Question

In a survey of $600$ students in a school, $150$ students were found to be taking tea and $225$ taking coffee, $100$ were taking both tea and coffee. Find how many students were taking neither tea nor coffee?

Answer 1

The total number of students in the survey is $600.$

Let $A$ and $B$ be the number of students who take coffee and tea respectively.

Therefore,

Number of students taking tea $=n\left(A\right)=225$

Number of students taking coffee $=n\left(B\right)=150$

Number of students both taking tea and coffee $=n(A\cap B)=100$

The total number of students who take either coffee or tea is $n(A\cup B)$

Formula of union of two sets $A$ and $B$ is

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

$n(A\cup B)=225+150-100=275$

$n(A\cup B)=275$

Total number of students $=$ Students who drink either coffee or tea $+$ Students who drink neither

Students who drink neither is $(A\cup B{)}^{\prime }$

Total number of students $=n(A\cup B)+n(A\cup B{)}^{\prime }$

$600=275+n(A\cup B{)}^{\prime }$

$n(A\cup B{)}^{\prime }=325$

Therefore, the total numbers of students who drink neither coffee nor tea are $325$.