A survey shows that `63%` of the people watch a news channel whereas `76%` watch another channel. If `x%` of the people watch both channel then,

Options

A. `x = 35`
B. `x = 63`
C. `39 le x le 63`
D. `x = 39`

Explanation

To solve the problem, we need to find the percentage of people who watch both news channels, denoted as \( x\% \). We can use the principle of inclusion-exclusion in set theory.

### Step-by-Step Solution:

1. **Define the Sets**:
- Let \( A \) be the set of people who watch the first news channel.
- Let \( B \) be the set of people who watch the second news channel.
- Given:
- \( |A| = 63\% \) (or \( \frac{63}{100} \))
- \( |B| = 76\% \) (or \( \frac{76}{100} \))

2. **Use the Inclusion-Exclusion Principle**:
- The formula for the union of two sets is given by:
\[ |A \cup B| = |A| + |B| - |A \cap B| \]
- Here, \( |A \cap B| \) represents the percentage of people who watch both channels, which is \( x\% \) (or \( \frac{x}{100} \)).

3. **Set Up the Equation**:
- Since the total percentage of people cannot exceed 100%, we can set up the equation:
\[ |A \cup B| \leq 100\% \]
- Substituting the values we have:
\[ |A| + |B| - |A \cap B| \leq 100 \]
- This translates to:
\[ 63 + 76 - x \leq 100 \]

4. **Simplify the Equation**:
- Combine the percentages:
\[ 139 - x \leq 100 \]

5. **Solve for \( x \)**:
- Rearranging the inequality gives:
\[ x \geq 139 - 100 \]
- Therefore:
\[ x \geq 39 \]

6. **Conclusion**:
- The minimum percentage of people who watch both channels is \( 39\% \).