A survey shows that 63% of the Indians like cheese, whereas 76% like apples. If x % of the Indians like both cheese and apples, then x can be

Options

A. 40
B. 65
C. 39
D. None of these

Answer

To solve the problem step by step, we will use the principle of inclusion-exclusion in set theory.

### Step 1: Define the Variables
Let:
- \( N(A) \) = percentage of people who like apples = 76%
- \( N(C) \) = percentage of people who like cheese = 63%
- \( N(C \cap A) \) = percentage of people who like both cheese and apples = \( x \% \)

### Step 2: Apply the Inclusion-Exclusion Principle
According to the principle of inclusion-exclusion, the percentage of people who like either cheese or apples can be expressed as:
\[ N(C \cup A) = N(C) + N(A) - N(C \cap A) \]
Substituting the known values:
\[ N(C \cup A) = 63 + 76 - x \]
This simplifies to:
\[ N(C \cup A) = 139 - x \]

### Step 3: Set Up the Inequalities
Since the total percentage of people cannot exceed 100%, we have:
\[ N(C \cup A) \leq 100 \]
Substituting the expression we found:
\[ 139 - x \leq 100 \]
This simplifies to:
\[ x \geq 39 \]

### Step 4: Establish Upper Bound for \( x \)
Additionally, the percentage of people who like both cheese and apples cannot exceed the percentage of people who like either:
- \( N(C \cap A) \leq N(C) \)
- \( N(C \cap A) \leq N(A) \)

Thus, we have:
\[ x \leq 63 \quad \text{(from cheese)} \]
\[ x \leq 76 \quad \text{(from apples)} \]
However, the more restrictive condition is \( x \leq 63 \).

### Step 5: Combine the Inequalities
From the above steps, we have established two inequalities:
1. \( x \geq 39 \)
2. \( x \leq 63 \)

Therefore, we conclude that:
\[ 39 \leq x \leq 63 \]

### Final Answer
The percentage \( x \) of Indians who like both cheese and apples can be between 39% and 63%.

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