In a survey of 400 students in a school , 110 were listed as taking Apple Juice , 140 as taking Orange juice and 85 were listed as taking both Apple as well as Orange juice . Find how many students were taking neither Apple juice nor Orange juice.

Answer

To solve the problem step by step, we will use the principle of inclusion-exclusion and the concept of complements in set theory.

### Step 1: Define the Sets
Let:
- \( A \) = the set of students taking Apple Juice
- \( O \) = the set of students taking Orange Juice

From the problem, we have:
- \( n(A) = 110 \) (students taking Apple Juice)
- \( n(O) = 140 \) (students taking Orange Juice)
- \( n(A \cap O) = 85 \) (students taking both Apple and Orange Juice)

### Step 2: Calculate the Union of the Sets
We need to find the number of students taking either Apple Juice or Orange Juice or both, which is represented by \( n(A \cup O) \). According to the principle of inclusion-exclusion:
\[ n(A \cup O) = n(A) + n(O) - n(A \cap O) \]

Substituting the values:
\[ n(A \cup O) = 110 + 140 - 85 \]
\[ n(A \cup O) = 250 - 85 = 165 \]

### Step 3: Calculate the Number of Students Taking Neither Juice
To find the number of students taking neither Apple Juice nor Orange Juice, we need to subtract the number of students taking either juice from the total number of students surveyed. The total number of students is given as 400.

Thus, the number of students taking neither juice is:
\[ n(A' \cap O') = \text{Total Students} - n(A \cup O) \]
\[ n(A' \cap O') = 400 - 165 \]
\[ n(A' \cap O') = 235 \]

### Final Answer
The number of students taking neither Apple Juice nor Orange Juice is **235**.

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