In a survery it is to be found that `70%` of employees like bananas and `64%` like apples. If `x%` like both bananas and apples, then

Options

A. `x ge34`
B. `xle64`
C. `34lexle64`
D. all of these

Answer

To solve the problem, we need to determine the possible values of \( x \), where \( x \) represents the percentage of employees who like both bananas and apples.

### Step-by-Step Solution:

1. **Define the Variables:**
- Let \( n(A) \) be the percentage of employees who like bananas, which is \( 70\% \).
- Let \( n(B) \) be the percentage of employees who like apples, which is \( 64\% \).
- Let \( n(A \cap B) \) be the percentage of employees who like both bananas and apples, which is \( x\% \).

2. **Use the Formula for Union of Two Sets:**
The formula for the union of two sets is given by:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
Substituting the known values:
\[ n(A \cup B) = 70 + 64 - x \]

3. **Determine the Maximum Value of the Union:**
Since \( n(A \cup B) \) cannot exceed \( 100\% \) (the total percentage of employees), we set up the inequality:
\[ 70 + 64 - x \leq 100 \]

4. **Simplify the Inequality:**
Simplifying the inequality:
\[ 134 - x \leq 100 \]
Rearranging gives:
\[ x \geq 34 \]

5. **Determine the Maximum Value of \( x \):**
The intersection \( n(A \cap B) \) cannot exceed the individual sets, so:
\[ n(A \cap B) \leq n(A) \quad \text{and} \quad n(A \cap B) \leq n(B) \]
This means:
\[ x \leq 70 \quad \text{and} \quad x \leq 64 \]

6. **Combine the Inequalities:**
From the inequalities derived:
- \( x \geq 34 \)
- \( x \leq 64 \)

Therefore, we can conclude:
\[ 34 \leq x \leq 64 \]

### Final Result:
The possible values of \( x \) range from \( 34\% \) to \( 64\% \).