Number of people who speak Tamil and Hindi n (A ∩ C) = 13

Number of people who speak English and Hindi n (B ∩ C) = 10

Let "x" be the number of people who speak all the three language.

Total number of people =

100 = 40 + x + 32 – x + x + 13 – x + 10 – x – 2 + x – 3 + x

100 = 40 + 32 + 13 + 10 – 2 – 3 + x

100 = 95 – 5 + x

100 = 90 + x

x = 100 - 90

x = 10 %

Let us see the next problem on "venn diagram word problems with 3 circles".

**Problem 7 : **

An advertising agency finds that, of its 170 clients,115 use Television,110 use Radio and 130 use Magazines. Also 85 use Television and Magazines,75 use Television and Radio,95 use Radio and Magazines,70 use all the three. Draw Venn diagram to represent these data. Find

(i) how many use only Radio?

(ii) how many use only Television?

(iii) how many use Television and Magazine but not radio?

**Solution:**

Let A,B and C are the people who speak Television, Radio and Magazines respectively.

Number of people who use Television n (A) = 115

Number of people who use Radio n (B) = 110

Number of people who use Magazine n (C) = 130

Number of people who use Television and Magazines

n (A ∩ C) = 85

Number of people who use Television and Radio

n (A ∩ B) = 75

Number of people who use Radio and Magazine

n (B ∩ C) = 95

Number of people who use all the three n (A ∩ B ∩ C) = 70

(i) Number of people who use only Radio = 10

(ii) Number of people who use only Television = 25

(iii) Number of people who use Television and Magazine but not radio = 15

The venn diagram word problems with 3 circles explained above will give clear idea to students on solving venn diagram word problems with 3 circles.

And also we hope that the word problems on sets and venn diagrams explained above would be much useful for the students who struggle to solve venn diagram word problems with 3 circles.

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