Question

in a survey of employees at a fast food restaurant, it was determined that 8 cooked food, 16 washed dishes, 19 operated the cash register, 5 cooked food and washed dishes, 2 cooked food and operated the cash register, 7 washed dishes and operated the cash register, 1 did all three jobs, and 3 did none of these jobs. complete parts a) through f) below.

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

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

Explanation:

Define sets and regions

Using the Venn Diagram and Set Cardinality knowledge points
Let:

• \(C\): Cooked food
• \(D\): Washed dishes
• \(R\): Operated cash register

We start from the innermost intersection:

• All three: \(|C \cap D \cap R| = 1\)
• Cooked and Register only: \(|C \cap R| - 1 = 2 - 1 = 1\)
• Cooked and Dishes only: \(|C \cap D| - 1 = 5 - 1 = 4\)
• Dishes and Register only: \(|D \cap R| - 1 = 7 - 1 = 6\)

Calculate single-job regions

Using the Venn Diagram and Set Cardinality knowledge points

• Only Cooked: \(|C| - 4 - 1 - 1 = 8 - 6 = 2\)
• Only Dishes: \(|D| - 4 - 1 - 6 = 16 - 11 = 5\)
• Only Register: \(|R| - 1 - 1 - 6 = 19 - 8 = 11\)
• None of these: \(3\)

Sum all disjoint regions

Using the Venn Diagram and Set Cardinality knowledge points
Total surveyed:

$$ 2 + 5 + 11 + 4 + 1 + 6 + 1 + 3 = 33 $$

Answer:

Question a

How many employees were surveyed?
<blank>33</blank>

Question b

How many of the employees only cooked food?
<blank>2</blank>