In a small sweet shop people usually buy either one cake or one box of chocolate. One day the shop sold 57 cakes and 36 boxes of chocolates. How many customers were there that day if 12 people bought both a cake and a box of chocolates?

Answer

To solve the problem step by step, we will use the principle of inclusion-exclusion to find the total number of customers in the sweet shop.

### Step-by-Step Solution:

1. **Identify the Variables:**
- Let \( C \) be the number of customers who bought cakes. From the problem, \( C = 57 \).
- Let \( B \) be the number of customers who bought boxes of chocolates. From the problem, \( B = 36 \).
- Let \( X \) be the number of customers who bought both cakes and boxes of chocolates. From the problem, \( X = 12 \).

2. **Use the Inclusion-Exclusion Principle:**
The total number of customers \( U \) can be calculated using the formula:
\[ U = C + B - X \]
This formula accounts for the overlap (customers who bought both items) by subtracting \( X \) from the total of \( C \) and \( B \).

3. **Substitute the Values:**
Now, substitute the known values into the formula:
\[ U = 57 + 36 - 12 \]

4. **Perform the Calculation:**
First, add \( C \) and \( B \):
\[ 57 + 36 = 93 \]
Then, subtract \( X \):
\[ 93 - 12 = 81 \]

5. **Conclusion:**
Therefore, the total number of customers that day was:
\[ U = 81 \]

### Final Answer:
The total number of customers in the sweet shop that day is **81**.