Question

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?

Answer 1

Let A be rock climbing hobby $=130$

Let B be the sky watching hobby $=180$

$n(A\cap B)=110$

Total students $\left(U\right)=220$

Hence

$n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\cap B)$

$=130+180-110$

$=200$

Let $n(A\cup B{)}^{\prime }=$ Who do not have either hobby

$n(A\cup B{)}^{\prime }=n(U)-n(A\cup B)$

$=220-200$

$=20$

Therefore 20 students do not have both hobby

Student's who only like rock climbing $=n\left(A\right)-n(A\cap B)$

$=130-110$

$=20$

Students who only like sky watching $=n\left(B\right)-n(A\cap B)$

$=180-110$

$=70$