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 step by step, we will use the principle of inclusion-exclusion. Let's break it down:

### Step 1: Identify the given data
- Total number of students surveyed (U) = 400
- Number of students who like apple juice (A) = 100
- Number of students who like orange juice (O) = 150
- Number of students who like both apple and orange juice (A ∩ O) = 75

### Step 2: Use the formula for the union of two sets
The formula for the number of elements in the union of two sets is given by:
\[ |A \cup O| = |A| + |O| - |A \cap O| \]

### Step 3: Substitute the values into the formula
Now we substitute the values we have into the formula:
\[ |A \cup O| = 100 + 150 - 75 \]

### Step 4: Calculate the union
Now perform the calculation:
\[ |A \cup O| = 100 + 150 - 75 = 175 \]
This means that 175 students like either apple juice or orange juice or both.

### Step 5: Find the number of students who like neither juice
To find the number of students who like neither apple juice nor orange juice, we subtract the number of students who like either or both juices from the total number of students:
\[ \text{Number of students who like neither} = |U| - |A \cup O| \]
Substituting the values:
\[ \text{Number of students who like neither} = 400 - 175 \]

### Step 6: Calculate the final answer
Now perform the calculation:
\[ \text{Number of students who like neither} = 400 - 175 = 225 \]

Thus, the number of students who like neither apple juice nor orange juice is **225**.

### Summary of Steps:
1. Identify the given data.
2. Use the union formula for two sets.
3. Substitute the values into the formula.
4. Calculate the union.
5. Find the number of students who like neither juice.
6. Calculate the final answer.