In a survey of employees at a fast food​ restaurant, it was determined that 14 cooked​ food, 16 washed​ dishes, 25 operated the cash​ register, 6 cooked food and washed​ dishes, 7 cooked food and operated the cash​ register, 9 washed dishes and operated the cash​ register, 4 did all three​ jobs, and 5 did none of these jobs. How many of the employees only operated the cash​ register?

To solve the problem, let's use the principle of inclusion-exclusion (PIE) and set notation.

### Given:
- \( |C| = 14 \) (employees who cooked food)
- \( |D| = 16 \) (employees who washed dishes)
- \( |R| = 25 \) (employees who operated the cash register)
- \( |C \cap D| = 6 \) (employees who cooked food and washed dishes)
- \( |C \cap R| = 7 \) (employees who cooked food and operated the cash register)
- \( |D \cap R| = 9 \) (employees who washed dishes and operated the cash register)
- \( |C \cap D \cap R| = 4 \) (employees who did all three jobs)
- \( |None| = 5 \) (employees who did none of these jobs)

We need to find the number of employees who only operated the cash register.

### Step 1: Total Number of Employees
First, find the total number of employees surveyed:
\[
|U| = |C \cup D \cup R| + |None|
\]
We use the principle of inclusion-exclusion to find \( |C \cup D \cup R| \):
\[
|C \cup D \cup R| = |C| + |D| + |R| - |C \cap D| - |C \cap R| - |D \cap R| + |C \cap D \cap R|
\]
\[
|C \cup D \cup R| = 14 + 16 + 25 - 6 - 7 - 9 + 4 = 37
\]
So, the total number of employees surveyed is:
\[
|U| = 37 + 5 = 42
\]

### Step 2: Calculate Number of Employees Who Only Operated the Cash Register
We now need to subtract those who operated the cash register but also did other jobs from the total number who operated the cash register:
\[
| \text{Only } R | = |R| - \left( |C \cap R| + |D \cap R| - |C \cap D \cap R| \right)
\]
\[
| \text{Only } R | = 25 - (7 + 9 - 4) = 25 - 12 = 13
\]

### Final Answer:
The number of employees who only operated the cash register is **13**.

Would you like more details or have any questions about this? 

Here are some related questions you might find interesting:
1. How many employees operated the cash register and washed dishes but did not cook food?
2. How many employees did exactly two jobs?
3. How many employees did at least one job?
4. What is the probability that a randomly selected employee cooked food?
5. How many employees cooked food but did not operate the cash register?

**Tip:** When solving problems involving multiple sets, Venn diagrams can be very helpful for visualizing the relationships and intersections among the sets.

Learn how to solve a set theory problem involving employees at a fast food restaurant. Discover how to apply the principle of inclusion-exclusion to determine the number of employees who only operated the cash register. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation