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

Let $U$ be the set of all students who took part in the survey.
Let $T$ be the set of students taking tea.
Let $C$ be the set of students taking coffee.
$n\left(U\right)=600,n\left(T\right)=150,n\left(C\right)=225,n(T\cap C)=100$
Where $n(T\cap C)$ means number of students who take both tea and coffee.
To find: Number of student taking neither tea nor coffee i.e., we have to find $n(T^{\prime }\cap C^{\prime })$.
$n(T^{\prime }\cap C^{\prime })=n(T\cup C{)}^{\prime }$ [From De Morgan's law]
$=n\left(U\right)-n(T\cup C)$
Where $n(T\cup C)$ is number of students who take either tea or coffee.
$=n\left(U\right)-\left[n\right(T)+n(C)-n(T\cap C\left)\right]$
$=600-[150+225-100]$ [ Substituting given values]
$=600-275$
$=325$
Hence, $325$ students were taking neither tea nor coffee.