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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.
Hence, data is inadequate.
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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
read History.
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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Answers and explanations
(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
answered.
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
64. 4;Adding up all the values, we get required answer
= 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.