Question

in a survey of employees at a fast food restaurant, it was determined that 11 cooked food, 13 washed dishes, 23 operated the cash register, 4 cooked food and washed dishes, 4 cooked food and operated the cash register, 8 washed dishes and operated the cash register, 2 did all three jobs, and 4 did none of these jobs. complete parts a) through f) below

a) how many employees were surveyed?
37 (simplify your answer.)

b) how many of the employees only cooked food?
5 (simplify your answer.)

c) how many of the employees only operated the cash register?
□ (simplify your answer.)

Explanation:

Step1: Identify the formula for only operating cash register

To find the number of employees who only operated the cash register, we use the principle of inclusion - exclusion for three sets. Let \( C \) be the set of employees who cooked food, \( D \) be the set of employees who washed dishes, and \( R \) be the set of employees who operated the cash register. The number of employees who only operated the cash register is given by \( n(R) - n(C\cap R)-n(D\cap R)+n(C\cap D\cap R) \) (we add back \( n(C\cap D\cap R) \) because we subtracted it twice when we subtracted \( n(C\cap R) \) and \( n(D\cap R) \)).

Step2: Substitute the given values

We know that \( n(R) = 23 \), \( n(C\cap R)=4 \), \( n(D\cap R) = 8 \) and \( n(C\cap D\cap R)=2 \).

Substitute these values into the formula:

\( n(\text{only } R)=23 - 4- 8 + 2 \)

First, calculate \( 23-4=19 \), then \( 19 - 8=11 \), and then \( 11 + 2 = 13 \).

Answer:

13