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?

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

Answer 1

The correct option is C

Both I and II are true

Step 1: Determining the number of student taking apple or orange juice

Let $P$ be the set of total student surveyed and $L$ denote the students taking apple juice and $M$ denote the students taking orange juice,

then , $n\left(P\right)=400,n\left(L\right)=100,n\left(M\right)=150,n\left(L\cap M\right)=75$

$$\begin{gathered} n\left(L\cup M\right)=n\left(L\right)+n\left(M\right)-n\left(L\cap m\right) \\ n\left(L\cup M\right)=100+150-75 \\ n\left(L\cup M\right)=175 \end{gathered}$$

so, $175$ students are taking atleast one juice, so statement i) is true.

Step 2: Determining the number of students that doesn't take any of the juices

$$\begin{gathered} n{\left(L\cup M\right)}^c=n\left(P\right)-n\left(L\cup M\right) \\ =400-175 \\ =225 \end{gathered}$$

Hence, 225 students were taking neither apple nor orange juice so statement ii) is also true..

Therefore, correct option is C