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.
Then, which of the following is/are true?
I. 150 students were taking atleast one juice.
II. 225 students were taking neither apple juice nor orange juice.

• A. Only I is true
• B. Only II is true
• C. Both I and II are true
• D. None of these

Answer 1

Answer: (B)

Let $U$ denote the set of surveyed students and $X$ denote the set of students taking apple juice and $Y$ denote the set of students taking orange juice. Then,
$n(U)=400,n(X)=100,n(Y)=150$
and $n(X\cap Y)=75$
$n(X\cup Y)=n(X)+n(Y)-n(X\cap Y)$
$=100+150-75$
$=175$
$\therefore 175$ students were taking atleast one juice.
$n\left(X^{\prime }\cap Y^{\prime }\right)=n(X\cup Y{)}^{\prime }$
$=n(U)-n(X\cup Y)$
$=400-175$
$=225$
Hence, 225 students were taking neither apple juice nor orange juice.