# Di Venn Diagram

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264 DATA INT ERPRETATION

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KUNDAN

Chapter-9

Venn Diagram

IntroductionPictorial representation of sets gives most of the ideas about sets and their properties in a much easier way

than the representation of sets given in language form. This pictorial representation is done by means of dia-

grams, known as Venn Diagram.

The objects in a set are called the members or elements of the set.

If A = {1, 2, 3, 4, 5, 6}, then 1, 2, 3, 4, 5 and 6 are the members or elements of the set A.

If B = { x  : x  is a positive integer divisible by 5 and x  < 25} or, B = {5, 10, 15, 20}, then 5, 10, 15 and 20 are the

elements of the set B.

A  B (read as set A intersection set B) is the set having the common elements of both the sets A and B.A  B (read as set A union set B) is the set having all the elements of the sets A and B. A - B (read as set A minusset B) is the set having those elements of set A which are not in set B.

In other words, A - B represents the set A exclusively, ie A – B have the elements which are only in A.

Similarly, B - A represents the set B exclusively. We keep it in mind that n(A  B) = n(B  A) and n(A B)= n(B  A).

The number of elements of a set A is represented by n(A), but n(A - B)  n(B - A)Now, by the above Venn diagram it is obvious that

n(A) = n(A - B) + n(A  B) ..... (1)n(B) = n(B - A) + n(A  B) ..... (2)n(A  B) = n(A - B) + n(A  B) + n(B - A) .... (i)Adding (1) and (2) we get,

n(A) + n(B) = n(A - B) + n(B - A) + n(A  B) + n(A  B)or, n(A) + n(B) - n(A  B) = n(A - B) + n(B - A) + n(A  B) ... (ii)From (i) and (ii), we have

n(A  B) = n(A) + n(B) – n(A  B) .... (3)Let us see some worked out examples given below:

Solved Examples

Ex. 1: I n a c l a s s of 7 0 s t u d en t s , 4 0 l i k e a c er t a i n m a g a z i n e a n d 3 7 l i k e a n o t h er c e r t a i n m a g a z i n e . F i n d

t h e num ber o f s t u d e n t s who l i k e bot h t h e maga z i n e s s im u l t a n e ou s l y .

Soln: We have, n(A  B) = 70, n(A) = 40, n(B) = 37Now, 70 = 40 + 37 – n(A  B)    n(A  B) = 77 – 70 = 7.

Ex. 2: I n a g r o u p o f 6 4 p er s o n s, 2 6 d r i n k t e a b u t n o t c of f e e a n d 3 4 d r i n k t e a . Fi n d h ow m a n y d r i n k (i ) t e a

an d co f f ee bo t h , (i i ) co f f ee bu t no t t ea .

Soln: (i) n(T  C) = 64, n(T - C) = 26, n(T) = 34We have, n(T) = n(T - C) + n(T  C)or, 34 = 26 + n(T  C)  n(T  C) = 34 – 26 = 8

(ii)Again, we have

n(T  C) = n(T) + n(C) – n(T  C)  or, 64 = 34 + n(C) – 8

n(C) = 38  Now, n(C) = n(C - T) + n(T  C)  or, 38 = n(C - T) + 8

n(C – T) = 38 - 8 = 30

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Ex. 3: I n a c l a s s of 3 0 s t u d en t s , 1 6 h a v e op t e d Ma t h ema t i c s a nd 1 2 ha v e op t e d Ma t h ema t i c s b u t n o t

B i o l o g y . F i n d t h e numbe r o f s t u d e n t s wh o ha v e op t e d B i o l o g y bu t n o t Ma t h ema t i c s .

Soln: n(M  B) = 30, n(M) = 16, n(M - B) = 12, n(B - M) = ?We have, n(M) = n(M - B) + n(M  B)or, 16 = 12 + n(M  B)   n(M  B) = 16 - 12 = 4Again, we have, n(M  B) = n(M) + n(B) - n(M  B)or, 30 = 16 + n(B) – 4

or, n(B) = 30 - 12 = 18

Now, n(B) = n(B – M) + n(M  B)or, 18 = n(B - M) + 4

n(B – M) = 18 – 4 = 14Ex. 4: I n a c l a s s o f 7 0 s t u d e n t s, 4 0 l i k e a cer t a i n m a g a z i n e a n d 3 7 l i k e a n ot h e r w h i l e 7 l i k e n ei t h e r .

(i ) F i n d t h e numbe r o f s t u d e n t s wh o l i k e a t l e a s t o n e o f t h e tw o maga z i n e s.

(i i ) F i n d t h e numbe r o f s t u d e n t s who l i k e bo t h t h e maga z i n e s sim u l t a n e ou s l y .

Soln: We have, total number of students = 70 in which 7 do not like any of the magazines.

For our consideration regarding liking of magazines, we are left with (70 – 7 =) 63 students.

Thus, n(A  B) = 63, n(A) = 40, n(B) = 37(i)  The number of students who like at least one of the two magazines = n(A  B) = 63.(ii) The number of students who like both the magazines simultaneously = n(A  B) = ?

We have, n(A  B) = n(A) + n(B) – n(A  B)  or, 63 = 40 + 37 – n(A  B)    n(A  B) = 77 – 63 = 14

Ex. 5: I n a s c h oo l , 4 5% o f t h e s t u d en t s p l a y c r i c k e t , 3 0% p l a y h o c k e y and 1 5% p l a y bo t h . Wha t p e r c en t

o f t h e s t u d e n t s p l a y n e i t h e r c r i c k et n o r h o ck ey ?

Soln: n(C) = 45, n(H) = 30, n(C  H) = 15

n(C  H) = 45 + 30 - 15 = 60ie, 60% of the students play either cricket or hockey or both.

So, the remaining (100 - 60 =) 40% students play neither cricket nor hockey.

Ex. 6: Ou t o f a t o t a l o f 3 6 0 m u s i c i a n s i n a c l u b 1 5% ca n p l a y a l l t h e t h r ee i n st r u m e n t s — g u i t a r , v i o l i n

a n d f l u t e . T h e n um b er o f m u s i ci a n s w h o c a n p l a y t w o a n d o n l y t w o o f t h e a bo ve i n st r u m e n t s i s 7 5 .

T h e n um b er o f m u s i c i a n s w h o c a n p l a y t h e gu i t a r a l o n e i s 7 3 .

(i) F i n d t h e t ot a l n um b e r o f m u s i ci a n s w h o ca n p l a y v i o l i n a l o n e a n d f l u t e a l o n e.

(ii) I f t h e n um b er o f m u s i c i a n s w h o c a n p l a y v i o l i n a l o n e b e t h e s am e a s t h e n um b e r o f m u s i ci a n s

w h o ca n p l a y g u i t a r a l o n e, t h e n f i n d t h e n um b er o f m u s i c i a n s w h o ca n p l a y f l u t e .

Soln: (i)  Total number of musicians = 360

15% of 360 = 54 musicians can play all the three instruments.

Given that x  + y  + z  = 75Now, 73 + f  + v  + (x  + y + z  =) 75 + 54 = 360

v  + f  = 360 – (73 + 75 + 54) = 158(ii)Now we have v  = 73

The number of musicians who can play flute alone,

f = (v  + f ) – v  = 158 – 73 = 85

and the number of musicians who can play flute =  f  + x  + y  + 54 = 85 + 54 + (x  + y )

We have x  + y  + z  = 75, x  + y  = 75 - z .

As either x   + y   or z is unknown, we cannot find out the number of musicians who can play flute.

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Ex. 7: Ou t o f a t o t a l 8 5 c h i l d r en p l a y i n g ba dm i n t o n o r t a b l e t en n i s o r b ot h , t o t a l n u m b er o f g i r l s i n t h e

g r o u p i s 7 0% o f t h e t o t a l n um ber o f b oy s i n t h e g r o u p . T he num ber o f b o y s p l a y i n g on l y b adm i n t o n

i s 5 0% of t h e numbe r o f b oy s and t h e t o t a l n um ber o f b o y s p l a y i n g badm i n t o n i s 6 0 % o f t h e t o t a l

n um ber o f b o y s. T he numbe r o f c h i l d r e n p l a y i n g on l y t a b l e t e n n i s i s 4 0% of t h e t o t a l n um ber o f

c h i l d r en a n d a t o t a l o f 1 2 c h i l d r en p l a y b a dm i n t o n a n d t a b l e t en n i s b ot h . Wh a t i s t h e n um b er o f

g i r l s p l a y i n g o n l y b a dm i n t o n ?

Soln: Let the number of boys be x , then x  +7

10

x  = 85 x  = 50

Number of girls = 85 - 50 = 35

Exercise

Directions (Q. 1-2): Study the following information carefully and answer accordingly:Out of a total of 240 musicians in a club, 7.5% can play all the three instruments — guitar, violin and flute.

The number of musicians who can play two and only two of the above instruments is 45. The number of musi-

cians who can play the guitar alone is 60.

1. Find the total number of musicians who can play flute alone and violin alone.

1) 115 2) 117 3) 118 4) 121 5) None of these

2. If the number of musicians who can play violin only be the same as the number of musicians who can play

only guitar, then find the number of musicians who can play flute.

1) 56 2) 57 3) 162 4) Cannot say 5) None of these

Directions (Q. 3-8): Study the following information carefully and answer accordingly: There are five high schools A, B, C, D and E in a certain town. Total number of high school students of the

town is 1800. The strength of school A is 20% and B is 37.5% of the total number of students of the town. D and

E have equal strengths. 40% students of A know only one language - Hindi. 60% students of D know only one

language - English. There are 111 more students in B who know Hindi exclusively than the number of students

of D who know English only. 55 students of C know Hindi but not English. 15 students of D know both the

languages. The strength of C is 37.5% of the strength of A. Two-fifths of students of B know both the languages.

The number of students of C who know English but not Hindi is 40 less than the number of the same category

of B. 97 students of E know only English and 20% students of A know both the languages. 28 students of E know

both the languages.

3. What is the percentage of the number of students who know both the languages?

1) 22.33 2) 22.66 3) 22.22 4) 22.5 5) None of these

4. What is the difference between the number of students who know English and those who know Hindi exclu-

sively?

1) 250 2) 200 3) 400 4) 360 5) None of these5. The number of students who know only Hindi of C is how many times those who know both the languages of

the same school?

1)2

43

2)1

33

3)1

43

4)2

33

5) None of these

6. Find the percentage of number of students who know Hindi exclusively.

1) 44.44 2) 55.55 3) 33.33 4) 66.33 5) None of these

7. What is the number of schools in which the number of students who know English only is more than the

average number of students who know English only?

1) 1 2) 2 3) 3 4) 4 5) None of these

8. What is the maximum difference between the number of students of a certain school who know only Hindi

and only English?

1) 195 2) 93 3) 165 4) 97 5) None of these

Directions (Q. 9-13): Study the following information carefully and answer accordingly:

In a group of 1440 persons,1

6  like Coca-Cola only, 37.5% like Pepsi only and 510 like Mirinda. 6.25% of

them like all the three drinks while 6 do not like even one of the drinks. The number of persons who like both

Mirinda and Pepsi only is half the number of persons who like both Coca-Cola and Pepsi only.1

8 like both Coca-

Cola and Mirinda only.

9. How many persons like Mirinda only?

1) 174 2) 160 3) 168 4) Data inadequate 5) None of these

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10. What is the difference between the number of persons who like Coca-Cola and those who like Pepsi only?

1) 300 2) 118 3) 192 4) Data inadequate 5) None of these

11. Find the percentage of number of persons who like more than one drink.

1) 27.5 2) 33.9 3) 33.75 4) Data inadequate 5) None of these

12. In a class of 55 students 35 take tea, 27 take coffee and 12 take both. Find the number of students who take

neither tea nor coffee.

1) 10 2) 5 3) 15 4) 8 5) None of these13. There are 1000 students, out of which 650 drink tea and 390 drink coffee and 30 students do not drink either

tea or coffee. How many students drink both tea and coffee?

1) 80 2) 90 3) 70 4) Data inadequate 5) None of these

Directions (Q. 14-18): Following Venn diagram shows the specialisation in different fields of some

players out of 120 players.

14. What is the percentage of those players who have specialised in bowling?

1) 12.50% 2) 30% 3) 37.50% 4) Can’t be determined 5) None of these

15. What is the percentage of those players who have specialised in any of the two departments?

1) 7.50% 2) 12.50% 3) 5.83% 4) 23.33% 5) None of these

16. What is the percentage of those players who have specialised in only one department?

1) 32.43% 2) 45.83% 3) 54.39% 4) 60% 5) None of these

17. In a class of 150 students, 65 play football, 50 play hockey, 75 play cricket, 35 play hockey and cricket, 20

play football and cricket, 42 play football and hockey and 8 play all the three games. Find the number of

students who do not play any of these three games.

1) 101 2) 49 3) 51 4) Can’t say 5) None of these

18. In a class there are 200 students. 70% of them like Hindi, 30% like English and 20% like Sanskrit. Find the

maximum possible percentage of students who like all the three languages.

1) 20 2) 10 3) 5 4) Can’t say 5) None of these

Directions (Q. 19-23): Study the following information carefully and answer accordingly:In the figure shown below circle I represents readers of BSC magazine, Circle II represents the students who

have joined Correspondence Course of BSC (Banking Services Chronicle), and circle III represents the students

who have joined Classroom Coaching of BSC Academy.

19. The students who have joined the Classroom Coaching but are neither readers of BSC nor associated with

BSC through Correspondence Course, are represented by the portion

1) G + D 2) G + F 3) C 4) C - (D + G + F)

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20. The portion which represents the students who are readers of BSC as well as are pursuing Correspondence

Course is

1) G 2) E + G 3) A + B 4) None of these

21. Ranjan Mukherjee is a regular reader of BSC Magazine and is pursuing its Correspondence Course too but has

not joined its Classroom Coaching. Then which of the following groups does he belong to?

1) A 2) G 3) E + G 4) E

22. Priya is a regular reader of BSC Magazine, is pursuing its Correspondence Course too and is determined toleave behind Ranjan Mukherjee after joining BSC Classroom Coaching. Then which of the following groups

does she belong to?

1) A 2) G 3) E + G 4) E

23. The readers of BSC Magazine have been represented by the portion

1) A + E + D + G 2) A + E + G 3) A 4) None of these

Directions (Q. 24-27): Study the following information carefully and answer accordingly:

Note: Use additional information given in any question for answering subsequent questions.

24. How many students study Geography or English?

1) 108 2) 91 3) 62 4) 130 5) 115

25. If 32 students study only Geography, how many students study English?

1) 63 2) 67 3) 52 4) 59 5) Can’t say

26. If there are 123 students in the class, how many students study Economics?

1) 67 2) 62 3) 63 4) 52 5) None of these

27. How many students study Economics or Geography or both but not all three?

1) 28 2) 60 3) 68 4) 54 5) None of these

Directions (Q. 28-29): Study the following information carefully and answer accordingly:

There are 120 students in a class, who read Maths or History or English. It is known that no student can readall three subjects. 24 read only Maths and History, 8 read only History and English and 21 read only Maths and

English. 32 read only Maths and 13 only History.

28. How many students read English?

1) 22 2) 30 3) 51 4) 54 5) None of these

29. If 9 of the students who read only Maths start to read all three subjects, find the percentage of students who

1) 50% 2) 53.33% 3) 60% 4) 40% 5) None of these

Directions (Q. 30-34): Study the following information carefully and answer accordingly:

A survey was conducted among 770 people who speak one or more languages from among Hindi, English and

Urdu. It was also found that 500 speak Hindi, 400 English and 300 Urdu.

(i) 30% of the Urdu-speaking people speak all three languages, which is 10% less than those who speak

Hindi and English both but not Urdu.

(ii) Number of people who speak Hindi and Urdu both but not English is

1

33 %3   less than the number of

people who speak only English.

(iii)Number of people who speak English and Urdu both but not Hindi is 30.

30. How many people speak only Hindi?

1) 190 2) 170 3) 120

4) Can’t be determined 5) None of these

31. How many people speak only English?

1) 190 2) 100 3) 90

4) Can’t be determined 5) None of these

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32. How many people speak Hindi and Urdu both but not English?

1) 180 2) 120 3) 90

4) 150 5) None of these

33. By what per cent the number of people who speak only Urdu is less than those who speak Hindi and English

both but not Urdu?

1)2

66 %

3

2)1

33 %

3

3) 40%

4) Can’t be determined 5) None of these

34. By what per cent the number of people who speak only English is more than those who speak Hindi and Urdu

both but not English?

1) 40% 2)2

66 %3

3) 50%

4) Can’t be determined 5) None of these

Directions (Q. 35-36): Study the following information carefully and answer accordingly:

There are 200 students in graduation. Out of these 165 are supposed to study at least one of the subjects

from among Physics, Chemistry and Mathematics. 110 students study Physics, 80 students study Chemistry and

90 students Mathematics. 40 students study Physics and Chemistry but not Mathematics, 35 students study

Physics and Mathematics but not Chemistry and 20 students study Chemistry and Mathematics but not Physics.

35. How many students study all three subjects?

1) 10 2) 12 3) 15 4) Can’t say 5) None of these

36. What is the percentage of those students who study all the three subjects with respect to those admitted ingraduation?

1) 5.40% 2) 6.06% 3) 4% 4) Can’t say 5) None of these

Directions (Q. 37-42): Study the following information carefully and answer accordingly:

There are three companies A, B and C. The employees of the company speak at least one of the three languages,

viz English, Hindi and French, in following manner:

(i) In company A, 700 employees speak Hindi, 600 speak English and 555 French. In company B, 650

speak Hindi, 580 speak English and 700 speak French. And in company C, 500 speak Hindi, 600

English and 700 French.

(ii)  The number of employees of company A who speak only Hindi is equal to that of company C who speak

English and French but not Hindi. It is also equal to that of company B who speak all the three lan-

guages.

(iii)  The number of employees of company C who speak only French is equal to 180, which is 20% more than

the number of employees of company B who speak only Hindi.

(iv)  The ratio of the number of employees of company C who speak only English to the number of employeesof company A who speak only French to the number of employees of company B who speak only

Hindi is 2 : 4 : 5.

(v)  The number of employees of company A who speak only English is equal to the number of employees of

company B who speak only French, which is equal to 180, which is also 25% less than those who speak

English and French but not Hindi in company C.

(vi)  The number of employees of company C who speak Hindi and French but not English is equal to the

number of employees of company A who speak Hindi and English but not French, which is equal to the

number of employees of company B who speak English and French but not Hindi.

(vii) The number of employees of company A who speak French and Hindi but not English is 165, which is

10% more than those who speak Hindi and French but not English in company C.

37. How many employees speak Hindi and English but not French in company C?

1) 130 2) 80 3) 150 4) 170 5) None of these

38. How many employees speak all the three languages in company A?

1) 145 2) 125 3) 130 4) 150 5) None of these

39. How many employees speak any two of the three languages in company B?

1) 540 2) 410 3) 670 4) Can’t say 5) None of these

40. The number of employees of company A who speak English and French but not Hindi is what per cent more

than the number of those who speak only Hindi in company C?

1) 125% 2) 60% 3) 150% 4) 100% 5) None of these

41. What is the difference between the number of employees of company C who speak all the three languages and

the number of employees of company B who speak only English?

1) 10 2) 20 3) 50 4) 110 5) None of these

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42. By what approximate per cent the number of employees of company B is more than that of C?

1) 4% 2) 6% 3) 8% 4) 10% 5) 12%

Directions (Q. 43-47): The following questions are based on the diagram given below:

P = Physics C = Chemistry M = Mathematics Class strength = 260.

Number of students passed in a subject

43. What is the percentage of students who have failed in all three subjects?

1) 5.8 2) 17.5 3) 35 4) 22.5 5) None of these

44. What is the percentage of students who have passed in two or more subjects?

1) 33 2) 29 3) 36 4) 25 5) 20

45. What is the percentage of students who have failed in at least one subject?

1) 96.5 2) 5.8 3) 65.0 4) 75.5 5) None of these

46. Taking any two subjects, which pair of subjects has the maximum number of students passed in at least one

of them?1) Physics, Chemistry 2) Physics, Mathematics 3) Chemistry, Mathematics

4) Cannot be determined 5) None of these

47. To be promoted to the next class it is essential to pass in Mathematics and at least in one of Physics and

Chemistry. How many students are likely to be promoted to the next class?

1) 245 2) 160 3) 97 4) 48 5) Can’t be determined

Directions (Q. 48-52): Answer these questions on the basis of the information given below:

(i) In a class of 80 students the girls and the boys are in the ratio of 3 : 5. The students can speak only Hindi

or only English or both Hindi and English.

(ii) The number of boys and the number of girls who can speak only Hindi is equal and each of them is 40%

of the total number of girls.

(iii)10% of the girls can speak both the languages and 58% of the boys can speak only English.

48. How many girls can speak only English?

1) 12 2) 29 3) 18 4) 15 5) None of these

49. In all how many boys can speak Hindi?1) 12 2) 9 3) 24 4) Data inadequate 5) None of these

50. What percentage of all the students (boys and girls together) can speak only Hindi?

1) 24 2) 40 3) 50 4) 30 5) None of these

51. In all how many students (boys and girls together) can speak both the languages?

1) 15 2) 12 3) 9 4) 29 5) None of these

52. How many boys can speak either only Hindi or only English?

1) 25 2) 38 3) 41 4) 29 5) None of these

Directions (Q. 53-55): Study the following information carefully and answer accordingly:

i) In a school, a total of 220 students are studying together in two sections A and B in the ratio of 5 : 6. The

students are studying only English or only Sanskrit or both English and Sanskrit.

ii)  The numbers of students studying only English from section A and of those studying both Sanskrit and

English from Section B are equal and each of them is 40% of the students who are studying only English

from section B.

iii) The number of students studying only Sanskrit from section A is 30% of the number of students studyingin section B and 60% of the students studying only English from section B.

53. How many students are studying both English and Sanskrit from section A?

1) 48 2) 16 3) 40 4) 36 5) None of these

54. How many students are studying only Sanskrit from section B?

1) 36 2) 10 3) 12 4) 24 5) None of these

55. Number of students studying only English from section B is what per cent more than that of the students

studying only English from section A?

1) 150% 2) 100% 3) 75% 4) 20% 5) None of these

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Directions (Q. 56-57): Study the following informations carefully and answer accordingly:

A survey was conducted by an agency in 25000 houses. It was found that 48% used Head & Shoulders, 48%

used Clinic Plus and 53% used Pentene Shampoo. 12% used both Head & Shoulders and Clinic Plus only and

10% used both Clinic Plus and Pentene only.

56. How many people used both Head & Shoulders and Pentene only if 8% used all the three?

1) 2750 2) 2500 3) 3000 4) 2000 5) Data inadequate

57. How many people used only Pentene if 8% used all the three shampoos?1) 5000 2) 6000 3) 8750 4) 8000 5) None of these

Directions (Q. 58-62): Read the following data to answer the questions that follow:In a class of 106 students, each student studies at least one of the three subjects Maths, Physics and

Chemistry. 48 of them study Maths, 51 Physics and 53 Chemistry. 16 study Maths and Physics, 17 study Maths

and Chemistry and 18 study Physics and Chemistry.

58. The number of students who study exactly two subjects is

1) 31 2) 32 3) 33 4) 36

59. The number of students who study more than one subject is

1) 39 2) 41 3) 40 4) 42

60. The number of students who study all the three subjects is

1) 5 2) 6 3) 7 4) 4

61. The number of students who study exactly one subject is

1) 45 2) 55 3) 65 4) 70

62. The number of students who study Physics and Maths but not Chemistry is1) 9 2) 11 3) 10 4) 12

Directions (Q. 63-67): Study the following Venn diagram and answer accordingly:

The following Venn diagram represents the results of a survey conducted by a market research firm NSD Ltd

to ascertain the profiles of a sample group. The diagram below shows the number of people who are Poets,

Sportsmen, Graduates or Orators. Refer to the diagram to answer the questions that follow:

Note:

(1) P = Poets, S = Sportsmen, G = Graduates, O = Orators

(2) The figures in any region of the above diagram pertain to the “only” value for that region. For example, 3

persons are only (Orators + Sportsmen + Graduates) etc.

63. Number of Sportsmen who have at least three specialities is

1) 12 2) 21 3) 9 4) 30

64. Total number of people having at least one speciality is

1) 403 2) 321 3) 343 4) 340

65. Number of people having only one speciality exceeded the number of people having exactly two specialities by

1) 113 2) 111 3) 112 4) 110

66. The number of people having at least one of the described specialities for what percentage of the total sample?

1) 38% 2) 62% 3) 44% 4) Cannot be determined

67. Orators who were neither Sportsmen nor Graduates exceeded Poets who were neither Orators nor Graduatesby a margin of

1) 32 2) 61 3) 43 4) 27

Directions (Q. 68-72): Refer to the following data to answer the questions that follow:

The result of an exam is given below:Out of 1000 students who appeared

(i) 658 failed in Physics

(ii) 166 failed in Physics and Chemistry

(iii) 372 failed in Chemistry, 434 failed in Physics and Maths

(iv) 590 failed in Maths, 126 failed in Maths and Chemistry

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68. The number of students who failed in all the three subjects is

1) 178 2) 73 3) 106 4) 126

69. The number of students who failed in Maths but not in Chemistry is

1) 464 2) 392 3) 387 4) 472

70. The number of students who failed in Physics but not in Maths is

1) 318 2) 224 3) 378 4) 232

71. The number of students who failed in Chemistry but not in Physics is1) 318 2) 198 3) 213 4) 206

72. The number of students who failed in Physics or Maths but not in Chemistry is

1) 558 2) 718 3) 628 4) 692

Directions (Q. 73-75): These questions are based on the following information:

A sports club has 80 members, out of which male and female members are in the ratio of 9 : 7 respectively.

All the members play either badminton or table tennis (TT) or both. 40% of the male members play only badminton.

20% of the female members play both the games, which is equal to the number of female members playing only

TT. Number of male members playing only TT is more than that of male members playing both the games by 3.

73. Number of female members playing badminton is what per cent of the total number of female members in the

club?

1) 80 2) 60 3) 75 4) 40 5) None of these

74. In all how many members play TT?

1) 39 2) 15 3) 22 4) 19 5) None of these

75. How many male members play both the games?

1) 17 2) 12 3) 19 4) 16 5) None of these

Directions (Q. 76-80): These questions are based on the following information:

In a class of 84 students boys and girls are in the ratio 5 : 7. Among the girls 7 can speak Hindi and English.

50 per cent of the total students can speak only Hindi. The ratio of the number of students speaking only Hindi

to that speaking only English is 21 : 16. The ratio of the number of boys speaking English only to that of girls

speaking English only is 3 : 5.

76. What is the number of boys who speak both the languages ?

1) 4 2) 5 3) 3 4) 2 5) None of these

77. What is the number of girls who speak English only ?

1) 12 2) 20 3) 22

4) Cannot be determined 5) None of these78. What is the ratio of the number of boys who speak Hindi only to that of girls who speak Hindi only?

1) 10 : 11 2) 11 : 10 3) 2 : 5

4) Cannot be determined 5) None of these

79. How many girls can speak Hindi ?

1) 29 2) 22 3) 27 4) 23 5) None of these

80. What is the ratio of the number of boys who speak English to that of girls who do so?

1) 3 : 5 2) 3 3) 5 : 8 4) 5 5) None of these

Directions (Q. 81-83): Study the following information to answer the given questions:

In a school, three languages are taught. Out of the total 600 students each one is required to study at least

one of the three, viz Gujarati, Tamil, Hindi. 20 students study all the three languages. 202 study only Hindi and

111 study only Gujarati. In all, 250 study Tamil. 57 study Hindi and Gujarati. 194 study only Tamil.

81. How many students, along with Tamil, study either Gujarati or Hindi (but not both)?

1) 36 2) 56 3) 16 4) Cannot be determined 5) None of these82. In all, how many students study Gujarati?

1) 199 2) 181 3) 163 4) Cannot be determined 5) None of these

83. Which of the following statements is definitely true?

1) The total number of students studying Hindi cannot be less than 290.

2) The total number of students studying Hindi cannot be less than 260.

3) The total number of students studying Gujarati cannot be more than 199.

4) Not more than 93 students study more than one language.

5) None of these

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Directions (Q. 84-88): Study the following information carefully to answer the questions:

The teachers’ colony has 2800 members, out of which 650 members read only English newspaper. 550 members

read only Hindi newspaper and 450 members read only Marathi newspaper. The number of members reading all the

three newspapers is 100. Members reading Hindi as well as English newspaper are 200. 400 members read Hindi as

well as Marathi newspaper and 300 members read English as well as Marathi newspaper.

84. Find the difference between the number of members reading English as well as Marathi newspaper and the

number of members reading English as well as Hindi newspaper.

1) 300 2) 200 3) 100

4) 50 5) None of these

85. How many members read at least two newspapers?

1) 600 2) 800 3) 500

4) 1000 5) None of these

86. Find the number of members reading Hindi newspaper.

1) 750 2) 980 3) 1000

4) 1020 5) None of these

87. How many members read only one newspaper?

1) 1560 2) 1650 3) 1640

4) 1540 5) None of these88. Find the number of members reading no newspaper.

1) 150 2) 460 3) 550

4) 750 5) None of these

Directions (Q. 89-93): Study the following information carefully and answer the questions given

below it:

There are 2500 residents in a village. 1,375 residents from this vil lage speak only their local language. 200

residents of the village speak the local language as well as English. The number of residents in the village who speak

the local language as well as Hindi is 625. 300 residents of the village speak all the three languages ie, English, Hindi

and the local language.

89. The number of residents who speak English as one of the languages forms what per cent of the total residents

in the village?

1) 12 2) 8 3) 20

4) 18 5) None of these

90. The number of residents who speak only the local language forms what per cent of the total number of

residents in the village?

1) 45 2) 55 3) 58

4) 40 5) None of these

91. The number of residents who speak Hindi as one of the languages is approximately what per cent of the

number of residents who speak only the local language?

1) 67 2) 70 3) 61

4) 59 5) 63

92. What is the ratio of the number of residents who speak all the three languages to the number of residents who

speak the local language as well as Hindi?

1) 12 : 55 2) 10 : 25 3)14 : 554) 12 : 25 5) None of these

93. If 25 more people who can speak all the three languages come to reside in the village and 45 more people who

can speak the local language and Hindi come to reside in the village, what would be the difference between

the number of residents who can speak all the three languages and the number of residents who can speak

the local language and Hindi?

1) 325 2) 330 3) 340

4) 355 5) None of these

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(1-2):

1. 2; 7.5% of 240 = 18

Given that x  + y  + z  = 45

Now, 60 + 18 + (x  + y  + z =) 45 + ( f  + v ) = 240

or, 123 + ( f  + v ) = 240

f  + v = 240 - 123 = 117

2. 4; Now, given that v = 60

f  = 117 - 60 = 57

But, the number of musicians who can play flute

= f +  (x  + y ) + 18 = 57 + 18 + (x + y ). Since x  + y is

not known so, the number of musicians who canplay flute cannot be determined.

(3-8): We symbolize the number of students who know

only Hindi, ie Hindi but not English by H - E, the

number of students who know only English by E

- H, the number of students who know both the

languages by H  E and the total strength of schools by T.

We have T = (H - E) + (E - H) + (H  E)Now collecting the given pieces of information and

using the above formula, we get

3. 3; Required percentage

=400 200

100 22.221800 9

4. 2; Required difference = (600 + 400) - 800 = 200

5. 4; 55 = x  × 15  x  =11 2

33 3

6. 5; Required percentage = 800 400 1200

1001800

=200 2

663 3

7. 2; Average number of students who know English

only =600

5 = 120.

So, A and D are the two desired schools.

8. 1; Clearly for B, the difference is maximum and it

is (300 – 105 =) 195

(9-13):

We have, 240 + x  + 540 + 510 = 1440 - 6

or, x  = 1434 – 1290 = 144 and

2

x  = 72

9. 3; The number of persons who like Mirinda only

= 510 – (180 + 90 + 72) = 168

10. 5;Required difference

= 240 + 144 + 180 + 90 – 540 = 654 – 540 = 114

11. 3;Total number of persons who like more than one

drink = 180 + 144 + 72 + 90 = 486

Required percentage =486

1001440

= 33.75%

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12. 2;Required number of students = 55 - (23 + 12 +

15) = 55 - 50 = 5

13. 3;Let x  be the number of students who drink both.

650 – x + x  + 390 –  x + 30 = 1000

or, – x   = 1000 – 1070 or, x  = 70

(14-18):

14. 3;Total number of players who have specialised in

bowling = 15 + 11 + 9 + 10 = 45

Required percentage =45

120 × 100 = 37.50%

15. 4;Required percentage =7 10 11

100120

=28

100120

= 23.33%

16. 2;Required percentage =22 18 15

100120

=55

100120

= 45.83%

17. 2; Number of students who play at least one game

= n(F H C) = 65 + 50 + 75 – 35 – 20 – 42 + 8= 101

Number of students who don’t play any of thethree games = 150 – 101 = 49.

18. 2;

S = 40

For x   to be maximum the other common

sections should be zero. Now,

(140 – x ) + (60 – x ) + (40 – x ) + x  = 200

x  = 20  Required % = 10

(19-23):

19. 3 20. 2 21. 4 22. 2 23. 1

(24-27):

24. 1;Students studying Geography or English

= (c  + 13 + a  + 14) + 16 + 25 = 67 + 16 + 25 = 108

25. 1;According to the question, c = 32

a  = 67 - (13 + 14 + 32) = 8  Students studying English= 14 + 8 + 16 + 25 = 63

26. 4;67 + b  + 16 + 25 = 123 or, b  = 123 - 108 = 15

Students studying Economics

= 13 + 15 + 8 + 16 = 52

(with the help of Q.No. 25)

27. 5;Students studying Economics or Geography or

both but not all three = (67 - 8) + 15 + 16 = 90

(28-29):

28. 3;32 + 24 + 13 + 8 + 0 + 21 + E = 120

E = Number of students who read only English

E = 120 - 98 = 22

total number of students who read English= 22 + 8 + 21 = 51

29. 5;Total number of students who read History

= 24 + 9 + 8 + 13 = 54

Required % =54

100120

= 45%

(30-34):

(i) 30% of Urdu = 30% of 300 = 90

Number of people who speak Hindi and

English both but not Urdu = 100

(ii) Number of people who speak English and

Urdu both but not Hindi = 30 Therefore, Number of people who speak only

English = 400 - (100 + 90 + 3) = 180 ... (A)

(iii) Now, with the help of (A),

Number of people who speak Hindi and Urdu

both but not English = 120 ... (B)

Therefore, number of people who speak only

Urdu = 300 – (120 + 90 + 30) = 60 ... (C)

Similarly, number of people who speak only

Hindi 500 – (100 + 90 + 120) = 190 ... (D)

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30. 1;From (D).

31. 5;From (A).

32. 2;From (B).

33. 3;Number of people who speak only Urdu

= 300 – (120 + 90 + 30) = 60

Required less % =100 60

100 40%100

34. 3;Required more % =180 120

100 50%120

.

(35-36):

Let x  be the number of students who study all

the three subjects. Then the number of students

who study only Physics = (35 – x )

Number of students who study only Chemistry

= (20 – x )

Number of students who study only Mathematics

= (35 – x )

Now, 110 + (20 – x ) + 20 + (35 – x ) = 165

or, x  = 10

35. 1

36. 5;Required % =10 100

5%200

(37-42):

Try to depict all the given informations in Venn-

diagram.

A B

C

From (iii),

Number of employees of company C who speak

only French = 180

Number of employees of company B who speak

only Hindi = 150100120

180

Combining (iii) and (iv), we have:5 = 150  :4 = 120 and :2 = 60From (v),

Number of employees of company A who speak

only English = Number of employees of company

B who speak only French = 180

Number of employees of company C who speakEnglish and French but not Hindi

= 24010075

180

Now, combining this with (ii), we have

Number of employees of company A who speak

only Hindi = Number of employees of company B

who speak all the three languages = Number of

employees of C who speak English and French

but not Hindi = 240

From (vii),

Number of employees of company A who speak

French and Hindi but not English = 165

Number of employees of company C who speakHindi and French but not English

= 150100110

165

Now, when we combine this with (vi), the rest of

our Venn-diagram will be filled.

37. 4;Number of employees of company C who speak

all the three languages = 700 - (180 + 240 + 150)

= 130Now, the number of employees of company C who

speak Hindi and English but not French

= 600 - (240 + 130 + 60) = 170

38. 1;Number of employees of company A who speak

all the three languages

= 700 - (240 + 150 + 165) = 145

39. 2;Number of employees of company B who speak

Hindi and English but not French

= 580 - (60 + 150 + 240) = 130

Number of employees of company B who speak

Hindi and French but not English

= 700 - (180 + 150 + 240) = 130

Total number of employees of company B who

speak any two of the three languages= 130 + 130 + 150 = 410

40. 3;Number of employees of company A who speak

English and French but not Hindi = 125

Number of employees of company C who speak

only Hindi = 50

Required % =125 50

100 150%50

41. 5;Required difference = 130 - 60 = 70

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42. 2;Number of employees in company B

= 700 + 60 + 130 + 150 = 1040

Number of employees in company C

= 700 + 60 + 170 + 50 = 980

Required % =1040 980

100 6%980

(43-47):Class strength = 260

Students passing in P + C + M = 9

Students pasing in P + C = 28 – 9 = 19

Students passing in P + M = 42 – 9 = 33

Students passing in M + C = 15 – 9 = 6

Students passing only in C = 63 – 19 – 6 – 9 = 29

Students passing only in M = 97 – 6 – 33 – 9 = 49

Students passing only in P = 85 – 9 – 19 – 33

= 24

Total students passing in at least one subject

= 63 + 97 + 85 – 28 – 42 – 15 + 9 = 169

43. 3;Students who have failed in all subjects

= 260 – 169 = 91

44. 4;Students who have passed in two or moresubjects = 9 + 19 + 33 + 6 = 67

Required67

% 100 25%260

45. 1;Total number of students who have failed in at

least one subject = 260 – 9 = 251

% value =251

100 96.5%260

46. 3;P, C = 19 + 9 + 24 + 29 + 33 + 6 = 120

P, M = 33 + 9 + 24 + 49 + 6 + 19 = 140

M, C = 6 + 9 + 49 + 29 + 33 + 19 = 145

47. 4;9 + 33 + 6 = 48

(48-52): Number of boys in the class = 5 80 508

Number of girls in the class = 80 – 50 = 30

48. 4 49. 5 50. 4

51. 2 52. 3

(53-55):Number of students in section A

=5

22011

= 100

Number of students in section B

= 220 - 100 = 120

53. 3 54. 1

55. 1;Required % =

60 24100

24

= 150%(56-57):

56. 1;Let x % people use both Head & Shoulders and

Pentene only.

Percentage of people who used only Head &Shoulders = (28 - x )Percentage of people who used only Pentene

= (35 - x )

28 – x  + 12 + 18 + 8 + 10 + x  + 35 – x  = 100or, 111 – x   = 100   x  = 11%

Number of people who used both Head Shoul-ders and Pentene only = 11% of 25000

= 2750

57. 2;Number of people who used only Pentene

= 24% of 25000 = 6000

(58-62): We have

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a + b + c + d + e + f + g   = 106

a + e + d + b  = 48

c + b + d + f = 51

g + e + d + f  = 53

b + d  = 16; d  + e  = 17; d + f  = 18

and from the standard formula,

n A B C n A n B n C n A B

n B C n C A n A B C We get, 106 = 48 + 51 + 53 – 17 – 18 – 16 + d

d  = 5. Now, all the values can be obtained as

shown in the figure and all the questions can be

58. 4;b  + e + f  = 36

59. 2;b  + d  + e  + f  = 41

60. 1;d  = 5

61. 3;a  + c  + g  = 65

62. 2;b  = 11

Note: This question, and its solution, is so mechanical

and direct that with proper practice, you should

be able to solve it very quickly.

(63-67):63. 2;2 + 7 + 9 + 3 = 21

= 340.

65. 3;Only one speciality = 19 + 63 + 101 + 28 = 211

Exactly two specialities = 53 + 11 + 23 + 12 = 99

Required answer = 211 – 99 = 112

66. 4; The number of people having at least one speci-

ality is 340. But the total number of people sur-

veyed is not known. Hence, percentage cannot

be determined.

67. 1; (53 + 19) – (28 + 12) = 32

(68-72):

Let P be the set of the students who failed in

Physics, C be the set of the students who failedin Chemistry, and M be the set of the students

who failed in Maths. Then

n(P) = 658, n(P  C) = 166,n(C) = 372, n(P  M) = 434n(M) = 590, n(M  C) = 126 andn(P  M  C) = 1000

68. 3;The number of students who failed in all the three

subjects = n(P  M  C)

n P M C n P n M n C

n P M n P C n M C  = 100 – 658 – 590 – 372 + 434 + 166 + 126

= 106

69. 1;Number of students who failed in Maths but not

in Chemistry

n M C n M n M C 590 126 464 70. 2;Number of students who failed in Physics but

not in Maths

n P M n P n P M 658 434 224. 71. 4;Number of students who failed in Chemistry but

not in Physics

n C P n C n C P 372 166 206

72. 3;Number of students who failed in Physics or

Maths but not in Chemistry

n P M C n C 1000 372 628 (73-75):

The whole information is as follows:

Total members : 80

73. 1: Required per cent =

%8010035

721

74. 5; 19 + 22 = 41

75. 2;It is obvious from the above figure.

(76-80):

76. 3 77. 2 78. 1 79. 1

80. 5; 12 + 3 : 20 +7 = 15 : 27 = 5 : 9

(81-83):

We have been given

A = 20, E = 111, F = 194, G = 202,

A + D = 57 and A + B + C + F = 250

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Here, we have

A + B + C + D + E + F + G = 600

B + C = 600 - (111 + 37 + 194 + 20 + 202)

= 600 - 564 = 36

We can get B + C through other ways also.

Note that A + B + C + F = 250

or 20 + B + C + 194 = 250

B + C = 250 - (194 + 20) = 36.

81. 1; Here we need to find out the values of B and C

together ie, 36.

82. 4; Here we need to find out the sum of the values of

A, B, D and E. Since value of B is not known,

hence sum of the values of A, B, D and E can’t

be determined.

83. 4; Total number of students who study more than

one language = A + B + C + D = 20 + 36 + 37 = 93

(84-88):

84. 3;  Difference = (E + M) - (H + E) = 300 - 200= 100

85. 4; Number of members who read at least 2 news-

papers = 400 + 300 + 200 + 100 = 1000.

86. 5; Number of members reading Hindi newspaper

= 550 + 400 + 200 + 100 = 1250

87. 2; Number of members who read only one news-

paper = 550 + 650 + 450 = 1650.

88. 1; Number of members reading no newspaper

= 2800 - (650 + 550 + 450 + 400 + 300 +

200 + 100)

= 150.

(89-93)

89. 3; L  local language, E  English, H  Hindi

Required percentage

500100 20%

2500

90. 2

91. 1;Required percentage925

100 671375

92. 4;Required ratio = 300 : 625 = 12 : 25

93. 5;After addition people who speak all the three

languages = 300 + 25 = 325

After addition people who speak local language

as well as Hindi = 625 + 45 = 670

Required difference = 670 - 325 = 345.