Students and Teachers Forum

1. In a group of 100 persons, 72 people can speak English and 43 can speak French. How many can speak English only? How many can speak French only and how many can speak both English and French? 

Let, E = set of people who speak English  F = set of people who speak French  Then, Total number of people, n(U) = 100 Number of people who speak English, n(E) = 72 Number of people who speak French, n(F) = 43 Number of people who speak .....

2. Among the total candidates in an examination, 40% students passed in Maths, 45% in Science and 55% in Health, 10% students passed both in Maths and Science, 20% in Science and Health and 15%  in Health and Maths. Then,   
(i)    Draw a Venn diagram to show the above information.
(ii)   Calculate the percentage passed in all three subjects.

Let S, H and M represent the sets of students passed in Science, Health and Maths respectively. Then, Percentage of students who passed in mathematics, n(M) = 40% Percentage of students who passed in science, n(S) = .....

3. In a group of 95 students, the ratio of the students who like Mathematics and Science is 4:5. If 10 of them like both subjects and 15 of them like none of the subjects then by drawing a venn-diagram, find how many of them 
(a) like only mathematics 
(b) like only science.
 

Let,  Total number of students, n(U) = 95  Let x be any constant. Then, No. of student who like Mathematics, n(M) = 4x No. of students who like Science, n(S) = 5x No. of students who like both subjects, n(M∩S) = 10 No. of students who .....

4. If n (A) = 40, n (B) = 60, n (A ∪ B) = 80, find the values of n (A ∩ B), n0 (A) and n0 (B). Draw a Venn-diagram to illustrate the above information.

Given, n (A) = 40 n (B) = 60 n (AUB) = 80 n (A ∩ B) = ? n0 (A) = ? n0 (B) = ? We know, n (AUB) = n(A) + n(B) – n(A∩B) Or, 80 = 40 + 60 – n(A∩B)  Or, 80 = 100 – n(A∩B) Or, n(A∩B) = 100 – .....

5. In a group of 70 people, 37 like tea, 52 like milk and each person likes at least one the two drinks. Using Venn-diagram, find how many people like both tea and milk. How many of them like tea only.

Given, Total number of people, n(U) = 70 Number of people who like tea, n(T) = 37 Number of people who like milk, n(M) = 52 Number of people who like both tea and milk, n(T∩M) = ? Number of people who like tea only, no(T) = ? Venn .....

6. In a survey, it was found that the ratio of the people who liked modern songs and folk songs is 8:9. Out of which 50 people liked both songs, 40 liked folk songs only and 80 liked none of the songs.
i)  Find the number of people who participate in the survey 
ii) Represent the above information in venn diagram.
 

Let x be any constant. Then,         Number of people who like modern songs, n(M) = 8x         Number of people who like folk songs, n(F) = 9x         Number of people who like both songs, .....

7. If n(A) = 12 and n(B) = 20, then the minimum value of n(A ∪ B) is

Given, n(A) = 12 and n(B) = 20, Minimum value of n(A ∪ B) = ? We know,  n(A ∪ B) = n(A) + n(B) - n(A ∩ B) We get the minumum value of n(A ∪ B) when n(A ∩ B) is maximum. And for set A and B, n(A ∩ B) is .....

8. In an examination 45% students passed in science only, 25% passed in English only and 5% students failed in both subjects. If 200 students passed in English then find the total number of students by using a Venn-diagram.

Let S and E be the set of students who passed in Science and English respectively.  Then, Total percentage of students, n (U) = 100%  Percentage of students who passed in Science only, no(S) = 45%  Percentage of students who passed .....

9. In a survey of 70 students, it was found that all students liked either football, or basketball or cricket. 40 students liked cricket, 35 students liked football, 34 students liked basketball, 15 students liked basketball and football, 16 students liked football and cricket, 18 students liked basketball and cricket. Find the number of students who liked all the three games. Also illustrate the above information in a Venn-diagram.

Let F, B and C be the set of students who liked football, basketball and cricket respectively.  Then, Total number of students, n (U) = 70  Number of students who like cricket, n (C) = 40  Number of students who like football, n(F) = .....

10. In a survey of 150 people, 30 liked milk but not tea and 20 liked tea but not milk, 40 like none of these. How many people liked both milk and tea? Represent the information in Venn-diagram.

Let M and T be the set of people who liked milk and tea respectively.  Then, Total number of people, n (U) = 150  Numbe of people who like milk only, n0 (M) = 30  Number of people who like tea only, n0 (T) = 20  Number of people .....

11. In a group of 95 students the ratio of students who like Mathematics and Science is 4:5. If 10 of them like both the subjects and 15 of them like none of the subjects then find how many of them
(a) like only Mathematics and
(b) like only Science.

Let M and S denote the sets of students who like Mathematics and Science respectively. From question, Total number of students, n(U) = 95 Ration of number of students who like mathematics and science, n(M): n(S) = .....

12. 131 students were asked how they travelled daily. 56 said they used the autorickshaw, 103 said they used the bus, 65 said they used the bus but never an autorickshaw.
(i) How many students used the autorickshaw but not the bus?
(ii)How many students used neither the autorickshaw nor the bus?

Let, A and B denote the set of students who used the autorickshaw and bus respectively. From the question, Total number of students, n(U) = 131 Number of students who used autorickshaw, n(A) = 56 Number of students who used bus, n(B) = 103 Number .....

13. In a survey of 100 people, 65 read Kantipur, 45 read Gorkhapatra, 40 read Himalayan times, 25 read Kantipur as well as Gorkhapatra, 20 read Kantipur as well as Himalayan times, 15 read Gorkhapatra as well as Himalayan times and 5 read all three newspaper. Draw a Venn diagram to represent the above information.

(a)    How many people read at least one of these newspapers?
(b)    How many people don’t read all three newspapers?
(c)    How many read Gorkhapatra only?
 

Let K, G and H represent the sets of people who read Kantipur, Gorkhapatra and Himalayan times respectively. Total number of students, n(U) = 100 Number of people who read Kantipur, n(K) = 65 Number of people who read Gorkhapatra, n(G) = 45 Number .....

14. In a survey of a village, it was found that 39% of the people liked only tea, 27% liked only coffee and 15% people did not like both. Find

(i) What percent liked tea?
(ii) What percent liked coffee?

Let, T and C denote the sets of people who like tea and coffee respectively. From the question, Percentage of people who like tea only, no(T) = 39%  Percentage of people who like coffee only, no(C) = .....

15. In a survey of 119 students, it was found that 16 liked neither swimming nor driving, 69 liked driving and 39 liked swimming
(a) How many students liked swimming only?
(b) How many students liked driving only?
(c) Represent the above information in venn-diagram.

Let,       Total number of students, n(U) = 119       No. of students who like neither swimming nor driving, n (SUD)ˈ = 16       No. of students who like driving, n(D) = 69       No. of .....

16. Some people were asked if they like to travel Europe, America or Japan. The results were 50% liked Europe, 60% America, 40% Japan, 20% liked Europe and America, 25% liked America and Japan, 25% liked Europe and Japan, 10% did not liked all. If 318 liked all three, how many people were asked? Also, find the people who liked to travel America only.

Let, Percentage of people who like to travel Europe, n(E) = 50% Percentage of people who like to travel America, n(A) = 60% Percentage of people who like to travel Japan, n(J) = 40% Percentage of people who like to travel Europe and America, .....

17. In a survey of 200 students, 70 are fond of playing crickets, 80 football and 100 basketball. Similarly 20 are found of playing cricket and football only, 30 cricket and basketball only, 40 are playing basketball only. If 20 students are fond of playing football and basketball only, by drawing venn-diagram find 
(i) number of students who play all games. 
(ii) how many students don’t play any game?
 

Let, Total number of students, n(U) = 200 Number of students who play cricket, n(C) = 70 Number of students who play football, n(F) = 80 Number of students who play basketball, n(B) = 100 Number of students who play cricket and football only, .....

18. If the number of elements in set A = 10 and the number of elements in set B = 25, find the maximum and the minimum number of elements in set A∪B.

Given,          n(A) = 10          n(B) = 25 For maximum number of element in set AUB, the set must be disjoint sets. Then, n(A∪B) = n(A) + n(B)               .....

19. Sets P and Q are subsets of a universal set U. If n(U) = 90, n(P) = 50, n(Q) = 35 and n(P ∪Q)ˈ = 15, find n (P ∩ Q).

Given:           n (U) = 90           n (P) = 50           n (Q) = 35           n (P ∪Q)ˈ = 15        .....

20. In a group of 95 students, the ratio of the students who like mathematics and science is 4:5. If 10 of them like both the subjects and 15 of them like none of the subjects. By drawing a Venn- diagram, find how many of them
a) like only mathematics?
b) like only science.

Let M and S be the set of students who like mathematics and science respectively.  Then, Total number of students, n(U) = 95  Ratio of students who like mathematics and science, n(M): n(S) = 4:5  So, if x is any constant, n(M) = 4x .....

21. Let T and M be the group of people, who like tea and milk respectively. If n(T) = 80, n(M) = 70, and n(T ∪ M) = 120, find the number of people who like tea but not milk.

Given,  n (T) = 80  n (M) = 70  n (T ∪ M) = 120  Number of people who like tea but not milk, n0 (T) = ? We have,  n (T ∪ M) = n (T) + n (M) – n (T ∩ M)  Or, 120 = 80 + 70 – n (T ∩ .....

22. In a group of 175 students, 90 like grapes and 50 like grapes but not mangoes. If every student likes at least one of them, find the number of students who like both mangoes and grapes. 

Let A and B be the group of students who like grapes and mangoes respectively.  Then, Total number of students, n (U) = 175  Number of students who like grapes, n (A) = 90  Number of students who like grapes only, n0 (A) = .....

23. Let M and E be the set of students, who like Mathematics and English respectively. If n0 (M) = 20, n (E) = 75 and n (M ∩ E) = 50. Find the number of students who like mathematics.

Given,  n0 (M) = 20  n (E) = 75 n (M ∩ E) = 50  Number of people who like mathematics, n (M) = ? We have,  n0 (M) = n (M) – n (M ∩ E)  Or, 20 = n (M) – 50  Or, n (M) = 50 + 20    .....

24. In a survey in a school, it was found that 60 like ice-cream, 50 like curd and 10 like both. If 5 like neither, find the total number of students surveyed.

Solution:  Given,  Let A and B be the set of students who like Ice-cream and curd respectively.  Now, Number of students who like ice-cream, n (A) = 60  Number of students who like curd, n (B) = 50  Number of students who .....

25. In n (A) = 80 and n (B) = 65, find the least value of n (A∪B) and the greatest value of n(A∩B).

Given,          n(A) = 80          n(B) = 65 For the least value of n (A∪B), the sets must be overlapping sets, with set B as the proper subset of A. Then, n(A∪B)= n(A)    .....

26. Out of 50 students in class, 10 students like Maths but not Science and 15 students like Science but not Maths. If 10 students like none of them.
i) Show the information in Venn diagram.
ii) Find the ratio of students who like Maths and Science.
 

Let A be the students who like Maths and B be the students who like Science.  We have given, no(A) = 10, no(B) = 15 n(AUB) ' = 10 So, n(AUB) = n(U) - n(AUB)'                     = 50 .....

27. Let U = {1, 2, 3, …….10}, A = {2, 4, 6, 8} and B = {1, 2, 3, 4, 5}. Verify that (A ∪ B )ˈ= Aˈ ∩  Bˈ.

Given,  U = {1, 2, 3, …….,10} A = {2, 4, 6, 8} B = {1, 2, 3, 4, 5} Now, A ∪ B  = {2, 4, 6, 8} ∪ {1, 2, 3, 4, 5}                 = {1, 2, 3, 4, 5, 6, 8} And  (A ∪B .....

28. If X, Y, Z are subsets of U, where n(X) = 48, n(Y) = 51, n(Z) = 40, n(X ∩ Y) = 11, n(Y ∩ Z) = 10, 
n(Z ∩ X) = 9 , n(X ∪ Y ∪ Z) = 180 and n(U) = 120, find n(X ∩ Y ∩ Z) .
 

Here, X, Y, Z are the subsets of U, n(X) = 48, n(Y) = 51, n(Z) = 40, n(X ∩ Y) = 11, n(Y ∩ Z) = 10,  n(Z ∩ X) = 9 , n(X∩ Y ∩ Z) = 180 and n(U) = 120. n(X ∩ Y∩ Z) = ? We know, n(X∪ Y ∪Z) = n(X) + n(Y) + n(Z) .....

29. Out of 100 students of a school, 65 liked Mathematics, 75 liked English and 50 liked both the subjects. How many students liked only one subject?

Let,  M = Mathematics and E = English Given,  Total no. of  student, n(U) = 100 Number of students who like Mathematics, n(M) = 65  Number of students who like English, n(E) = 75 Nuber of students who like both .....

30. In a survey of a community of 350 people, it was found that 150 people like tea, 200 like coffee and 50 people like none of these two. 
i.    Find the number of people who like both tea and coffee?
ii.   Illustrate the above information in a Venn-diagram. 
 

Let T and C be the set of people who like tea and coffee respectively.  Then,  Total number of people, n (U) = 350  Number of people who like tea, n (T) = 150  Number of people who like coffee, n (C) = 200  Number of people .....

31. Out of 200 students in a class, 150 like football and 120 like cricket. If the number of students who like cricket only is one third of the number of students who like football only, then using Venn-diagram find the number of students who like 
i.  both the games,      
ii. football only,         
iii.none of these games.
 

Let F and C be the set of students who like football and cricket respectively.  Then, Total number of students, n(U) = 200 Number of students who like football, n (F) = 150  Number of students who like cricket, n (C) = 120  According .....

32. In a survey, 45% of students of Padmapur play basketball, 40% play cricket and 30% play both. If 360 students play neither of them, find the number of students in the school. And also find the number of students who play basketball only.

Let A and B be the percentage of students playing basketball and cricket respectively. Given, Percentage of students who play basketball, n(A)          = 45% Percentage of students who play cricket, n(B)      .....

33. If n(A) = 40, n(B) = 60, and n(A∪B) = 80, find the value of n(A∩B).

Here,  n(A) = 40 n(B) = 60 n(A∪B) = 80 n(A∩B) = ? We know, n(A∪B) = n(A) + n(B) -n(A∩B) or, 80 = 40 + 60 – n(A∩B) or, n(A∩B) = 100 – 80 or, n(A∩B) = 20 Hence, n(A∩B) = .....

34. In a group of students, 18 liked football, 19 liked volleyball and 16 liked basketball. Six liked football only, 9 liked volleyball only, 5 liked football and volleyball only and 2 liked volleyball and basketball only. Using venn-diagram find 
(a) how many students liked all three games 
(b) how many liked football and basketball only 
(c) how many liked basketball only 
(d) how many students are there altogether
 

Let, No. of students who like football, n(F) = 18 No. of students who like volleyball, n(V) = 19 No. of students who like basketball, n(B) = 16 No. of students who like football only, no(F) = 6 No. of students who like volleyball only, no(V) = 9 No. .....

35. Out of 120 students appeared in an examination the number of students who passed in mathematics only is twice the number of students who passed in science only. If 50 students passed in both subjects and 40 students failed in both subjects then;
i.   Find the number of students who passed in mathematics. 
ii.  Find the number of students who passed in science. 
iii. Show the result in a Venn-diagram. 
 

Let M and S be the set of students who passed in mathematics and science respectively.  Then Total number of student, n (U) = 120  Number of students who passed in science only, no(S) = x (say) Then, number of students who passed in .....

36. In a survey of 100 people, it was found that 65 liked singing, 55 liked dancing, 35 liked singing as well as dancing, find how many people don’t like both singing and dancing.

Let, S and D denote the sets of people who like singing and dancing respectively.   n(U) = 100   n(S) = 65   n(D) = 55   n(S∩D) = 35   n(S∪D)ˈ = ? The given information are shown in the Venn diagram. Now, from .....

37. In an examination, out of 200 students 70 succeeded in English, 80 in Mathematics, 60 in Nepali, 35 in English as well as Mathematics, 25 in English as well as Nepali, 35 in Nepali as well as Mathematics and 10 succeeded in all three subjects. 
(a) Draw a venn-diagram showing all the above information 
(b) Find the number of students passing at least one of the subjects. 
(c) Find the number of students who unsucceeded in all three subjects. 
 

Let, Total number of students, n(U) = 200 No. of students who succeeded in English, n(E) = 70 No. of students who succeeded in Mathematics, n(M) = 80 No. of students who succeeded in Nepali, n(N) = 60 No. of students who succeeded in English and .....

38. Let A and B are two non-empty sets A ⊂ B and B ⊂ A. What will be true for A and B?

Given, A and B are two non-empty sets such that A ⊂ B and B ⊂ A. Here, A ⊂ B means that all elements of set A are also in set B. Similarly, B ⊂ A means that all elements of set B are also in set A. The conditions hold true .....

39. If A and B are two disjoint sets, what relation is true for n (A ∪ B) ?

Given, A and B are two disjoint sets, We know, n (A ∪ B) = n(A) + n(B) - n(A ∩ B) For, disjoint sets, n(A ∩ B) is zero. So, the relation is given by,  n (A ∪ B) = n(A) + n(B),  which is the true relation for .....

40. Define union of two sets with example.

Let A and B be any two sets. Then the union of two sets A and B is a set of elements which are both in sets A and B. It is written as A ∪ B.  For example, Let A = {2, 4, 6, 8} and B = {1, 2, 3, 4, 5} then, A ∪ B = {1, 2, 3, 4, 5, 6, .....

41. If A ⊂ B, n(A) = 50 and n(B) = 80, what is the possible value of n(A ∪ B)?

Given, n(A) = 50 and n(B) = 80, Since, A ⊂ B, all the elements of set A is also the elements of set B. But n(A) is less than n(B). So, A is a proper subset of B. In that case, n(A) = n(A∩B) Then, n(A ∪ B) = n(B)    .....

42. In a group of 50 students, 35 students like only one subject out of English and science. The ratio of students who like both and who do not like both is 1:2. Find the number of students who like both the subjects.

Let E and S be the set of students who like English and Science.  Then, Total no.of students, n(U) = 50 Number of students who like only one subject, n0 (E) + n0 (S) = 35  Ratio of number of students who like both and who do not .....

43. Out of the total candidates in an examination, 60% of the students liked mathematics and 75% students liked English. If all the students liked one of the two subjects, find the percentage of students who liked both the subjects.
 

Let M and E be the group of students who liked mathematics and English respectively.  Then,  Total percentage of students, n (U) = 100%  Percentage of students who liked mathematics, n (M) = 60%  Percentage of students who liked .....

44. Let A and B are two sets, such that n(A ∩ B) = 10, n(A ∪ B) = 50 and n(A)/n(B)  = 2/3. Find n (A). 

Given, n(A ∩ B) = 10  n(A ∪ B) = 50  n(A)/n(B)  = 2/3 ==> n(B) = 3/2 n(A)  n(A) = ? We know,      n(A ∪ B) = n(A) + n(B) – n(A ∩ B)  Or, 50 = n(A) + 1.5 n(A) – 10  Or, .....

45. If n(A) = 12 and n(B) = 20, then find the maximum value of n(A∪ B).

Given, n(A) = 12 n(B) = 20,  Maximum value of n(A∪ B) = ? We know,  n(A∪ B) = n(A) + n(B) - n(A ∩ B) We get the maximum value of n(AUB), when n(A ∩ B) is zero. So, the maximum value of n(A∪ B) = n(A) + n(B)   .....

46. In a group of 60 people, 27 like cold drinks and 42 like hot drinks and each person likes at least one of the two drinks. How many like both coffee and tea?

Let A = Set of people who like cold drinks.       B = Set of people who like hot drinks. Then, Total number of people, n(A ∪ B) = 60 Number of people who like cold drinks, n(A) = 27     Number of people who like hot .....

47. A and B are two subset of a universal set U in which n(U) = 70, n(A) = 40, n(B) = 20 and n(A∪B)ˈ = 15, find the value of n(A∩B).

Here, n(U) = 70 n(A) = 40 n(B) = 20 n(A∪B)ˈ = 15 n(A∩B) = ? We know, n(AUB) = n(U) - n(A∪B)ˈ Or, n(AUB) = 70 - 15 Or, .....

48. In a survey of 120 students, it was found that 17 drink neither tea nor coffee, 80 drink tea and 26 drink coffee. By drawing a venn diagram, find the number of students who drink both tea and coffee.

Let A be the number of students drinking tea and B be students drinking coffee. Let ‘x’ be students drinking both tea and coffee. From Venn diagram, Or, 80 -x + x + 26 – x + 17 = 120 Or,  123 – x = 120 Or,, x = 3 So, .....

49. In a survey, it was found that 45% of the people like Dashain festival, 65% like Tihar festival and 28% like both festivals. 
(a) Represent the above information in venn-diagram. 
(b) Find what percent of the people like at least one of the festivals.
 

Let,     No. of people who like Dashain, n(D) = 45%     No. of people who like Tihar, n(T) = 65%     No. of people who like both festivals, n(D∩T) = 28%      No. of people who like atleast one of the .....

50. If n(A - B) = 18, n(A ∪ B) = 70 and n(A ∩ B) = 25, then find n(B). 

Given, n(A - B) = 18 n(A ∪ B) = 70 and n(A ∩ B) = 25, n(B) = ? We know, n(A∪B)     = n(A - B) + n(B - A) + n(A ∩ B) Or, 70        = 18 + 25 + n(B - A) Or, 70        = .....