SET THEORY WORD PROBLEMS AND SOLUTIONS WITH 2 CIRCLES

Problem 1 :

In a class, all students take part in either music or drama or both. 25 students take part in music, 30 students take part in drama and 8 students take part in both music and drama. Find

(i) The number of students who take part in only music.

(ii) The number of students who take part in only drama.

(iii) The total number of students in the class.

Solution :

Let "A" and "B" represents the students who participated music and drama respectively.

n(A)  =  25

n(B)  =  30

n(A n B)  =  8

(i) The number of students who take part in only music

=  17

(ii) The number of students who take part in only drama

=  22

(iii) The total number of students in the class

  =  17 + 8 + 22

  =  47

Problem 2 :

In a party of 45 people, each one likes tea or coffee or both. 35 people like tea and 20 people like coffee. Find the number of people who (i) like both tea and coffee. (ii) do not like Tea. (iii) do not like coffee.

Solution :

Let "A" and "B" represents the people who drink tea and coffee respectively. 

n(A)  =  35

n(B)  =  20

n(A U B)  =  45

Total number of students  =  35 - x + x + 20 - x

55 - x  =  45

 -x  =  4 - 55

x  =  10

(i) like both tea and coffee  =  10

(ii) do not like Tea  =  20 - x (like coffee)

  =  20 - 10 

  =  10

(iii) do not like coffee  =  35 - x

  =  35 - 10 

  =  25

Problem 3 :

In an examination 50% of the students passed in Mathematics and 70% of students passed in Science while 10% students failed in both subjects. 300 students passed in both the subjects. Find the total number of students who appeared in the examination, if they took examination in only two subjects

Solution :

Number of students passed in both the subjects  =  300

Percentage of students passed in math  =  50% 

Percentage of students passed in science  =  70%

Percentage of students failed  =  10%

Percentage of students passed  =  90%

percentage of students passed in atleast one subject

  =  (50 + 70) - 90

  =  120 - 90

  =  30%

30% of x  =  300

x  =  300(100/30)

x  =  1000

So, the total number of students is 1000.

Problem 4 :

A and B are two sets such that n(A – B) = 32 + x, n(B – A) = 5x and n(A∩B) = x. Illustrate the information by means of a Venn diagram. Given that n(A) = n(B), calculate the value of x.

Solution :

Formulas for n(A) and n(B):

n(A)  =  n(A–B) + n(A∩B)

n(B)  =  n(B–A) + n(A∩B)

n(A – B) = 32 + x, n(B – A) = 5x and n(A∩B) = x

n(A)  =  32 + x + x

n(A)  =  32 + 2x  ----(1)

n(B)  =  5x + x

n(B)  =  6x  ------(2)

(1)  =  (2)

32 + 2x  =  6x

32  =  6x - 2x

4x  =  32

x  =  8

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