Question

1. At an ice cream store, there are 112 customers. If 43 have purchased sundaes, 57 have purchased milkshakes, and 38 have purchased neither, how many purchased both sundaes and milkshakes

Answer 1

To solve this problem, we can use the principle of inclusion-exclusion.

Let:
- \( S \) be the number of customers who purchased sundaes.
- \( M \) be the number of customers who purchased milkshakes.
- \( N \) be the number of customers who purchased neither.

From the problem, we have:
- \( S = 43 \)
- \( M = 57 \)
- \( N = 38 \)
- Total customers = 112

First, we can find the number of customers who purchased at least one of the two options (either sundaes or milkshakes):

\[
\text{Customers who purchased at least one} = \text{Total customers} - \text{Customers who purchased neither} = 112 - 38 = 74
\]

Now we use the principle of inclusion-exclusion to find the number of customers who purchased both sundaes and milkshakes (denoted as \( B \)):
\[
\text{Customers who purchased at least one} = S + M - B
\]
Substituting the values we know:
\[
74 = 43 + 57 - B
\]
Now simplify the equation:
\[
74 = 100 - B
\]

To isolate \( B \), we will rearrange the terms:
\[
B = 100 - 74
\]
\[
B = 26
\]

Thus, the number of customers who purchased both sundaes and milkshakes is \( \boxed{26} \).