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 in set theory. Here's a step-by-step solution:

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

### Step 2: Understand the total percentage
Since we are dealing with percentages, the total percentage of people surveyed cannot exceed 100%. Therefore, we can say that:
- The percentage of people who like at least one of the fruits (mangoes or apples) is equal to 100%.

### Step 3: Use the principle of inclusion-exclusion
According to the principle of inclusion-exclusion, the percentage of people who like at least one of the fruits can be expressed as:
\[ M + A - \text{(percentage of people who like both fruits)} \]
Let \( B \) be the percentage of people who like both mangoes and apples. Thus, we can write:
\[ M + A - B = 100 \]

### Step 4: Substitute the known values
Substituting the values of \( M \) and \( A \):
\[ 60 + 70 - B = 100 \]

### Step 5: Simplify the equation
Now, simplify the equation:
\[ 130 - B = 100 \]

### Step 6: Solve for \( B \)
Rearranging gives:
\[ B = 130 - 100 \]
\[ B = 30 \]

### Conclusion
Thus, the percentage of Americans who like both mangoes and apples is **30%**.

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