Question

A group of 140 students is polled to see how many watched three TV shows, Awesome, Bongo and Crikey. The results showed that 66 watched Awesome, 63 watched Bongo, 63 watched Crikey, 34 watched Awesome and Bongo, 31 watched Awesome and Crikey, 26 watched Bongo and Crikey, and 27 did not watch any of the three. Let A denote the set of students who watched Awesome, and similarly define sets B and C. (a) Calculate the number of students in each of the eight subsets shown in the Venn diagram. Copy the Venn diagram and enter the number of students in each subset. (b) Hence find how many students watched: (i) Awesome and Crikey, but not Bongo; (ii) Bongo only; (iii) only two of the three shows; (iv) at least two of the shows.

Answer 1

I have provided the Venn diagram in the image.

Now we can answer the questions as follows from the venn diagram.

a) Bongo only :

63 - (26 + 34) = 3

b) Only two of the three shows

It could be :

(A and B) or (A and C) or (C and B)

= 34 + 26 + 31 = 91

c) At least two of the three shows :

It means :

Those who watched two or three shows.

Those who watched none were 27

This leaves us with 140 - 27 = 113

Those who watched 2 or 1 show are :

(91 + 1 + 3 + 6) = 101

Those who watched two = 91

Those who watched three are thus :

113 - 101 = 12

Those who watched at least two are thus :

91 + 12 = 103

= 103 students