**For previous exercise click Sets – Exercise 1.1 – Class X**

**Solve the following problems and verify the data In each case by drawing Venn diagrams. If A and B are the sets such that n(A) = 37, n(B) = 26 and n(A∪B) = 51, find n(A∩B).**

Solution:

Given: n(A) = 37, n(B) = 26, n(A∪B) = 51,

We have to find n(A∩B)

n(A∪B)= n(A) + n(B) – n(A∩B)

51 = 37 + 26 – n(A∩B)

∴ (A∩B)=12

**In a group of 50 persons, 30 like tea, 25 like coffee and 16 like both. How many like (i) either tea or coffee (ii) neither tea nor coffee**

Solution:

Number of persons who like Tea(T) = 30

Number of persons who (C) = 25

Number of persons who like both (T∩C) = 16

i) W.K.T n(T∪C) = n(T) + n(C) – n(T∩C)

= 30 + 25 – 16

= 55 – 16

= 39

ii) Number of people who like only coffee = n(C) \ n(T∩C)

= 25 – 16

= 9

∴ Number of people who like only coffee = 9

iii) Number of people who like only tea.

n(T-C) = n(T) \ n(T∩C)

= 30 – 16

= 14

∴ No of people who like only tea = 14

**In a group of passengers, 100 know Kannada, 50 know English and 25 know both. If passengers know either Kannada or English, how many passengers are in the group?**

Solution:

Number of passengers who know Kannada n(K) =100

Number of passengers who know English n(E) = 50

Number of passengers who know both n(K∩E) = 25

n(K∪E) = n(K) + n(E) – n(K∩E)

= 100 + 50 – 25

= 150 – 25

= 125

∴ There are 125 passengers in the group.

**In a class, 50 students offered Mathematics, 42 offered Biology and 24 offered both the subjects. Find the number of students, who offer**

**i) Mathematics only **

**(ii) Biology only. Also find the total number of students in the class.**

Solution:

Number of students who offered Mathematics n(M) = 50

Number of students who offered Mathematics n(B) = 42

Number of students who offered Mathematics n(M∩B) = 24

n(M∪B) = n(M) + n(B) – n(M∩B)

= 50 + 42 – 24

= 92 – 24

= 68

∴Total number of students in the class = 68

i) No. of students who offer Mathematics only n(M – B) = n(M) \ n(M∩B)

= 50 – 24

= 26

ii) No. of students who offer Biology only n(B-M) = n(B) \ n(M∩B)

= 42 – 24

= 18

**In a medical examination of 150 people, it was found that 90 had eye problem, 50 had heart problem and 30 had both complaints. What percentage of people had either eye problem or heart problem?**

Solution:

Number of people who have eye problem n(E) = 90

Number of people who have heart problem n(E) = 50

Number of people who complain both n(E) = 30

We have to find, number of people who have either eye trouble or heart trouble ie., n(E∪H) *[this should be expressed in** percentage.]*

n(E∪H) = n(E) + n(H) – n(E∩H)

= 90 + 50 – 30

= 140 – 30

= 110

Out of 150 people examined, 110 have either eye or heart trouble.

Then, out of 100 people number of people examined = ^{110}/_{150} x ^{100}/_{1} = ^{220}/_{3} = 73.33%

II. Solve by drawing Venn diagrams only.

**A radio station surveyed 190 audience to determine the types of music they liked. The survey revealed that 114 liked rock music, 50 liked folk music, 41 liked classical music, 14 liked rock music and folk music, 15 liked rock music and classical music, and 11 liked classical music and folk music. 5 liked all the three types of music. Find**

**(i) How many did not like any of the 3 types?**

**(ii) How many liked any two types only?**

**(iii) How many liked folk music but not rock music?**

Solution:

N(R∪F∪C) = 155

i) Number of Audience who did not like any of these 3 types = 190 – 140 = 50

**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 three. Find**

**i) How many use only television?**

**ii) How many use only radio?**

**iii) How many use television and magazines but not radio?**

Solution:

i) Number of clients who use only television = 25

ii) Number of clients who use only radio = 10

iii) Number of clients who use television and magazine but not radio = 15

**In a village, out of 120 farmers, 93 farmers have grown vegetables, 63 have grown flowers, 45 farmers have sugarcane, 45 farmers have grown vegetables and flowers, 24 farmers have grown flowers and sugarcane, 27 farmers have grown vegetables and sugarcane. Find how many farmers have grown vegetables, flowers and sugarcane.**

Solution:

Total 120 = P + Q + R + (45 – x) + (24 – x) + (27 – x) + x

120 = P+Q+R+96-2x……………………(1)

Total vegetables = 93 = P + 45 – x + 27 – x + x

Total flowers = 63 = Q + 24 – x + 45 – x + x

Total sugarcane = 45 = R + 24 – x + 27 – x + x

201 = (P+Q+R) + 192 – 3x P+Q+R

= 201 – 192 + 3x P+Q+R

= 9 + 3x

From equation (1), P+Q+R = 9+3x

120 = 9 + 3x + 96 – 2x

120 = 105 + x

x = 120 – 105

x = 15

**Progressions – Exercise 2.1 – Class X**