In a survey of 400 students of a school, it is found that 100 students like apple juice, 150 like orange juice and 75 like both. Find how many students do like neither apple not orange juice ?

Answer

To solve the problem, we can use the principle of inclusion-exclusion to find the number of students who like either apple juice or orange juice, and then subtract that from the total number of students to find those who like neither.

### Step-by-step Solution:

1. **Identify the given values:**
- Let \( A \) be the set of students who like apple juice.
- Let \( O \) be the set of students who like orange juice.
- We have:
- \( |A| = 100 \) (students who like apple juice)
- \( |O| = 150 \) (students who like orange juice)
- \( |A \cap O| = 75 \) (students who like both apple and orange juice)

2. **Use the inclusion-exclusion principle:**
The number of students who like either apple juice or orange juice is given by:
\[ |A \cup O| = |A| + |O| - |A \cap O| \]
Substituting the values:
\[ |A \cup O| = 100 + 150 - 75 \]
\[ |A \cup O| = 250 - 75 = 175 \]

3. **Calculate the number of students who like neither:**
To find the number of students who like neither apple nor orange juice, we subtract the number of students who like either from the total number of students:
\[ \text{Students who like neither} = \text{Total students} - |A \cup O| \]
Substituting the values:
\[ \text{Students who like neither} = 400 - 175 = 225 \]

4. **Final Answer:**
Therefore, the number of students who like neither apple nor orange juice is **225**.