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Answer the next 5 questions based on the information given below.
A survey was conducted in a class to find out the preference of its students. A total of 230 students liked Cricket, 240 students liked Football, while 290 people liked playing Hockey. Among these, 128 students liked only Cricket and Football, 50 liked playing only cricket and Hockey, while 22 liked only Hockey and Football. There were 50 students who liked all three types of sport. A total of 500 students were surveyed.
How many students liked cricket only?
Answer: B
From the given data, we get the following diagram.
Number of students who preferred only Cricket = 230 – (128 + 50 + 50) = 2.
Number of students who preferred only Football = 240 – (128 + 50 + 22) = 40.
Number of students who preferred only Hockey = 290 – (22 + 50 + 50) = 168.
∴ We get the following Venn Diagram.
∴ Number of students who preferred only cricket = 2
Hence, option (b).
How many students liked at least two forms of sport?
Answer: C
Consider the solution to the first question of this set.
Number of students who liked atleast two forms of sport = students who liked exactly two forms of sport + students who liked all three forms of sport
= (128 + 50 + 22) + 50 = 250.
Hence, option (c).
How many students liked Hockey or football but not cricket?
Answer: A
In a 3-set Venn, A or B but not C implies (only A) + (only B) + (only A and B but not C).
Hence, in this case, it is equivalent to: (only Hockey) + (only Football) + (only Hockey and Football).
Consider the solution to the first question.
∴ Required count, as explained above = 168 + 40 + 22 = 230.
Hence, option (a).
How many people surveyed did not like playing any of the three forms of sport?
Answer: C
Consider the solution to the previous questions.
Total number of people who like atleast one form of sport = (168 + 2 + 40) + (22 + 50 + 128) + 50 = 460.
Total people surveyed = 500.
∴ Number of people who did not prefer any form of sport = 500 − 460 = 40.
Hence, option (c).
How many students liked exactly one of the three forms of sport?
Answer: A
Consider the solution to the third question of this set.
Number of students who liked exactly one form of sport = 2 + 40 + 168 = 210.
Hence, option (a).
Answer the next 5 questions based on the information given below.
A survey was conducted among 500 employees of an office regarding the types of transports that they use for traveling. It is known that 250 people use Bike, 170 people use Car and 300 people use Bus. 65 people use both Bike and Car. 136 people use both Bike and Bus, 74 use both Car and Bus. It is also known that, 15 people use none among these three.
How many employees use all the three types of transports?
Answer: 40
Let the number of people using all three types of transports = x.
∴ n(Bi ∪ Ca ∪ Bu) = n(Bi) + n(Ca) + n(Bu) – [n(Bi ∩ Ca) + n(Ca ∩ Bu) + n(Bu ∩ Bi)] + n(Bi ∩ Ca ∩ Bu)
⇒ (500 - 15) = 250 + 170 + 300 – [65 + 74 + 136] + x
⇒ 485 = 720 – 275 + x
⇒ x = 40
We can make the following Venn Diagram.
∴ Number of people using all three types of transports = 40
Hence, 40.
How many employees use at most one type of transport?
Answer: 305
Consider the solution to first question of this set.
Number of employees using at most one type of transport = exactly one transport + no transport
= (89 + 71 + 130) + 15 = 305
Hence, 305.
How many employees use either Bike or Bus?
Answer: 414
Consider the solution to first question of this set.
Number of employees using Bike or Bus = n(Bi) + n(Bu) – n(Bi ∩ Bu)
= 250 + 300 - 136
= 414
Hence, 414.
How many employees use at least two types of transports?
Answer: 195
Consider the solution to first question of this set.
Number of employees that use at least two types of transports = exactly 2 transports + exactly 3 transports
= (25 + 34 + 96) + 40
= 155 + 40 = 195
Hence, 195.
How many use neither Bike nor Car?
Answer: 145
Consider the solution to first question of this set.
Employees that use neither Bike nor Car = Total – n(Bi ∪ Ca)
= 500 – (250 + 170 - 65)
= 500 – 355
= 145
Alternately,
Employees that use neither Bike nor Car = Employees that use only Bus + Employees that use none of the three
= 130 + 15
= 145
Hence, 145.