Exam practice

Every question from the end of every unit, with hints and answers to open. After you check an answer, mark how you did, and come back to the ones you didn’t get.

44 questions

Unit 1 · Sets

Worksheet to print (PDF, 5 pages) Answer key for teachers

Harder worksheet (PDF, 4 pages) Answer key for teachers

Exercise 1.1

Question 1

Give examples of the cardinality of sets, and show them to your teacher.

Answer

Any correct examples, such as:
A = {vowels of the English alphabet} = {a, e, i, o, u}, so n(A) = 5.
B = {days of the week}, so n(B) = 7.
C = {prime numbers less than 10} = {2, 3, 5, 7}, so n(C) = 4.
And for two sets: if P = {1, 2, 3, 4} and Q = {3, 4, 5}, then n(P ∪ Q) = 4 + 3 − 2 = 5.

For two sets A and B with A ⊂ B, find n(A ∪ B) and n(A ∩ B).

Answer

Every element of A is already in B, so the A circle sits inside the B circle.
n(A ∪ B) = n(B) and n(A ∩ B) = n(A).

If A and B are overlapping sets, write the formula for n(A ∪ B).

Answer

n(A ∪ B) = n(A) + n(B) − n(A ∩ B), or n(A ∪ B) = n₀(A) + n₀(B) + n(A ∩ B).

The sets A and B have 12 and 8 elements. Find the least possible number of elements in A ∪ B.

Answer

12, when all 8 elements of B are also in A (B ⊂ A): 12 + 8 − 8 = 12.

Question 2

In Figure 1.8, 80 people are in set M, 90 are in set E and 15 are in neither. Find:

Figure 1.8 Exercise 1.1, question 2.

a.n₀(M)
b.n₀(E)
c.n(M)
d.n(E)
e.n(M ∪ E)
f.n(M ∩ E)
g.n(M ∪ E)
h.n(U)

Answer

a.20
b.30
c.20 + 60 = 80
d.60 + 30 = 90
e.20 + 60 + 30 = 110
f.60
g.15
h.110 + 15 = 125

Question 3

If n(U) = 200, n₀(M) = 2x, n₀(E) = 3x, n(M ∩ E) = 60 and n(M ∪ E) = 40, find x.

Answer

2x + 60 + 3x + 40 = 200, so 5x = 100 and x = 20.

If n(U) = 350, n(A) = 200, n(B) = 220 and n(A ∩ B) = 120, find n(A ∪ B) and n(A ∪ B).

Answer

n(A ∪ B) = 200 + 220 − 120 = 300.
n(A ∪ B) = 350 − 300 = 50.

If n(A) = 35 and n(A) = 25, find n(U).

Answer

Every element of U is either in A or not, so n(U) = n(A) + n(A) = 35 + 25 = 60.

Of two sets P and Q, P has 40 elements, P ∪ Q has 60 and P ∩ Q has 10. How many elements has Q?

Answer

n(P ∪ Q) = n(P) + n(Q) − n(P ∩ Q), so 60 = 40 + n(Q) − 10 and n(Q) = 30.

Question 4

In a survey of 180 students of a school, 45 like Nepali only and 60 like English only, but 15 like neither subject.

i.Show the information in a Venn diagram.
ii.Find the number of students who like both subjects.
iii.Find the number who like at least one subject.

Answer

Let N and E be the students who like Nepali and English, and n(N ∩ E) = x.

Figure 1.9 Exercise 1.1, 4a.

ii.45 + x + 60 + 15 = 180, so x = 60.
iii.45 + 60 + 60 = 165 (or 180 − 15).

In a survey of 1,200 students of a school, 100 like Mathematics only and 200 like Science only, but 700 like neither.

i.Show the information in a Venn diagram.
ii.Find the number who like both subjects.
iii.Find the number who like at least one subject.

Answer

Let M and S be the students who like Mathematics and Science, and n(M ∩ S) = x.

Figure 1.10 Exercise 1.1, 4b.

ii.100 + x + 200 + 700 = 1200, so x = 200.
iii.1200 − 700 = 500.

In a survey of 60 students, 10 play football only and 20 play volleyball only, but 12 play neither game.

i.Show the information in a Venn diagram.
ii.Find the number who play both games.
iii.Find the number who play at least one game.

Answer

Let F and V be the students who play football and volleyball, and n(F ∩ V) = x.

Figure 1.11 Exercise 1.1, 4c.

ii.10 + x + 20 + 12 = 60, so x = 18.
iii.60 − 12 = 48.

Question 5

A survey of 900 people of a community found that 525 read Madhupark, 450 read Yuba Manch, and 75 read neither magazine.

i.Show the information in a Venn diagram.
ii.Find the number who read both.
iii.Find the number who read only one.

Answer

Let M and Y be the readers of Madhupark and Yuba Manch. At least one: 900 − 75 = 825.
ii. Both: 525 + 450 − 825 = 150.

Figure 1.12 Exercise 1.1, 5a: 525 − 150 = 375 and 450 − 150 = 300.

iii. Only one: 375 + 300 = 675.

In a survey of 150 people, 90 like modern songs and 70 like folk songs, but 30 like neither.

i.Show the information in a Venn diagram.
ii.Find the number who like both kinds of song.
iii.Find the number who like only modern songs.

Answer

Let M and F be the people who like modern songs and folk songs.
At least one: 150 − 30 = 120. Both: 90 + 70 − 120 = 40.

Figure 1.13 Exercise 1.1, 5b. M: modern songs, F: folk songs.

Only modern: 90 − 40 = 50.

In a survey of 360 players, 210 liked to play volleyball and 180 liked football, but 30 liked neither.

i.Show the information in a Venn diagram.
ii.Find the number who like both games.
iii.Find the number who like only one game.

Answer

Let V and F be the players who like volleyball and football.
At least one: 360 − 30 = 330. Both: 210 + 180 − 330 = 60.

Figure 1.14 Exercise 1.1, 5c.

Only one: 150 + 120 = 270.

Question 6

Of the students who sat an examination, 70% passed English and 60% passed Mathematics, 20% failed both, and 550 students passed both.

i.Show the information in a Venn diagram.
ii.Find the total number of students who sat the examination.
iii.Find how many passed English only.

Answer

Let E and M be the students who passed English and Mathematics. In percentages: at least one passed 100 − 20 = 80%, so both: 70 + 60 − 80 = 50%.

Figure 1.15 Exercise 1.1, 6a, in percentages.

ii.50% of the total is 550, so the total is $550 \times \frac{100}{50} =$ 1,100.
iii.English only is 70 − 50 = 20% of 1,100 = 220.

In a survey of students who sat the Grade 10 examination, 60% wanted to study Science and 70% Management, while 10% wanted neither. 400 students wanted to study both.

i.Show the information in a Venn diagram.
ii.Find how many students took part in the survey.
iii.Find the number who wanted to study Science only.

Answer

Let S and M be the students who wanted to study Science and Management.
At least one: 100 − 10 = 90%. Both: 60 + 70 − 90 = 40%.

Figure 1.16 Exercise 1.1, 6b, in percentages.

ii.40% is 400, so the total is 1,000.
iii.Science only: 60 − 40 = 20% of 1,000 = 200.

In a survey of the people of a community, 65% ride a motorcycle and 35% ride a scooter, while 20% ride both. 200 people ride both.

i.Show the information in a Venn diagram.
ii.Find how many people took part in the survey.
iii.Find the number who ride motorcycles only.

Answer

Let M and S be the people who ride a motorcycle and a scooter.
At least one: 65 + 35 − 20 = 80%, so 20% ride neither.

Figure 1.17 Exercise 1.1, 6c, in percentages.

ii.20% is 200, so the total is 1,000.
iii.Motorcycle only: 65 − 20 = 45% of 1,000 = 450.

Question 7

Among 95 people of a community, the numbers who drink tea and coffee are in the ratio 4 : 5. 10 people drink both, and 15 drink neither.

i.Show the information in a Venn diagram.
ii.Find the number who drink exactly one of tea or coffee.
iii.Find the number who drink at least one.

Answer

Let T and C be the people who drink tea and coffee.
Let n(T) = 4x and n(C) = 5x.

Figure 1.18 Exercise 1.1, 7a.

(4x − 10) + 10 + (5x − 10) + 15 = 95, so 9x + 5 = 95 and x = 10: 40 drink tea, 50 coffee.

ii.Exactly one: (40 − 10) + (50 − 10) = 30 + 40 = 70.
iii.At least one: 95 − 15 = 80.

In a class of 64 students, each likes milk or curd or both. The numbers who like only milk and only curd are in the ratio 2 : 1, and 16 like both.

i.Show the information in a Venn diagram.
ii.Find the number who like milk.
iii.Find the number who like only one.

Answer

Let M and C be the students who like milk and curd.
Let n₀(M) = 2x and n₀(C) = x. Everyone likes at least one, so the three regions inside the circles make 64:

Figure 1.19 Exercise 1.1, 7b.

2x + 16 + x = 64, so 3x = 48 and x = 16.

ii.Milk: 2 × 16 + 16 = 48.
iii.Only one: 32 + 16 = 48.

At a conference of 320 participants, 60 only sing and 100 only dance. The number who do neither is three times the number who do both.

i.Show the information in a Venn diagram.
ii.Find how many do neither.
iii.Find the number who do at most one of the two.

Answer

Let S and D be the participants who sing and dance.
Let n(S ∩ D) = x, so neither is 3x.

Figure 1.20 Exercise 1.1, 7c.

60 + x + 100 + 3x = 320, so 4x = 160 and x = 40.

ii.Neither: 3 × 40 = 120.
iii.At most one (everyone except those who do both): 320 − 40 = 280.

Questions 8 to 11

In a survey of 200 people of a community, the numbers who use only a laptop and only a mobile were in the ratio 2 : 3. 30% use both and 15% use neither.

i.Show the information in a Venn diagram.
ii.Find the number who use a laptop.
iii.Find how many use at most one of the two.

Answer

Let L and M be the people who use a laptop and a mobile.
Both: 30% of 200 = 60. Neither: 15% of 200 = 30. Let n₀(L) = 2x, n₀(M) = 3x.

Figure 1.21 Exercise 1.1, question 8.

2x + 60 + 3x + 30 = 200, so x = 22.

ii.Laptop: 44 + 60 = 104.
iii.At most one: 200 − 60 = 140.

In a survey of 300 players, one-third play volleyball only. 60% of the rest play football only, and 60 play neither. Using a Venn diagram, find the ratio of the number who play volleyball to the number who play football.

Answer

Let V and F be the players who play volleyball and football.
Volleyball only: $\frac{1}{3} \times 300 = 100$ . The rest: 200. Football only: 60% of 200 = 120. Neither: 60.
Both: 300 − 100 − 120 − 60 = 20.

Figure 1.22 Exercise 1.1, question 9.

n(V) = 120, n(F) = 140: the ratio is 120:140 = 6 : 7.

10.

Of 65 players in a survey, 11 play volleyball only and 33 play cricket only. The number who play cricket is twice the number who play volleyball. Using a Venn diagram, find the number who play both and the number who play neither.

Answer

Let V and C be the players who play volleyball and cricket.
Let n(V ∩ C) = x. Then n(V) = 11 + x and n(C) = 33 + x.
33 + x = 2(11 + x), so 33 + x = 22 + 2x and x = 11.

Figure 1.23 Exercise 1.1, question 10.

Both: 11. Neither: 65 − 11 − 11 − 33 = 10.

11.

In a survey of 80 people, 60 like only oranges and 10 like both oranges and apples. The number who like oranges is 5 times the number who like apples. Using a Venn diagram, find the number who like apples only and the number who like neither fruit.

Answer

Let O and A be the sets of people who like oranges and apples.
n(O) = 60 + 10 = 70 = 5 × n(A), so n(A) = 14.
Apples only: 14 − 10 = 4.

Figure 1.24 Exercise 1.1, question 11.

Neither: 80 − (60 + 10 + 4) = 6.

Exercise 1.2

Question 1

Figure 1.32 shows the elements of the sets P, Q and R. Find:

Figure 1.32 Exercise 1.2, question 1.

a.n(P)
b.n(Q)
c.n(P ∪ Q ∪ R)
d.n₀(P)
e.n₀(R)
f.n(P ∩ R)
g.n(P ∪ Q ∪ R)
h.n₀(P ∩ Q)
i.n(P ∩ Q ∩ R)

Answer

P = {1, 2, 3, 4, 6, 9, 18}, Q = {5, 6, 9, 12, 13, 15}, R = {4, 6, 11, 12, 16, 17}.

a.7
b.6
c.14
d.4 (1, 2, 3, 18)
e.3 (11, 16, 17)
f.2 (4, 6)
g.4 (7, 8, 10, 14)
h.1 (9)
i.1 (6)

Question 2

If U = {positive integers less than 30}, P = {multiples of 2 less than 30}, Q = {multiples of 3 less than 30} and R = {multiples of 5 less than 30}, show the sets in a Venn diagram and verify:

a.n(P ∪ Q) = n(P) + n(Q) − n(P ∩ Q)
b.n(P ∪ Q ∪ R) = n(P) + n(Q) + n(R) − n(P ∩ Q) − n(Q ∩ R) − n(R ∩ P) + n(P ∩ Q ∩ R)
c.n(P ∪ Q ∪ R) = n(P − Q) + n(Q − R) + n(R − P) + n(P ∩ Q ∩ R)

Answer

P = {2, 4, 6, …, 28}, n(P) = 14. Q = {3, 6, 9, …, 27}, n(Q) = 9. R = {5, 10, 15, 20, 25}, n(R) = 5.
P ∩ Q = {6, 12, 18, 24} (4), Q ∩ R = {15} (1), R ∩ P = {10, 20} (2), P ∩ Q ∩ R = {} (0: the first number divisible by 2, 3 and 5 is 30).

Figure 1.33 Exercise 1.2, question 2: multiples of 2, 3 and 5 below 30. The middle is empty.

a.P ∪ Q has 8 + 4 + 4 + 2 + 1 = 19 elements (from the diagram), and 14 + 9 − 4 = 19. ✓
b.P ∪ Q ∪ R has 8 + 4 + 2 + 4 + 1 + 2 + 0 = 21 elements, and 14 + 9 + 5 − 4 − 1 − 2 + 0 = 21. ✓
c.n(P − Q) = 14 − 4 = 10, n(Q − R) = 9 − 1 = 8, n(R − P) = 5 − 2 = 3: 10 + 8 + 3 + 0 = 21. ✓ (This works for any three sets: every region of the union lies in exactly one of P − Q, Q − R, R − P and P ∩ Q ∩ R.)

Question 3

If n(U) = 100, n(M) = 45, n(E) = 50, n(S) = 35, n(M ∩ E) = 20, n(E ∩ S) = 20, n(S ∩ M) = 15 and n(M ∩ E ∩ S) = 5, find n(M ∪ E ∪ S).

Answer

n(M ∪ E ∪ S) = 45 + 50 + 35 − 20 − 20 − 15 + 5 = 80, so n(M ∪ E ∪ S) = 100 − 80 = 20.

If n(U) = 105, n(A) = 40, n(B) = 35, n(C) = 30, n(A ∩ B) = 15, n(B ∩ C) = 12, n(A ∩ B ∩ C) = 6 and n(A ∪ B ∪ C) = 30, find n(A ∩ C).

Answer

n(A ∪ B ∪ C) = 105 − 30 = 75. So 75 = 40 + 35 + 30 − 15 − 12 − n(A ∩ C) + 6 = 84 − n(A ∩ C), and n(A ∩ C) = 9.

If n(U) = 120, n(M) = 50, n(E) = 40, n(S) = 45, n(M ∩ E) = 15, n(E ∩ S) = 15, n(S ∩ M) = 15 and n(M ∪ E ∪ S) = 15, find n(M ∩ E ∩ S).

Answer

n(M ∪ E ∪ S) = 120 − 15 = 105. So 105 = 50 + 40 + 45 − 15 − 15 − 15 + n(M ∩ E ∩ S) = 90 + n(M ∩ E ∩ S), and n(M ∩ E ∩ S) = 15.

If n(A ∪ B ∪ C) = 105, n₀(A) = 25, n₀(B) = 25, n₀(C) = 15, n₀(A ∩ B) = 15, n₀(A ∩ C) = 10 and n(A ∩ B ∩ C) = 10, find n₀(B ∩ C).

Answer

The seven regions inside the circles make the union: 25 + 25 + 15 + 15 + 10 + n₀(B ∩ C) + 10 = 105, so n₀(B ∩ C) = 5.

Question 4

Of 90 students who sat an examination, 43 passed Science, 40 Mathematics and 38 Nepali. 13 passed Science and Mathematics, 18 Mathematics and Nepali, and 16 Science and Nepali; 6 passed all three.

i.Show the information in a Venn diagram.
ii.Find the number who passed none of the subjects.

Answer

Let S, M and N be the students who passed Science, Mathematics and Nepali.
Pairs only: 13 − 6 = 7, 18 − 6 = 12, 16 − 6 = 10. Only one: Science 43 − (6 + 7 + 10) = 20, Mathematics 40 − (6 + 7 + 12) = 15, Nepali 38 − (6 + 12 + 10) = 10.

Figure 1.34 Exercise 1.2, 4a.

ii. n(S ∪ M ∪ N) = 43 + 40 + 38 − 13 − 18 − 16 + 6 = 80, so none: 90 − 80 = 10.

In a survey of a group, 60 like tea, 45 coffee and 30 milk; 25 like coffee and tea, 20 milk and tea, 15 coffee and milk, and 10 like all three. Everyone surveyed likes at least one of them.

i.Show the information in a Venn diagram.
ii.Find how many people were surveyed.

Answer

Let T, C and M be the people who like tea, coffee and milk.
Pairs only: tea and coffee 25 − 10 = 15, coffee and milk 15 − 10 = 5, tea and milk 20 − 10 = 10. Only one: tea 60 − 35 = 25, coffee 45 − 30 = 15, milk 30 − 25 = 5.

Figure 1.35 Exercise 1.2, 4b.

ii. 60 + 45 + 30 − 25 − 20 − 15 + 10 = 85 people.

In a survey of 60 students, 23 played volleyball, 15 basketball and 20 cricket. 7 played volleyball and basketball, 5 basketball and cricket, and 4 volleyball and cricket, while 15 played none of the games.

i.Show the information in a Venn diagram.
ii.Find how many played all three games.
iii.How many played volleyball and cricket only?

Answer

Let V, B and C be the students who played volleyball, basketball and cricket, and n(V ∩ B ∩ C) = x.

ii.n(V ∪ B ∪ C) = 60 − 15 = 45. So 45 = 23 + 15 + 20 − 7 − 5 − 4 + x = 42 + x, and x = 3.
iii.Volleyball and cricket only: 4 − 3 = 1.

Figure 1.36 Exercise 1.2, 4c. Volleyball only: 23 − (3 + 4 + 1) = 15; basketball only: 15 − (3 + 4 + 2) = 6; cricket only: 20 − (3 + 2 + 1) = 14.

Question 5

Of the students who sat an examination, 40% passed Science, 45% Mathematics and 50% Nepali. 10% passed Science and Mathematics, 20% Mathematics and Nepali, and 15% Science and Nepali, while 5% failed all three.

i.Find the percentage who passed all three subjects.
ii.Find the percentage who passed only one subject.
iii.Find the percentage who passed exactly two subjects.
iv.Find the percentage who passed at least one subject.
v.Show the information in a Venn diagram.

Answer

Let S, M and N be the students who passed Science, Mathematics and Nepali, and let x% pass all three.

iv.At least one: 100 − 5 = 95%.
i.95 = 40 + 45 + 50 − 10 − 20 − 15 + x = 90 + x, so all three: 5%.
iii.Exactly two: (10 − 5) + (20 − 5) + (15 − 5) = 5 + 15 + 10 = 30%.
ii.Only one: 95 − 30 − 5 = 60% (20% each: Science 40 − 20 = 20, Mathematics 45 − 25 = 20, Nepali 50 − 30 = 20).

Figure 1.37 Exercise 1.2, question 5, in percentages.

Question 6

A survey asked some people of a community whether they read Yuba Manch, Madhupark or Muna. 30 read Yuba Manch and 25 Madhupark; 15 read both Yuba Manch and Muna, and 12 both Yuba Manch and Madhupark. 9 read Madhupark only, 11 Muna only, 5 Yuba Manch and Madhupark only, and 10 read none of them.

a.Show the information in a Venn diagram.
b.Find the total number of people surveyed.
c.Find the number who read exactly two magazines.
d.Find the number who read Muna.

Hint

Start from what is “only”: 12 read Yuba Manch and Madhupark, and 5 of those read only those two, so the other 7 read all three.

Answer

Let Y, D, M be the readers of Yuba Manch, Madhupark and Muna.
All three: n(Y ∩ D) − n₀(Y ∩ D) = 12 − 5 = 7.
Y and M only: 15 − 7 = 8.
Y only: 30 − (5 + 7 + 8) = 10.
D and M only: 25 − (9 + 5 + 7) = 4.

Figure 1.38 Exercise 1.2, question 6. Y: Yuba Manch, D: Madhupark, M: Muna.

b.10 + 9 + 11 + 5 + 4 + 8 + 7 + 10 = 64.
c.Exactly two: 5 + 4 + 8 = 17.
d.Muna: 11 + 4 + 8 + 7 = 30.

Question 7

90 people were asked which films they like by language: 48 like Nepali films, 40 English and 31 Hindi; 24 like Nepali and English, 19 Hindi and English, and 6 like all three. 21 like none of them.

a.How many like both Nepali and Hindi films?
b.How many do not like Hindi films?
c.How many like neither Nepali nor Hindi films?

Answer

Let N, E and H be the people who like Nepali, English and Hindi films.

a.n(N ∪ E ∪ H) = 90 − 21 = 69. So 69 = 48 + 40 + 31 − 24 − 19 − n(N ∩ H) + 6 = 82 − n(N ∩ H), and n(N ∩ H) = 13 (of whom 7 like only those two, and 6 English as well).
b.90 − 31 = 59.
c.n(N ∪ H) = 48 + 31 − 13 = 66, so neither: 90 − 66 = 24.

Figure 1.39 Exercise 1.2, question 7. Nepali only: 48 − (18 + 6 + 7) = 17; English only: 40 − (18 + 6 + 13) = 3; Hindi only: 31 − (13 + 6 + 7) = 5.

Mixed exercise

Two overlapping sets A and B have n₀(A) = 16 + x, n₀(B) = 5x, n(A ∩ B) = y and n(A ∪ B) = x.

a.Show the information in a Venn diagram.
b.If n(A) = n(B), find n(A ∪ B).
c.If also n(U) = 50, find the ratio of n(A ∩ B) to n(A ∪ B).

Answer

Figure 1.42 Mixed exercise, question 1.

b.n(A) = 16 + x + y and n(B) = 5x + y. Equal, so 16 + x = 5x and x = 4: n(A ∪ B) = 4.
c.(16 + 4) + y + 20 + 4 = 50, so y = 6. The ratio is 6:4 = 3 : 2.

A and B are subsets of the universal set U, with n(U) = 100, n(A − B) = 32 + x, n(B − A) = 5x, n(A ∩ B) = x and n(A ∪ B) = y.

a.Show the information in a Venn diagram.
b.If n(A) = n(B), find n(A ∩ B).
c.Find n(A ∪ B).
d.By what percentage is n(A ∪ B) more or less than n(A ∩ B)?

Answer

Figure 1.43 Mixed exercise, question 2.

b.n(A) = 32 + 2x and n(B) = 6x. Equal, so 4x = 32 and x = 8: n(A ∩ B) = 8.
c.40 + 8 + 40 + y = 100, so y = 12.
d.12 is 4 more than 8, and $\frac{4}{8} \times 100\% = 50\%$ : n(A ∪ B) is 50% more than n(A ∩ B). (Turned round, 8 is 4 less than 12, which is $33\tfrac{1}{3}\%$ less: always divide by the number you compare with.)

In a survey of 93 women of a community, 80 work in farming and 71 in sewing, while 10 do other work.

a.Find the cardinality of each region and show it in a Venn diagram.
b.Find how many women do both.
c.How many times the number who only farm is the number who only sew?

Answer

Let F and S be the women who work in farming and in sewing.
At least one: 93 − 10 = 83. b. Both: 80 + 71 − 83 = 68.

Figure 1.44 Mixed exercise, question 3.

c. Farming only 12, sewing only 3: 12 ÷ 3 = 4 times.

In a survey of 1,000 farmers in a community, 800 grew potatoes and 500 tomatoes, while 50 grew other crops only.

a.Find the cardinality of each region and show it in a Venn diagram.
b.Find the number who grow both.
c.Write the numbers who grow only potatoes and only tomatoes as a ratio.

Answer

Let P and T be the farmers who grow potatoes and tomatoes.
At least one: 1000 − 50 = 950. b. Both: 800 + 500 − 950 = 350.

Figure 1.45 Mixed exercise, question 4.

c. 450:150 = 3 : 1.

In a survey of 400 people of a community, the numbers with only a motorcycle license and only a car license were in the ratio 5 : 3. One-fourth of the people had both licenses, and 60 had none.

a.Show the information in a Venn diagram.
b.How many people had each license?
c.Find the number who had only a motorcycle license.

Answer

Let M and C be the people with a motorcycle license and a car license.
Both: $\frac{1}{4} \times 400 = 100$ . Let n₀(M) = 5x, n₀(C) = 3x: 5x + 100 + 3x + 60 = 400, so 8x = 240 and x = 30.

Figure 1.46 Mixed exercise, question 5.

b.Motorcycle: 150 + 100 = 250; car: 90 + 100 = 190.
c.Only motorcycle: 150.

In a school, 100 students like football, 80 volleyball and 120 cricket. 30 like football and volleyball, 20 volleyball and cricket, and 35 football and cricket. 10 like all three games, and 5 like none of them.

a.Write the information in cardinality notation.
b.Show it in a Venn diagram.
c.Find the total number of students in the school.
d.What percentage of the students like football only?

Answer

Let F, V and C be the students who like football, volleyball and cricket.

a.n(F) = 100, n(V) = 80, n(C) = 120, n(F ∩ V) = 30, n(V ∩ C) = 20, n(F ∩ C) = 35, n(F ∩ V ∩ C) = 10, n(F ∪ V ∪ C) = 5.
b.Pairs only: football and volleyball 30 − 10 = 20, volleyball and cricket 20 − 10 = 10, football and cricket 35 − 10 = 25. Only one: football 100 − 55 = 45, volleyball 80 − 40 = 40, cricket 120 − 45 = 75.

Figure 1.47 Mixed exercise, question 6.

c.100 + 80 + 120 − 30 − 20 − 35 + 10 + 5 = 230.
d. $\frac{45}{230} \times 100\% \approx$ 19.57%.

A survey of 45 people of a community: 25 speak Nepal Bhasa, 23 Tamang and 15 Maithili. 12 speak Nepal Bhasa and Tamang, 5 Nepal Bhasa and Maithili, and 10 Tamang and Maithili; 4 speak all three.

a.Show the information in a Venn diagram.
b.How many speak none of these three languages?
c.How many speak only one of them?
d.How many speak both Nepal Bhasa and Tamang but not Maithili?

Answer

Let N, T and M be the people who speak Nepal Bhasa, Tamang and Maithili.
Pairs only: 12 − 4 = 8, 10 − 4 = 6, 5 − 4 = 1. Only one: Nepal Bhasa 25 − 13 = 12, Tamang 23 − 18 = 5, Maithili 15 − 11 = 4.

Figure 1.48 Mixed exercise, question 7.

b.n(N ∪ T ∪ M) = 25 + 23 + 15 − 12 − 5 − 10 + 4 = 40, so none: 5.
c.12 + 5 + 4 = 21.
d.12 − 4 = 8.