VENN DIAGRAMS – 3 SETS (REAL WORLD)

Junior Cert Revision

Each of the students in sixth year in a particular school has WhatsApp (𝑊), Instagram (𝐼), or Snapchat (𝑆). The numbers who have each app are as follows:

36 students have WhatsApp40 students have Instagram54 students have Snapchat14 students have WhatsApp and Instagram24 students have Instagram and Snapchat𝑥 students have WhatsApp and Snapchat, but not Instagram8 students have all three apps.

Use this information to fill in the Venn diagram below, in terms of 𝑥.

2017 JCHL Paper 1 – Question 6 (a)

36 40

54

8𝑥

6

16

1022 − 𝑥

30 − 𝑥

14 students have WhatsApp and Instagram# 𝑊 ∩ 𝐼 \ 𝑆 = 624 students have Instagram and Snapchat# 𝐼 ∩ 𝑆 \ 𝑊 = 16

There are 80 students in total in sixth year in the school.Find the value of 𝑥.

2017 JCHL Paper 1 – Question 6 (b)

22 − 𝑥 + 6 + 8 − 𝑥 + 10 + 16 + 30 − 𝑥 = 80−𝑥 = −12𝑥 = 12

The sum of all the sections of the Venn Diagram is equal to the total number of students.

36 40

54

8𝑥

6

16

1022 − 𝑥

30 − 𝑥

36 + 10 + 16 + 30 − 𝑥 = 80−𝑥 = −12𝑥 = 12

Alternate Method:We know that there are 36 in the WhatApp Circle so:

The table below shows four statements. Each statement is written in mathematical notation and in English. Complete the table.

2017 JCHL Paper 1 – Question 6 (c)

8 students have WhatsApp, Instagram and Snapchat

24 = # 𝐼 ∩ 𝑆

10 students have Instagram but NOT WhatsApp or Snapchat

#𝑆 > #𝑊

A researcher has found old census data about Measles (𝑀), Chickenpox (𝐶),and Whooping cough (𝑊) among 12-year-old children. In a group of 100 children:31 had none of these diseases2 had all three diseases2 had Measles and Chickenpox, but not Whooping cough6 had Whooping cough and Chickenpox11 had at least two diseases18 had Measles40 had Chickenpox.

Use this data to fill in the Venn diagram.

2015 JCHL Paper 1 – Question 10 (a)

At least two includes the sections of the Venn Diagram that are exactly 2 as well as the section that includes all 3.

31

2

24

100

1840

3

11 32

1524

100 − 18 + 32 + 4 + 31 = 15

Find the probability that a child chosen at random from the group had Chickenpox.

2015 JCHL Paper 1 – Question 10 (b)

𝑃 Chickenpox =40

100

=2

5 31

2

24

100

1840

3

11 32

1524

Complete the table.

2015 JCHL Paper 1 – Question 10 (c)

36 had Chickenpox but not Measles

2 = # 𝑀 ∩ 𝐶 \ 𝑊

31

2

24

100

1840

3

11 32

1524

𝐶𝑇

𝑈

𝐷

4

24

8 5

326 22

8

A group of 100 students were surveyed to find out whether they drank tea (𝑇), coffee (𝐶), or a soft drink (𝐷) at any time in the previous week. These are the results:

24 had not drunk any of the three51 drank tea or coffee, but not a soft drink41 drank tea8 drank tea and a soft drink, but not coffee9 drank a soft drink and coffee20 drank at least two of the three4 drank all three.

Represent the above information on the Venn diagram.

2015 Sample JCHL Paper 1 – Question 3 (a)

41

At least two includes the sections of the Venn Diagram that are exactly 2 as well as the section that includes all 3.

20 − 8 + 4 + 5 = 3

51 − 26 + 3 = 22

51 drank tea or coffee, but not a soft drink.

100

The amount in D but not in C or T.

100 − 41 + 22 + 5 + 24 = 8

68

100=

17

25

3

100

Find the probability that a student chosen at random from the group had drunk tea and coffee but not a soft drink.

(c)

𝐶𝑇

𝑈

𝐷

4

24

8 5

326 22

8

Find the probability that a student chosen at random from the group had drunk tea or coffee.

2015 Sample JCHL Paper 1 – Question 3 (b)

A group of 100 students were asked if they had a presence on particular socialnetworking websites 𝐴, 𝐵 and 𝐶.24 students had a presence on 𝐴 only, 40 had a presence on 𝐵 and 50 had a presence on 𝐶.14 students had a presence on 𝐴 and 𝐵 but not on 𝐶.18 students had a presence on 𝐴 and 𝐶 but not on 𝐵.8 students had a presence on 𝐵 and 𝐶 but not on 𝐴.4 students stated that they did not have a presence on any of the websites.

Using 𝑥 to represent the number of students who had a presence on all three websites, construct a Venn diagram and solve for 𝑥.

2010 JCHL Paper 1 – Question 2 (b) (i)

𝐵𝐴

𝑈

𝐶

𝑥

4

18 8

1424 18 − 𝑥

24 − 𝑥

100

40

50

50 + 24 + 14 + 18 − 𝑥 + 4 = 100

−𝑥 = −10

𝑥 = 10

We know that there are 50 in C so:

Hence, calculate the ratio of students with a presence on 𝐵 only to the students with a presence on 𝐶 only.

2010 JCHL Paper 1 – Question 2 (b) (ii)

𝐵 Only: 𝐶 Only

8: 14

4: 7

𝐵𝐴

𝑈

𝐶

10

4

18 8

1424 8

14

100

40

50

Start by updating the diagram now that we know that 𝑥 = 10

A group of 49 students was asked which fruit each liked.28 said they liked apples. 25 said they liked pears while 26 said they liked oranges.8 said they liked all three types of fruit.17 said they liked pears and oranges. 11 said they liked apples and oranges.5 said they did not like any of the three types of fruit.Let 𝑥 represent those students who liked apples and pears but not oranges.

Represent the above information on a Venn diagram.

2009 JCHL Paper 1 – Question 2 (b) (i)

𝐴 28 𝑃 25

𝑂 26

8

17 − 𝑥 𝑥8 − 𝑥

6

3 9

𝑈 49

5

Calculate the value of 𝑥.

2009 JCHL Paper 1 – Question 2 (b) (ii)

𝐴 28 𝑃 25

𝑂 26

8

17 − 𝑥 𝑥8 − 𝑥

6

3 9

𝑈 49

528 + 8 − 𝑥 + 9 + 6 + 5 = 49

−𝑥 = −7

𝑥 = 7

Shorter Method:We know that there are 28 in the Apple Circle so:

Calculate the percentage of students who liked one type of fruit only.Give your answer correct to the nearest whole number.

2009 JCHL Paper 1 – Question 2 (b) (ii)

𝐴 28 𝑃 25

𝑂 26

8

10 71

6

3 9

𝑈 49

5

Start by updating the diagram now that we know that 𝑥 = 7

P 1 One Fruit Only =10 + 1 + 6

49

P 1 One Fruit Only =17

49

A survey of 40 students was carried out to find how many owned an MP3 player, a digital camera or a CD player.1 student does not own any of these.𝑥 students own all three, while 2𝑥 own an MP3 player and a digital camera but not a CD player.10 own an MP3 player and a CD player, while 11 own a digital camera and a CD player.22 own an MP3 player, 22 own a digital camera and 24 own a CD player.

Construct a Venn diagram and solve for 𝑥.

2007 JCHL Paper 1 – Question 2 (c) (i)

𝑀 22 𝐷𝐶 22

𝐶𝐷 24

𝑥

12 − 2𝑥2𝑥

11 − 2𝑥

3 + 𝑥

10 − 𝑥

𝑈 40

111 − 𝑥

Just 𝐶𝐷= 24 − 10 − 𝑥 + 𝑥 + 11 − 𝑥= 24 − 21 − 𝑥= 3 + 𝑥

22 + 11 − 2𝑥 + 11 − 𝑥 + 3 + 𝑥 + 1 = 40−2𝑥 = −8𝑥 = 4

Just 𝑀𝑃3= 22 − 10 − 2𝑥= 12 − 2𝑥

Just 𝐷𝐶= 22 − 11 − 2𝑥= 11 − 2𝑥

We know that there are 28 with MP3’s so:

Hence, calculate the percentage of students who own one item only.

2007 JCHL Paper 1 – Question 2 (c) (ii)

𝑀 22 𝐷𝐶 22

𝐶𝐷 24

4

4 8 3

7

6

𝑈 40

1

7One item only 4 + 3 + 7 = 14

14

40× 100 = 35%

Start by updating the diagram now that we know that 𝑥 = 4

70 teenagers responded to a survey about holiday destinations.30 had travelled to France,26 had travelled to Spainand 28 had travelled to Italy.12 had travelled to both France and Spain,8 had travelled to both Spain and Italy,while 𝑥 had travelled to France and Italy only.4 teenagers had travelled to all three countries.Twice as many had never travelled to any of these destinations as had travelled to France and Italy only.

Represent the above information on a Venn diagram.

2006 JCHL Paper 1 – Question 2 (b) (i)

𝐹 30 𝑆 26

𝐼 28

4

18 − 𝑥 810

20 − 𝑥

𝑥 4

𝑈 70

2𝑥

Find the number of teenagers who had travelled to France only.

2006 JCHL Paper 1 – Question 2 (b) (ii)

30 + 10 + 4 + 20 − 𝑥 + 2𝑥 = 70

2𝑥 − 𝑥 = 70 − 30 − 10 − 4 − 20

𝑥 = 6

Shorter Method:We know that there are 30 in the France Circle so:

𝐹 30 𝑆 26

𝐼 28

4

18 − 𝑥 810

20 − 𝑥

𝑥 4

𝑈 70

2𝑥