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 \( x \), which represents 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 Given Percentages**:
- Let \( A \) be the set of students who like mathematics. We know that \( |A| = 69\% \).
- Let \( B \) be the set of students who like chemistry. We know that \( |B| = 75\% \).
- Let \( x \) be the percentage of students who like both subjects, i.e., \( |A \cap B| = x\% \).

2. **Use the Inclusion-Exclusion Principle**:
According to the principle of inclusion-exclusion for two sets, we have:
\[ |A \cup B| = |A| + |B| - |A \cap B| \]
Here, \( |A \cup B| \) represents the percentage of students who like either mathematics or chemistry or both.

3. **Set the Maximum Condition**:
The maximum percentage of students who can like either subject cannot exceed 100%. Therefore:
\[ |A \cup B| \leq 100\% \]
Substituting the values we have:
\[ 69\% + 75\% - x \leq 100\% \]

4. **Solve the Inequality**:
Rearranging the inequality gives:
\[ 144\% - x \leq 100\% \]
\[ 144\% - 100\% \leq x \]
\[ x \geq 44\% \]

5. **Determine the Maximum Value of \( x \)**:
Since \( x \) represents the percentage of students who like both subjects, the maximum value of \( x \) occurs when all students who like mathematics also like chemistry. This means:
\[ x \leq |A| = 69\% \]

6. **Conclusion**:
Therefore, the maximum value of \( x \) is:
\[ \boxed{69\%} \]