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(a) x = 35`(b)x = 63`(c)` 39≤x≤63(d) `x = 39`

Options

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

Explanation

To solve the problem, we need to use the principle of set theory, specifically focusing on the union and intersection of two sets. Let's denote:

- \( A \): the set of people who watch the first news channel.
- \( B \): the set of people who watch the second news channel.
- \( |A| = 63\% \) (the percentage of people who watch the first channel).
- \( |B| = 76\% \) (the percentage of people who watch the second channel).
- \( |A \cap B| = x\% \) (the percentage of people who watch both channels).

### Step 1: Set up the equation for the union of two sets
According to the principle of inclusion-exclusion for sets, the union of two sets can be expressed as:

\[ |A \cup B| = |A| + |B| - |A \cap B| \]

### Step 2: Substitute the known values into the equation
We know that the total percentage of people cannot exceed 100%. Thus, we can write:

\[ |A \cup B| \leq 100\% \]

Substituting the values we have:

\[ |A| + |B| - |A \cap B| \leq 100 \]

This translates to:

\[ 63 + 76 - x \leq 100 \]

### Step 3: Simplify the inequality
Now, simplify the left side:

\[ 139 - x \leq 100 \]

### Step 4: Solve for \( x \)
Rearranging the inequality gives:

\[ -x \leq 100 - 139 \]
\[ -x \leq -39 \]
\[ x \geq 39 \]

### Step 5: Set up the second inequality
Next, we need to consider that the number of people who watch both channels cannot exceed the number of people who watch either channel. Therefore:

\[ |A \cap B| \leq \min(|A|, |B|) \]

Since \( |A| = 63\% \) and \( |B| = 76\% \), we have:

\[ x \leq 63 \]

### Step 6: Combine the inequalities
From the two inequalities we derived:

1. \( x \geq 39 \)
2. \( x \leq 63 \)

We can combine these to get:

\[ 39 \leq x \leq 63 \]

### Conclusion
Thus, the correct answer for \( x \) is:

**(c) \( 39 \leq x \leq 63 \)**