DilipKumar

Two Sets | Grade 10 Maths

Word Problems

Q. 4 Survey – Educational Tour (Lumbini & Pokhara)

A survey was conducted among 125 students of class 10 studying in Gyan Mandir Secondary School regarding the places for educational tour. It was found that 65 students preferred to visit Lumbini (L), 75 preferred to visit Pokhara (P), and 25 preferred both the places.

a)Write the cardinality of set of students who preferred to visit both Pokhara and Lumbini in set notational form.

b)Show the above information in a Venn-diagram.

c)Find the number of students who preferred neither of two places.

d)Find the ratio of the number of students who preferred Lumbini only and Pokhara only.

Solution

Given n(U) = 125 | n(L) = 65 | n(P) = 75 | n(L ∩ P) = 25

The cardinality of the set of students who preferred to visit both Pokhara and Lumbini:

n(P ∩ L) = 25

∴ n(P ∩ L) = 25

Venn Diagram

n_o(L) = 40 | n(L∩P) = 25 | n_o(P) = 50 | outside = 10

n(P ∪ L) = n(P) + n(L) − n(P ∩ L)
= 75 + 65 − 25 = 115

n(P̄ ∩ L̄) = n(U) − n(P ∪ L) = 125 − 115 = 10

∴ 10 students preferred neither of the two places.

n_o(L) = n(L) − n(L∩P) = 65 − 25 = 40

n_o(P) = n(P) − n(L∩P) = 75 − 25 = 50

Ratio n_o(L) : n_o(P) = 40 : 50 = 4 : 5

Q. 4(b) Survey – Local Languages (Bhojpuri & Maithili)

A survey was carried out in a village of Chandrapur municipality of Rautahat district regarding the local languages. Among 1,400 people who participated in the survey, it was found that 600 people can speak Bhojpuri, 900 can speak Maithili, and 350 people can speak both Bhojpuri and Maithili.

a)If B and M represent the sets of people who can speak Bhojpuri and Maithili respectively, write the cardinality of n(B ∩ M).

b)Draw a Venn-diagram to show the above data.

c)How many people cannot speak both the languages?

d)How many more or less people can speak Maithili only than Bhojpuri only?

Solution

Given n(U) = 1400 | n(B) = 600 | n(M) = 900 | n(B ∩ M) = 350

n(B ∩ M) = 350

∴ n(B ∩ M) = 350

Venn Diagram

n_o(B) = 250 | n(B∩M) = 350 | n_o(M) = 550 | outside = 250

n(B ∪ M) = n(B) + n(M) − n(B ∩ M)
= 600 + 900 − 350 = 1150

n(B̄ ∩ M̄) = n(U) − n(B ∪ M) = 1400 − 1150 = 250

∴ 250 people cannot speak both languages.

n_o(B) = 600 − 350 = 250

n_o(M) = 900 − 350 = 550

Difference = 550 − 250 = 300

∴ The number of people who can speak Maithili only is 300 more than those who can speak Bhojpuri only.

Q. 4(c) Visit Nepal 2020 – Chinese Tourists (Bhutan & Sri Lanka)

During 'Visit Nepal 2020', among 7,500 Chinese tourists who visited Nepal, 60% had already visited Bhutan, 50% had visited Sri Lanka, and 30% had visited both the countries.

a)If B and S denote the sets of tourists who visited Bhutan and Sri Lanka respectively, write the number of tourists who visited Bhutan in set notation.

b)Illustrate this information in a Venn-diagram.

c)How many tourists have visited neither Bhutan nor Sri Lanka?

d)How many tourists have already visited only one of these countries?

Solution

Given n(U) = 7500 | n(B) = 60% of 7500 = 4500 | n(S) = 50% of 7500 = 3750 | n(B ∩ S) = 30% of 7500 = 2250

n(B) = 60% of 7500 = 4500

∴ n(B) = 4500

Venn Diagram

n_o(B) = 2250 | n(B∩S) = 2250 | n_o(S) = 1500 | outside = 1500

n(B ∪ S) = 4500 + 3750 − 2250 = 6000

n(B̄ ∩ S̄) = 7500 − 6000 = 1500

∴ 1500 tourists visited neither country.

n_o(B) = 4500 − 2250 = 2250

n_o(S) = 3750 − 2250 = 1500

Only one country = 2250 + 1500 = 3750

Q. 5(a) Survey – Cultural Dances (Sorathi & Ghatu)

In a survey conducted in a community of Gorkha district regarding cultural dances, it was found that out of 500 people, 180 people can perform Sorathi dance (S), 250 can perform Ghatu dance (G) and 120 can perform neither of these two dances.

a)Write the set notation that represents the number of people who can perform neither Sorathi nor Ghatu dance.

b)Represent the above information in a Venn-diagram.

c)How many people can perform both the dances?

d)Compare the number of people who can perform Sorathi only and Ghatu only.

Solution

Given n(U) = 500 | n(S) = 180 | n(G) = 250 | n(S̄ ∩ Ḡ) = 120

The set notation is n(S̄ ∩ Ḡ) = 120

Venn Diagram

n_o(S) = 130 | n(S∩G) = 50 | n_o(G) = 200 | outside = 120

Let n(S ∩ G) = x

n(U) = n_o(S) + n_o(G) + n(S∩G) + n(S̄∩Ḡ)
500 = (180−x) + (250−x) + x + 120
500 = 550 − x ⟹ x = 50

∴ 50 people can perform both dances.

n_o(S) = 180 − 50 = 130

n_o(G) = 250 − 50 = 200

Difference = 200 − 130 = 70

∴ Sorathi only is 70 less than Ghatu only. Ratio = 13 : 20

Q. 5(b) School's Day – Athletics & Music

Last Wednesday, a School successfully accomplished its school's day. For this program, every student participated in at least one of the activities, athletics or music. In a class of 45 students, 21 participated in athletics (A) and 29 participated in music (M).

a)Write the cardinality of n(Ā ∩ M̄).

b)Draw a Venn-diagram to represent the above information.

c)How many students participated in both the activities?

d)Compare the number of students who participated in athletics only and music only.

Solution

Given n(U) = 45 | n(A) = 21 | n(M) = 29 | n(Ā ∩ M̄) = 0 (every student participates in at least one)

Since every student participated in at least one activity, n(Ā ∩ M̄) = 0

Venn Diagram

n_o(A) = 16 | n(A∩M) = 5 | n_o(M) = 24 | no outside region (n(Ā∩M̄) = 0)

Let n(A ∩ M) = x

45 = (21−x) + (29−x) + x + 0
45 = 50 − x ⟹ x = 5

∴ 5 students participated in both activities.

n_o(A) = 21 − 5 = 16

n_o(M) = 29 − 5 = 24

Difference = 24 − 16 = 8 | Ratio = 2 : 3

∴ Music only is 8 more than Athletics only.

Q. 5(c) Survey – Cricket & Basketball

In a survey conducted among 150 students of a school, it was found that 70 students liked cricket, 62 liked basketball and 30 students did not like any of these two games.

a)If C and B denote the set of students who liked cricket and basketball respectively, write the cardinality of n(B̄ ∩ C̄).

b)Present the information in a Venn-diagram.

c)Find the number of students who liked cricket only.

d)Compare the number of students who liked both games and who liked except these two games.

Solution

Given n(U) = 150 | n(C) = 70 | n(B) = 62 | n(B̄ ∩ C̄) = 30

n(B̄ ∩ C̄) = 30

Venn Diagram

n_o(B) = 50 | n(B∩C) = 12 | n_o(C) = 58 | outside = 30

Let n(B ∩ C) = x

150 = (62−x) + (70−x) + x + 30
150 = 162 − x ⟹ x = 12

n_o(C) = 70 − 12 = 58

n(B ∩ C) = 12 n(B̄ ∩ C̄) = 30

Difference = 30 − 12 = 18

∴ Students who liked both is 18 less than those who liked neither. Ratio = 2 : 5

Q. 6(a) SEE Students – Mathematics & English (A+ Grade)

Among 54 SEE appeared students from a school, 18 students got 'A+' grade in Mathematics only, 25 got 'A+' grade in English only and 7 students did not get 'A+' grade in either of these two subjects.

a)Write the cardinalities of the given sets in set notational forms.

b)Show the above information in a Venn-diagram.

c)How many students got 'A+' grade in Mathematics?

d)How many more or less students got 'A+' grade not in English than not in Mathematics?

Solution

Given n(U) = 54 | n_o(M) = 18 | n_o(E) = 25 | n(M̄ ∩ Ē) = 7

Let M = set of students who got A+ in Mathematics, E = set who got A+ in English.

n(U) = 54 | n_o(M) = 18 | n_o(E) = 25 | n(M̄ ∩ Ē) = 7

Venn Diagram

n_o(M) = 18 | n(M∩E) = 4 | n_o(E) = 25 | outside = 7

Let n(M ∩ E) = x

54 = 18 + 25 + x + 7
54 = 50 + x ⟹ x = 4

n(M) = n_o(M) + n(M∩E) = 18 + 4 = 22

n(E) = 25 + 4 = 29 → n(Ē) = 54 − 29 = 25

n(M) = 22 → n(M̄) = 54 − 22 = 32

Difference = 32 − 25 = 7

∴ Students not getting A+ in English is 7 less than those not getting A+ in Mathematics.

Q. 6(b) Survey – Cellular Data & Wi-Fi Usage

In a survey of a group of people, 20% are using cellular data but not Wi-Fi, 65% are using Wi-Fi but not cellular data and 5% of them use neither cellular data nor Wi-Fi.

a)If C and W denote the sets of people using cellular data and Wi-Fi respectively, write the cardinality of n(C̄ ∩ W̄).

b)Represent the above information in a Venn-diagram.

c)Find the percent of people who are using Wi-Fi.

d)By what percent is the number of people not using cellular data more or less than those not using Wi-Fi?

Solution

Given (let n(U) = 100) n_o(C) = 20 | n_o(W) = 65 | n(C̄ ∩ W̄) = 5

n(C̄ ∩ W̄) = 5

Venn Diagram (n(U) = 100)

n_o(C) = 20 | n(C∩W) = 10 | n_o(W) = 65 | outside = 5

Let n(C ∩ W) = x

100 = 20 + 65 + x + 5 = 90 + x ⟹ x = 10

n(W) = 65 + 10 = 75%

n(C) = 20 + 10 = 30% → n(C̄) = 70%

n(W) = 75% → n(W̄) = 25%

Difference = 70 − 25 = 45% more

∴ People not using cellular data is 45% more than those not using Wi-Fi.

Q. 8(a) Survey – Comedy & Action Movies

In a survey of a group of people, it was found that 65% liked comedy movies, 63% liked action movies, 33% liked both types of movies, and 150 people did not like either type of movie.

a)Draw a Venn-diagram to illustrate the above information.

b)Find the number of people who participated in the survey.

c)Find the number of people who liked both types of movies.

d)Amrita calculated that the number of people who did not like comedy is 150 more than the number who liked action only. Justify her calculation.

Solution

Given (let n(U) = x) n(C) = 0.65x | n(A) = 0.63x | n(C ∩ A) = 0.33x | n(C̄ ∩ Ā) = 150

Venn Diagram (n(U) = x)

x = 0.32x + 0.30x + 0.33x + 150
x = 0.95x + 150
0.05x = 150 ⟹ x = 3000

∴ 3000 people participated in the survey.

n(C ∩ A) = 0.33 × 3000 = 990

∴ 990 people liked both types of movies.

n(C̄) = 3000 − 0.65×3000 = 3000 − 1950 = 1050

n_o(A) = 0.30 × 3000 = 900

1050 − 900 = 150 ✓

∴ Amrita's calculation is correct.

Q. 8(b) Examination – English & Mathematics Pass

In an examination, 80% examinees passed in English, 70% in Mathematics, 60% passed in both the subjects, and 45 examinees failed in both subjects.

a)Draw a Venn-diagram to represent the above information.

b)Find the number of examinees who participated in the examination.

c)Find the number of examinees who passed in both subjects.

d)Ram calculated that the number of examinees who failed in English is double the number who passed Mathematics only. State whether he is right or wrong.

Solution

Given (let n(U) = x) n(E) = 0.8x | n(M) = 0.7x | n(E ∩ M) = 0.6x | n(Ē ∩ M̄) = 45

Venn Diagram (n(U) = x)

x = 0.2x + 0.1x + 0.6x + 45
x = 0.9x + 45
0.1x = 45 ⟹ x = 450

∴ 450 examinees participated.

n(E ∩ M) = 0.6 × 450 = 270

∴ 270 examinees passed both subjects.

n(Ē) = 450 − 0.8×450 = 450 − 360 = 90

n_o(M) = 0.1 × 450 = 45

n(Ē) ÷ n_o(M) = 90 ÷ 45 = 2 ∴ n(Ē) = 2 × n_o(M) Ram is correct ✓

Q. 9(a) Survey – Folk Songs & Modern Songs

In a survey of a community, it was found that 65% of people liked folk songs, 55% liked modern songs and 10% of people did not like both types of songs.

a)Illustrate the above information in a Venn-diagram.

b)What percentage of people liked either folk songs or modern songs?

c)If 360 people liked both types of songs, how many people were surveyed?

d)How many times is the number of people who liked both types of songs compared to those who did not like either?

Solution

Given (let n(U) = 100) n(F) = 65 | n(M) = 55 | n(F̄ ∩ M̄) = 10

Venn Diagram (n(U) = 100)

n_o(F) = 35 | n(F∩M) = 30 | n_o(M) = 25 | outside = 10

n(F ∪ M) = n(U) − n(F̄ ∩ M̄) = 100 − 10 = 90

∴ 90% of people liked either folk or modern songs.

Let n(F ∩ M) = x

100 = (65−x) + (55−x) + x + 10
100 = 130 − x ⟹ x = 30%

30% of total = 360 ⟹ Total = 360 ÷ 0.30 = 1200 people

n(F ∩ M) = 30% of 1200 = 360

n(F̄ ∩ M̄) = 10% of 1200 = 120

360 ÷ 120 = 3 times

∴ Those liking both is 3 times those liking neither.

Q. 9(b) Survey – Rice & Wheat Cultivation

In a survey of some farmers in a community, 70% are found cultivating rice, 60% cultivating wheat, 20% are not cultivating both the crops, and 450 farmers are cultivating both the crops.

a)Draw a Venn-diagram to illustrate the above information.

b)Find the percentage of farmers who are cultivating either rice or wheat.

c)Find the total number of farmers who participated in the survey.

d)By how many times is the number of farmers cultivating both crops more than those cultivating none?

Solution

Given (let n(U) = 100) n(R) = 70 | n(W) = 60 | n(R̄ ∩ W̄) = 20 | n(R ∩ W) = 450 (actual)

Venn Diagram (n(U) = 100)

n_o(R) = 20 | n(R∩W) = 50 | n_o(W) = 10 | outside = 20

n(R ∪ W) = 100 − n(R̄ ∩ W̄) = 100 − 20 = 80

∴ 80% of farmers cultivate either rice or wheat.

Let n(R ∩ W) = x

100 = (70−x) + (60−x) + x + 20
100 = 150 − x ⟹ x = 50%

50% of total = 450 ⟹ Total = 900 farmers

n(R ∩ W) = 450

n(R̄ ∩ W̄) = 20% of 900 = 180

450 ÷ 180 = 2.5 times

∴ Farmers cultivating both is 2.5 times those cultivating neither.

Dilip Kumar Khatiwada

Exercise 1.1

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