In a survey of 600 students in a school, (i) 160 students were found to be taking tea , 215 taking coffee , 150 were taking both tea and cofee.
(ii) 150 students were found to be taking tea and 225 taking coffee , 100 were taking both Tea and Coffee
Find how many students were taking neither tea nor coffee.

Answer

To solve the problem step by step, we will analyze the data provided for both parts of the question.

### Part (i)

1. **Identify the given values:**
- Total number of students (Universal set, U) = 600
- Number of students taking tea (n(T)) = 160
- Number of students taking coffee (n(C)) = 215
- Number of students taking both tea and coffee (n(T ∩ C)) = 150

2. **Use the formula for the union of two sets:**
\[ n(T ∪ C) = n(T) + n(C) - n(T ∩ C) \]
Substituting the values:
\[ n(T ∪ C) = 160 + 215 - 150 \]

3. **Calculate n(T ∪ C):**
\[ n(T ∪ C) = 375 \]

4. **Find the number of students taking neither tea nor coffee:**
\[ n(T' ∩ C') = U - n(T ∪ C) \]
Substituting the values:
\[ n(T' ∩ C') = 600 - 375 \]

5. **Calculate n(T' ∩ C'):**
\[ n(T' ∩ C') = 225 \]

So, **225 students were taking neither tea nor coffee** in the first part.

### Part (ii)

1. **Identify the given values:**
- Total number of students (Universal set, U) = 600
- Number of students taking tea (n(T)) = 150
- Number of students taking coffee (n(C)) = 225
- Number of students taking both tea and coffee (n(T ∩ C)) = 100

2. **Use the formula for the union of two sets:**
\[ n(T ∪ C) = n(T) + n(C) - n(T ∩ C) \]
Substituting the values:
\[ n(T ∪ C) = 150 + 225 - 100 \]

3. **Calculate n(T ∪ C):**
\[ n(T ∪ C) = 275 \]

4. **Find the number of students taking neither tea nor coffee:**
\[ n(T' ∩ C') = U - n(T ∪ C) \]
Substituting the values:
\[ n(T' ∩ C') = 600 - 275 \]

5. **Calculate n(T' ∩ C'):**
\[ n(T' ∩ C') = 325 \]

So, **325 students were taking neither tea nor coffee** in the second part.

### Summary of Results:
- Part (i): 225 students taking neither tea nor coffee.
- Part (ii): 325 students taking neither tea nor coffee.