A survey shows that 74% of the Indians like apples, whereas 68% like oranges. Then, percentage of the Indians like both apples and oranges is

Options

A. 42
B. 43
C. 45
D. 44

Answer

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

### Step 1: Define the given percentages
Let:
- \( A \) = percentage of Indians who like apples = 74%
- \( B \) = percentage of Indians who like oranges = 68%

### Step 2: Use the principle of inclusion-exclusion
According to the principle of inclusion-exclusion for two sets, the percentage of people who like either apples or oranges (or both) can be expressed as:
\[ A \cup B = A + B - A \cap B \]
where \( A \cap B \) is the percentage of people who like both apples and oranges.

### Step 3: Set up the equation
We know that the total percentage of people cannot exceed 100%. Therefore, we can write:
\[ A \cup B \leq 100 \]
Substituting the values we have:
\[ 74 + 68 - A \cap B \leq 100 \]

### Step 4: Simplify the equation
Combine the percentages:
\[ 142 - A \cap B \leq 100 \]

### Step 5: Solve for \( A \cap B \)
Rearranging the equation gives us:
\[ A \cap B \geq 142 - 100 \]
\[ A \cap B \geq 42 \]

### Step 6: Conclusion
Thus, the percentage of Indians who like both apples and oranges is at least 42%.

### Final Answer
The percentage of Indians who like both apples and oranges is **42%**.

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