Test Level 1: Set Theory- 1 (May 14) - CAT MCQ

Let A be the set of persons who take tea and B be the set of those who take coffee. We are given that n(A ∪ B) = 50, n(A) = 35 and n(B) = 25.
⇒ 50 = 35 + 25 – n(A ∩ B)
⇒ (A ∩ B) = 10
Hence, n(A – B) = n(A) – n(A ∩ B) = 35 – 10 = 25

A ∩ B = Elements that belong to both A and B
∴ Elements of A ∩ B should be multiples of 3 and 5 or we can say that elements of A ∩ B are multiples of 15.
Hence, A ∩ B = {x | x is a multiple of 15} = {15, 30, 45, ...}

n(A) = [1000/12] = 83 (A is a set of integers between 1 to 1000 is divisible by 12.) 
n(B) = [1000/18] = 55 (B is a set of integers between 1 to 1000 divisible by 18, where [x] is the greatest integer function.)
A ∩ B = Set of integers between 1 and 1000 which are divisible by 12 and 18 both, i.e. divisible by 36
n(A ∩ B) = [1000/36] = 27
n(A  B) = n(A) + n(B) – n(A ∩ B) = 83 + 55 – 27 = 111

Number of people who eat non-vegetarian food, but do not eat mutton or chicken = 600 – 200 – 200 – 100 = 100

Since the dailies are sold in equal number, the total sale of each may be taken as x.
Hence, from the diagram, if we add up all the ''only'' regions, we get:
(x - 54) + (x - 48) + (x - 39) + 9 + 27 + 18 + 12 = 906
⇒ x = 327

Number of persons buying Hindu and Express only = 18
Number of persons buying Hindu and Mail only = 27
Number of persons buying Express and Mail only = 12
Required ratio = 18 : 27 : 12 = 6 : 9 : 4

Number of people in the group = 400
270 + 50 + x = 400
x = 80

Total number of egg or meat eaters
= 1700 + 1500 + 1000 = 4200
Number of pure vegetarians = 5000 – 4200 = 800

Total number of people in the group = 70
Number of people who like coffee = 37
Number of people who like tea = 52
Number of people who like both coffee and tea = x
So, 70 = [37 - x + x + 52 - x]
x = 19

Highest number of families not using either a mixer or an air conditioner = 100 - 66 = 34%

From the Venn diagram,
(24 + 25 + 10) i.e. 59% passed in one or both.
∴ 41% failed in both.