Question

in a survey of employees at a fast food restaurant, it was determined that 11 cooked food, 9 washed dishes, 19 operated the cash register, 2 cooked food and washed dishes, 4 cooked food and operated the cash register, 6 washed dishes and operated the cash register, 1 did all three jobs, and 4 did none of these jobs. complete parts a) through f) below.

a) how many employees were surveyed?

b) how many of the employees only cooked food?

c) how many of the employees only operated the cash register?

d) how many of the employees washed dishes and operated the cash register but did not cook food?

e) how many of the employees washed dishes or operated the cash register but did not cook food?

f) how many of the employees did at least two of these jobs?

Explanation:

Define sets and intersections

Using the Venn Diagram and Set Cardinality knowledge points
Let:

• \(C\) be the set of employees who cooked food.
• \(D\) be the set of employees who washed dishes.
• \(R\) be the set of employees who operated the cash register.
• \(N\) be the set of employees who did none of these jobs, so \(|N| = 4\).

We are given:

• Total in each category:

\(|C| = 11\)
\(|D| = 9\)
\(|R| = 19\)

• Intersections:

\(|C \cap D \cap R| = 1\) (all three jobs)
\(|C \cap D| = 2\) (cooked food and washed dishes)
\(|C \cap R| = 4\) (cooked food and operated cash register)
\(|D \cap R| = 6\) (washed dishes and operated cash register)

Calculate disjoint regions

Using the Venn Diagram and Set Cardinality knowledge points
We find the number of employees in each disjoint region of the Venn diagram starting from the center:

• Only all three:

$$ n(C \cap D \cap R) = 1 $$

• Cooked and washed, but did not operate register:

$$ n(C \cap D \cap R') = |C \cap D| - n(C \cap D \cap R) = 2 - 1 = 1 $$

• Cooked and operated register, but did not wash dishes:

$$ n(C \cap R \cap D') = |C \cap R| - n(C \cap D \cap R) = 4 - 1 = 3 $$

• Washed dishes and operated register, but did not cook food:

$$ n(D \cap R \cap C') = |D \cap R| - n(C \cap D \cap R) = 6 - 1 = 5 $$

• Only cooked food:

$$ n(C \text{ only}) = |C| - n(C \cap D \cap R') - n(C \cap R \cap D') - n(C \cap D \cap R) = 11 - 1 - 3 - 1 = 6 $$

• Only washed dishes:

$$ n(D \text{ only}) = |D| - n(C \cap D \cap R') - n(D \cap R \cap C') - n(C \cap D \cap R) = 9 - 1 - 5 - 1 = 2 $$

• Only operated cash register:

$$ n(R \text{ only}) = |R| - n(C \cap R \cap D') - n(D \cap R \cap C') - n(C \cap D \cap R) = 19 - 3 - 5 - 1 = 10 $$

Solve parts a, b, and c

Using the Venn Diagram and Set Cardinality knowledge points

• Part a: Total employees surveyed is the sum of all disjoint regions plus those who did none:

$$ \text{Total} = n(C \text{ only}) + n(D \text{ only}) + n(R \text{ only}) + n(C \cap D \cap R') + n(C \cap R \cap D') + n(D \cap R \cap C') + n(C \cap D \cap R) + |N| $$

$$ \text{Total} = 6 + 2 + 10 + 1 + 3 + 5 + 1 + 4 = 32 $$

• Part b: Employees who only cooked food:

$$ n(C \text{ only}) = 6 $$

• Part c: Employees who only operated the cash register:

$$ n(R \text{ only}) = 10 $$

Solve parts d, e, and f

Using the Venn Diagram and Set Cardinality knowledge points

• Part d: Employees who washed dishes and operated the cash register but did not cook food:

$$ n(D \cap R \cap C') = 5 $$

• Part e: Employees who washed dishes or operated the cash register but did not cook food:

$$ n(D \text{ only}) + n(R \text{ only}) + n(D \cap R \cap C') = 2 + 10 + 5 = 17 $$

• Part f: Employees who did at least two of these jobs:

$$ n(C \cap D \cap R') + n(C \cap R \cap D') + n(D \cap R \cap C') + n(C \cap D \cap R) = 1 + 3 + 5 + 1 = 10 $$

Answer:

Question a

In a survey of employees at a fast food restaurant, it was determined that 11 cooked food, 9 washed dishes, 19 operated the cash register, 2 cooked food and washed dishes, 4 cooked food and operated the cash register, 6 washed dishes and operated the cash register, 1 did all three jobs, and 4 did none of these jobs.

How many employees were surveyed? <blank>32</blank>

Question b

How many of the employees only cooked food? <blank>6</blank>

Question c

How many of the employees only operated the cash register? <blank>10</blank>

Question d

How many of the employees washed dishes and operated the cash register but did not cook food? <blank>5</blank>

Question e

How many of the employees washed dishes or operated the cash register but did not cook food? <blank>17</blank>

Question f

How many of the employees did at least two of these jobs? <blank>10</blank>