Questions: In a survey of employees at a fast food restaurant, it was determined that 14 cooked food, 14 washed dishes, 28 operated the cash register, 7 cooked food and washed dishes, 9 cooked food and operated the cash register, 9 washed dishes and operated the cash register, 5 did all three jobs, and 3 did none of these jobs. Complete parts a) through 7 ) below. 3 (Simplify your answer.) c) How many of the employees only operated the cash register? 15 (Simplify your answer.) d) How many of the employees washed dishes and operated the cash register but did not cook food? 4 (Simplify your answer.) e) How many of the employees washed dishes or operated the cash register but did not cook food? 22 (Simplify your answer.) f) How many of the employees did at least two of these jobs? 15 (Simplify your answer.)

In a survey of employees at a fast food restaurant, it was determined that 14 cooked food, 14 washed dishes, 28 operated the cash register, 7 cooked food and washed dishes, 9 cooked food and operated the cash register, 9 washed dishes and operated the cash register, 5 did all three jobs, and 3 did none of these jobs. Complete parts a) through 7 ) below.

3 (Simplify your answer.)
c) How many of the employees only operated the cash register?

15 (Simplify your answer.)
d) How many of the employees washed dishes and operated the cash register but did not cook food?

4 (Simplify your answer.)
e) How many of the employees washed dishes or operated the cash register but did not cook food?

22 (Simplify your answer.)
f) How many of the employees did at least two of these jobs?

15 (Simplify your answer.)

Transcript text: In a survey of employees at a fast food restaurant, it was determined that 14 cooked food, 14 washed dishes, 28 operated the cash register, 7 cooked food and washed dishes, 9 cooked food and operated the cash register, 9 washed dishes and operated the cash register, 5 did all three jobs, and 3 did none of these jobs. Complete parts a) through 7 ) below. 3 (Simplify your answer.) c) How many of the employees only operated the cash register? 15 (Simplify your answer.) d) How many of the employees washed dishes and operated the cash register but did not cook food? 4 (Simplify your answer.) e) How many of the employees washed dishes or operated the cash register but did not cook food? 22 (Simplify your answer.) f) How many of the employees did at least two of these jobs? 15 (Simplify your answer.)

Solution

Solution Steps

Solution Approach

To solve these problems, we will use the principle of inclusion-exclusion and set operations. We will define sets for each job category and their intersections based on the given data. Then, we will calculate the required values by applying set operations and the principle of inclusion-exclusion.

Step 1: Calculate Employees Who Only Operated the Cash Register

To find the number of employees who only operated the cash register, we can use the formula: \[ \text{Only Cash Register} = \text{Operated Cash Register} - (\text{Cooked and Operated} + \text{Washed and Operated} - \text{All Three}) \] Substituting the values: \[ \text{Only Cash Register} = 28 - (9 + 9 - 5) = 28 - 13 = 15 \]

Step 2: Calculate Employees Who Washed Dishes and Operated the Cash Register but Did Not Cook Food

To find the number of employees who washed dishes and operated the cash register but did not cook food, we use: \[ \text{Washed and Operated Not Cooked} = \text{Washed and Operated} - \text{All Three} \] Substituting the values: \[ \text{Washed and Operated Not Cooked} = 9 - 5 = 4 \]

Step 3: Calculate Employees Who Washed Dishes or Operated the Cash Register but Did Not Cook Food

To find the number of employees who washed dishes or operated the cash register but did not cook food, we can use: \[ \text{Washed or Operated Not Cooked} = (\text{Washed Dishes} + \text{Operated Cash Register} - \text{Washed and Operated}) - (\text{Cooked and Washed} + \text{Cooked and Operated} - 2 \times \text{All Three}) \] Substituting the values: \[ \text{Washed or Operated Not Cooked} = (14 + 28 - 9) - (7 + 9 - 2 \times 5) = 33 - 6 = 27 \]

Final Answer

a) Only operated the cash register: \( \boxed{15} \)
d) Washed dishes and operated the cash register but did not cook food: \( \boxed{4} \)
e) Washed dishes or operated the cash register but did not cook food: \( \boxed{27} \)

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