WORD PROBLEMS ON VENN DIAGRAM OF 2 CIRCLES

Problem 1 :

A group of theater majors were asked whether they were in the Drama club or Music club. The results are summarized in the following Venn diagram.

a) Describe what each region in the Venn diagram represents.

b) How many students were in only one of the two clubs ?

c) How many students were the drama club or the music club ?

d) How many students were surveyed ?

Solution :

a) 

Region 1 : In drama club, not in music club.

Region 2 : In both clubs.

Region 3 : In music club, not in drama.

Region 4 : Not in either clubs.

b)

Drama = 9

Music = 7

Number of students who were in one and only of two clubs

= 9 + 7

= 16

c) The number of students who were the drama club or the music club

= 9 + 15 + 7

= 31

d)  Number of students surveyed 

= 9 + 15 + 7 + 12

= 43 

Problem 2 :

A group of students were asked what pet they had at home. The results are summarized in the following Venn diagram.

a) Describe what each region in the Venn diagram represents.

b) How many students had exactly one of the two ?

c) How many students had at least one of the two ?

Solution :

a) 

Region 1 : In cats, not in dogs

Region 2 : In both pets.

Region 3 : In dogs, not in cats.

Region 4 : Not in either pets.

b)

Cats = 6

Dogs = 8

Number of students who have one of the pet 

= 6 + 8

= 14

c)

Number of student who has atleast one of it

= 6 + 8 + 10

= 24

Problem 3 :

A group of coffee drinkers were asked what the added to their coffee. The results are summarized in the following Venn diagram.

a) Describe what each region in the Venn diagram represents.

b) How many coffee drinkers didn't use either cream or sugar ?

c) How many coffee drinkers used cream or sugar ?

d) How many coffee drinkers only used one of the two ?

Solution :

a) 

Region 1 : In cream, not in sugar.

Region 2 : In both cream and sugar

Region 3 : In sugar, not in cream.

Region 4 : Not in either

b) Coffee drinkers are outside the circles.

So, 0.

c)

16 + 20 + 35 = 71

d)

cream = 16

sugar = 35

= 16 + 35

= 51

Problem 4 :

A gym runs two fitness classes, spinning and circuits.

On Saturday 100 people visited the gym.

18 people attended the spinning class. 10 people attended both classes. 56 people did not attend either class.

a) Represent this information on a Venn diagram. A person who attended the gym is selected at random. Find the probability that this person.

b) attended only circuits.

c) attended exactly one class.

d) attended spinning, given that they attended circuits. 

Solution :

a)

b) Number of people = 100

18 people attended the spinning class.

10 people attended both classes.

56 people did not attend either class.

= 100 - 8 - 10 - 56

= 26

To find the probability :

= 26/100

= 13/50

c) Exactly one class = 8 + 26

= 34

To find the probability :

= 34/100

= 17/50

d) 10 people attended both classes.

Attended circuits = 10 + 26

= 36

= 10/36

= 5/18

Problem 5 :

In a class of 24 students, 12 students play the piano. 13 students play the guitar. 4 students play neither instrument.

a) Represent this information on a Venn diagram

A student is selected at random.

b) Work out the probability that the student only plays the guitar.

Solution :

a)

b) Number of students = 24

Guitar = 8 students

= 8/24

= 1/3

So, the probability that the student only plays the guitar is 1/3.

Problem 6 :

In a company there are 110 workers. 90 workers like tea. 41 workers like coffee. 25 workers like both tea and coffee.

Work out how many workers like neither tea or coffee.

Solution :

So, 4 workers like neither tea or coffee.

Problem 7 :

A group of friends have been surveyed. 38% have been to Canada. 80% have been to France. 11% have been to neither Canada or France.

Find the percentage of the group that have been to Canada and France.

Solution :

So, 29% percentage of the group that have been to Canada and France.

Problem 8 :

The Venn diagram shows information about the cars in a car park.

ξ = 150 cars in the car park

R = red cars

J = cars manufactured in Japan

A car is chosen at random.

Work out the probability that it is red.

Solution :

150 cars in the car park.

2x + 5 + x - 7 + 21 + 68 = 150

3x + 87 = 150

3x = 63

x = 63/3

x = 21

x = 21 substitute the equation x - 7. 

= 21 - 7

= 14

21  + 14 = 35 

To find the probability :

= 35/150

= 7/30