A survey shows that `69%` students like mathematics, whereas `75%` like chemistry. If `x%` students like both the subjects, then the maximum value of x is

Answer

To find the maximum value of \( x \) (the percentage of students who like both Mathematics and Chemistry), we can use the principle of inclusion-exclusion in set theory.

### Step-by-Step Solution:

1. **Define the Percentages**:
- Let \( A \) be the set of students who like Mathematics. According to the survey, \( |A| = 69\% \).
- Let \( B \) be the set of students who like Chemistry. According to the survey, \( |B| = 75\% \).
- Let \( x \) be the percentage of students who like both subjects, i.e., \( |A \cap B| = x\% \).

2. **Use the Inclusion-Exclusion Principle**:
The principle states that:
\[ |A \cup B| = |A| + |B| - |A \cap B| \]
Here, \( |A \cup B| \) represents the percentage of students who like at least one of the subjects.

3. **Set the Upper Limit**:
Since the total percentage of students cannot exceed 100%, we have:
\[ |A \cup B| \leq 100\% \]
Substituting the values we have:
\[ |A| + |B| - |A \cap B| \leq 100\% \]
This translates to:
\[ 69\% + 75\% - x \leq 100\% \]

4. **Combine Like Terms**:
Simplifying the equation gives:
\[ 144\% - x \leq 100\% \]

5. **Isolate \( x \)**:
Rearranging the equation to solve for \( x \):
\[ x \geq 144\% - 100\% \]
\[ x \geq 44\% \]

6. **Determine the Maximum Value of \( x \)**:
Since \( x \) represents the percentage of students who like both subjects, the maximum possible value of \( x \) occurs when all students who like Mathematics also like Chemistry. Therefore, the maximum value of \( x \) is limited by the smaller percentage of students who like Mathematics:
\[ x \leq 69\% \]

### Conclusion:
Thus, the maximum value of \( x \) is:
\[ \boxed{69} \]