Question

In a survey of $400$ students in a school, $100$ were listed as taking apple juice, $150$ as taking orange juice and $75$ were listed as taking both apple as well as orange juice. Find how many students were taking neither apple juice nor orange juice?

Answer 1

Given data:
Let number of students who take apple juice $n\left(A\right)=100$
Number of students who take orange juice $=n\left(O\right)=150$
Number of students who take both orange juice and apple juice $=n(O\cap A)=75$

Solve for $n(A\cup O)$
We know that,
$n(A\cup O)=n\left(A\right)+n\left(O\right)-n(A\cap O)$
$=100+150-75=175$
$\therefore$ Number of people who take either apple or orange juice $=175$

Solve for students who take neither apple nor orange juice.
Number of students who take neither apple juice nor orange juice
$=\text{Total number of students}-n(A\cup O)$
$=400-175$
$=225$

Final answer:
Hence, $225$ students were taking neither apple juice nor orange juice.