Word problems on Sets, venn diagrams

Question 21

Question:   In a class of 80 boys, there are 60 boys who play chess and 35 play table tennis. Use Venn diagrams to show how many boys play both the games? How many play chess only and how many play tennis only?

Answer:   
Let n(C) = number of boys of who play chess
n(T) = number of boys who play table tennis
Let x = number of boys who play both the games
From the above diagram it is clear that,

Those who play chess only = 60 - x = 60 - 15 = 45
Those who play table tennis only = 35 - 15 = 20
To find:

Question 22

Question:   At dinner, seven people ordered the special of the day. Three people order from the list of the menu, one person orders only salad. Find the number 'm' (dinners ordered by the people).

Answer:    The sets are disjoint. (assuming no one orders more than one dinner) m = 7 + 3 + 1 = 10, the total number of dinners ordered.

Question 23

Question:   A survey shows that 71% of Indians like to watch cricket, whereas 64% like to watch hockey. What percentage of Indians like to watch both cricket and hockey? (Assuming that every indian watches at least one of these games)

Answer:    Let n(C) = percentage of Indians who watch cricket and
n (H) = percentage of Indians who watch hockey.

Hence 35% of Indians like to watch both cricket and hockey.

Question 24

Question:   In a group of 80 persons, 30 drink Fanta but not Limca and 41 drink Fanta.
1. How many drink Fanta and Limca both?
2. How many drink Limca, but not Fanta?

Answer:   
Let n(F) = number who drink Fanta
n(L) = number who drink Limca


Now,


9 persons drink Fanta and Limca both.

\ 39 people drink Limca but not Fanta.

Question 25

Question:   Out of 120 students who secured first class marks in Mathematics or English, 70 obtained first class in Mathematics and 31 in English and Mathematics. How many students secured first class marks in English only?

Answer:   


Number of people who secured first class marks in English only is

Question 26

Question:   In a group of 1500 people, 950 can speak Hindi and 650 can speak both Hindi and Kannada. How many can speak Hindi only? How many can speak Kannada only?

Answer:    Let




Number who can speak Hindi only=

Number who can speak Kannada only=

Question 27

Question:   A class has 175 students. The following is the description of students studying one or more of the following subjects in this class. Mathematics 100, Physics 70, Chemistry 46, Mathematics and Physics 30; Mathematics and Chemistry 28; Physics and Chemistry 23; Mathematics, Physics and Chemistry 18. How many students are enrolled in Mathematics alone, Physics alone and Chemistry alone?

Answer:   
Let M = Set of students who study Mathematics
P = Set of students who study Physics
C = Set of students who study Chemistry
Let us denote the number of students in each bounded region by the letter a, b, c, d, e, f, g as shown in the figure.





Solving these equations we get,


Number of students who have not offered any of these three subject = 175 - 153 = 22
Students enrolled in Mathematics only = e = 60
Students enrolled in Chemistry only = g = 13

Question 28

Question:   Represent the following using Venn diagram
A = {2, 3, 4, 5} B= {2, 3}

Answer:   

A is a subset of B.

Question 29

Question:   Represent the following using Venn diagrams:
(i)
(ii)
(iii) A - B and B - A
(iv) A^l

Answer:   



(iii) A - B and B - A

(iv) A^l

Question 30

Question:  

Answer:   

Furthermore,

Hence from (1) and (2),