It is known from a survey that 60% American like mangoes and 70% like apples. How many per cent of Americans like both mangoes and apples ?

Answer

To solve the problem of how many percent of Americans like both mangoes and apples, we can use the principle of inclusion-exclusion. Here’s the step-by-step solution:

### Step 1: Define the given percentages
Let:
- \( P(M) \) = Percentage of Americans who like mangoes = 60%
- \( P(A) \) = Percentage of Americans who like apples = 70%

### Step 2: Use the principle of inclusion-exclusion
According to the principle of inclusion-exclusion, the formula for the union of two sets is given by:
\[ P(M \cup A) = P(M) + P(A) - P(M \cap A) \]
Where:
- \( P(M \cup A) \) = Percentage of Americans who like either mangoes or apples (or both)
- \( P(M \cap A) \) = Percentage of Americans who like both mangoes and apples

### Step 3: Determine the maximum possible percentage
Since the total percentage cannot exceed 100%, we have:
\[ P(M \cup A) \leq 100\% \]
Substituting the known values into the equation:
\[ 100\% \geq 60\% + 70\% - P(M \cap A) \]

### Step 4: Simplify the inequality
Now, simplify the right side:
\[ 100\% \geq 130\% - P(M \cap A) \]

### Step 5: Rearrange to find \( P(M \cap A) \)
Rearranging the inequality gives:
\[ P(M \cap A) \geq 130\% - 100\% \]
\[ P(M \cap A) \geq 30\% \]

### Conclusion
Thus, at least 30% of Americans like both mangoes and apples.