Set theory Level - 1
Q1–10 of 45In a survey of a city, it was found that 90 percent of the people in the city own a refrigerator and 15 percent own a washing machine. If everybody owns at least one appliance, what percentage owns both?
n(A) = 90, n(B) = 15, n(A ∪ B) = 100
n(A ∩ B) = n(A) + n(B) − n(A ∪ B)
= 90 + 15 − 100
= 5
Percentage of people who own both = 5%
In an examination, 34% of the students failed in Mathematics and 42% failed in English. If 20% of the students failed in both the subjects, then the percentage of students who passed in both the subjects was
n(A) = 34, n(B) = 42, n(A ∩ B) = 20
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
= 34 + 42 − 20
= 56
Percentage failed in either or both subjects = 56%
Percentage passed = (100 − 56)% = 44%
40% of the people read newspaper X, 50% read newspaper Y and 10% read both the papers. What percentage of the people read neither newspaper?
n(A) = 40, n(B) = 50, n(A ∩ B) = 10
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
= 40 + 50 − 10
= 80
Percentage reading either or both newspapers = 80%
Percentage reading neither newspaper = (100 − 80)% = 20%
Out of 450 students of a school, 325 play football, 175 play cricket and 50 neither play football nor cricket. How many students play both football and cricket?
n(A) = 325, n(B) = 175, n(A ∪ B) = 400
n(A ∩ B) = n(A) + n(B) − n(A ∪ B)
= 325 + 175 − 400
= 100
In a hotel, 60% had vegetarian lunch while 30% had non-vegetarian lunch and 15% had both types of lunch. If 96 people were present, how many did not eat either type of lunch?
Total people = 96
People having Lunch A = 60%
People having Lunch B = 30%
People having both lunches = 15
n(A) = 60% of 96 = 288/5
n(B) = 30% of 96 = 144/5
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
= 288/5 + 144/5 − 72/5
= 360/5
= 72
People who had either or both types of lunch = 72
People who had neither type of lunch = 96 − 72 = 24
There are 600 boys in a hostel. Each plays either hockey or football or both. If 75% play hockey and 45% play football, how many play both?
n(A) = 450
n(B) = 270
n(A ∪ B) = 600
n(A ∩ B) = n(A) + n(B) − n(A ∪ B)
= 450 + 270 − 600
= 120
In a certain office, 72% of the workers prefer tea and 44% prefer coffee. If each of them prefers tea or coffee and 40 like both, the total number of workers in the office is
Let total number be x
n(A) = 72% of x = 18x/25
n(B) = 44% of x = 11x/25
n(A ∩ B) = 40
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
18x/25 + 11x/25 − 40 = x
29x/25 − x = 40
4x/25 = 40
x = 250
In an examination, 30% and 35% students respectively failed in History and Geography while 27% students failed in both the subjects. If the number of students passing the examination is 248, find the total number of students who appeared in the examination.
Percentage of failed candidates
= (30 + 35 − 27)% = 38%
Percentage of passed candidates
= (100 − 38)% = 62%
Let total number of students appeared be x
62% of x = 248
x = (248 × 100) / 62
x = 400
In an examination, 35% candidates failed in one subject and 42% failed in another subject while 15% failed in both the subjects. If 2500 candidates appeared at the examination, how many passed in either subject but not in both?
Total students = 2500
Failed in 1st subject = 35% of 2500 = 875
Failed in 2nd subject = 42% of 2500 = 1050
Failed in both subjects = 15% of 2500 = 375
Failed in 1st subject only = 875 − 375 = 500
Failed in 2nd subject only = 1050 − 375 = 675
Passed in 1st only + Passed in 2nd only
= 500 + 675
= 1175
In a town, 65% people watched the news on television, 40% read a newspaper and 25% read a newspaper and watched the news on television also. What percent of the people neither watched the news on television nor read a newspaper?
Percentage of people watching news on television = 65%
Percentage of people reading newspaper = 40%
Percentage of people doing both = 25%
Using the formula:
Total = TV + Newspaper − Both
Total = 65 + 40 − 25
Total = 80%
Therefore, percentage of people who neither watched TV nor read newspaper:
= 100 − 80
= 20%
Answer: 20%