A survey show that 63% of the Indians like cheese whereas 76% like apples. If x% of the Indians like both cheese and apples, then find the range of x.

Options

A. `35 le x le 63%`
B. `23 le x le 76%`
C. `39 le x le 63%`
D. `63 le x le 76%`

Answer

To solve the problem, we need to determine the range of \( x \), which represents the percentage of Indians who like both cheese and apples. We will use the principle of inclusion-exclusion in set theory.

### Step-by-Step Solution:

1. **Identify the Given Percentages:**
- Let \( N_C \) be the percentage of Indians who like cheese = 63%
- Let \( N_A \) be the percentage of Indians who like apples = 76%
- Let \( x \) be the percentage of Indians who like both cheese and apples.

2. **Use the Inclusion-Exclusion Principle:**
The formula for the union of two sets is given by:
\[ N_{C \cup A} = N_C + N_A - N_{C \cap A} \]
Substituting the known values:
\[ N_{C \cup A} = 63 + 76 - x \]
This simplifies to:
\[ N_{C \cup A} = 139 - x \]

3. **Set Up the Inequality for the Union:**
Since the total percentage cannot exceed 100%, we have:
\[ N_{C \cup A} \leq 100 \]
Thus:
\[ 139 - x \leq 100 \]
Rearranging gives:
\[ x \geq 39 \]

4. **Establish the Upper Bound for \( x \):**
The percentage of people who like both cheese and apples cannot exceed the percentage of people who like either cheese or apples. Therefore:
\[ x \leq N_C \quad \text{and} \quad x \leq N_A \]
This means:
\[ x \leq 63 \quad \text{and} \quad x \leq 76 \]
The more restrictive condition is \( x \leq 63 \).

5. **Combine the Inequalities:**
From the inequalities derived, we have:
\[ 39 \leq x \leq 63 \]

### Final Answer:
The range of \( x \) is \( [39, 63] \).