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A survey was conducted to know the hobby of 220 students of class IX. Out of which 130 students informed about their hobby as ’rock climbing and 180 students informed about their hobby as sky watching. There are 110 students who follow both the hobbies. Then how many students do not have any of the two hobbies? How many of them follow the hobby of rock climbing only? How many students follow the hobby of sky watching only?

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i. Let U be the set of students of class IX, 

R be the set of students who follow the hobby of rock climbing and 

S be the set of students who follow the hobby of sky watching. 

∴ n (U) = 220, n (R) = 130, n (S) = 180, 

110 students follow both the hobbies 

∴ n (R ∩ S) = 110 

ii. n(R ∪ S) = n (R) + n (S) – n (R ∩ S) 

= 130 + 180 – 110 

∴ n (R ∪ S) = 200 

∴ 200 students follow the hobby of rock climbing or sky watching.

iii. Total number of students = 220. 

Number of students who do not follow the hobby of rock climbing or sky watching 

= n (U) – n (R ∪ S) 

= 220 – 200 = 20 

iv. Number of students who follow the hobby of rock climbing only 

= n (R) – n(R ∩ S) = 130 – 110 = 20 v. 

v. Number of students who follow the hobby of sky watching only 

= n (S) – n (R ∩ S) = 180 – 110 = 70 

Alternate Method: 

Let U be the set of students of class IX, R be the set of students who follow the hobby of rock climbing and S be the set of students who follow the hobby of sky watching.

From the Venn diagram 

i. Students who follow the hobby of rock climbing or sky watching = n(R ∪ S) = 20 + 110 + 70 = 200 

ii. Number of students who do not follow the hobby of rock climbing or sky watching 

= n (U) – n(R ∪S) = 220 – 200 = 20 

iii. Number of students who follow the hobby of rock climbing only 

= n (R) – n(R ∩S) = 130 – 110 = 20 

iv. Number of students who follow the hobby of sky watching only 

= n (S) – n (R ∩ S) 

= 180 – 110 

= 70

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