Of 320 consumers, 200 east strawberries and 300 eat oranges. If all 320 eat at least one of the fruits, how many eat both?

Answer

To solve the problem of how many consumers eat both strawberries and oranges, we can use the principle of inclusion-exclusion. Let's denote:

- \( n(A) \): the number of consumers who eat strawberries = 200
- \( n(B) \): the number of consumers who eat oranges = 300
- \( n(A \cup B) \): the number of consumers who eat at least one of the fruits = 320
- \( n(A \cap B) \): the number of consumers who eat both strawberries and oranges (this is what we need to find).

According to the principle of inclusion-exclusion, we have:

\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]

Substituting the known values into the formula:

\[ 320 = 200 + 300 - n(A \cap B) \]

Now, simplify the equation:

\[ 320 = 500 - n(A \cap B) \]

To isolate \( n(A \cap B) \), we can rearrange the equation:

\[ n(A \cap B) = 500 - 320 \]

Calculating the right side gives:

\[ n(A \cap B) = 180 \]

Thus, the number of consumers who eat both strawberries and oranges is **180**.