In a survey 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 ge 34`
B. `x le 64`
C. `34 le x le 64`
D. all of these

Answer

To solve the problem, we need to analyze the given percentages of employees who like bananas and apples, as well as those who like both. We can use the principle of inclusion-exclusion to find the relationships between these percentages.

### Step-by-Step Solution:

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

2. **Use the Inclusion-Exclusion Principle:**
The formula for the union of two sets is:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
In terms of percentages, this becomes:
\[ n(A \cup B) = 64 + 70 - x \]

3. **Set Up the Inequality:**
Since the total percentage of employees cannot exceed 100%, we have:
\[ n(A \cup B) \leq 100 \]
Substituting the expression from the previous step:
\[ 64 + 70 - x \leq 100 \]

4. **Simplify the Inequality:**
Combine the constants:
\[ 134 - x \leq 100 \]
Rearranging gives:
\[ -x \leq 100 - 134 \]
\[ -x \leq -34 \]
Multiplying through by -1 (which reverses the inequality):
\[ x \geq 34 \]

5. **Conclusion:**
The percentage of employees who like both bananas and apples, \( x \), must be at least 34%. Therefore, we conclude:
\[ x \geq 34 \]