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?
□ (simplify your answer)

Explanation:

Step1: Define the formula for only one job

To find the number of employees who only cooked food, we use the principle of inclusion - exclusion for three - set intersections. The number of employees who only cooked food (\(n(\text{only cook})\)) is given by the total number of employees who cooked food (\(n(C)\)) minus the number of employees who cooked food and did one other job minus the number of employees who did all three jobs.

Mathematically, if we 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. Then \(n(\text{only }C)=n(C)-n(C\cap D)-n(C\cap R)+n(C\cap D\cap R)\) (we add back \(n(C\cap D\cap R)\) because we subtracted it twice when we subtracted \(n(C\cap D)\) and \(n(C\cap R)\)).

Step2: Substitute the given values

We know that \(n(C) = 11\), \(n(C\cap D)=4\), \(n(C\cap R)=4\), and \(n(C\cap D\cap R) = 2\).

Substitute these values into the formula:

\(n(\text{only }C)=11 - 4-4 + 2\)

First, calculate \(11-4 = 7\), then \(7 - 4=3\), and then \(3 + 2=5\).

Answer:

5