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?

Explanation:

Define the sets and given values

Let the three sets of employees be:

• \(C\): Cooked food, with cardinality \(|C| = 11\)
• \(D\): Washed dishes, with cardinality \(|D| = 9\)
• \(R\): Operated the cash register, with cardinality \(|R| = 19\)

The intersections are given as:

• All three jobs: \(|C \cap D \cap R| = 1\)
• Cooked food and washed dishes: \(|C \cap D| = 2\)
• Cooked food and operated register: \(|C \cap R| = 4\)
• Washed dishes and operated register: \(|D \cap R| = 6\)
• None of these jobs: \(| (C \cup D \cup R)' | = 4\)

Determine disjoint regions of the Venn diagram

We calculate the number of employees in each disjoint region starting from the innermost intersection:

• Only all three:

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

• Cooked and washed, but did not operate register:

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

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

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

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

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

Calculate single-job regions

Now we find the number of employees who performed only one job:

• 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 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 the target question

The question asks for part e): "How many of the employees washed dishes or operated the cash register but did not cook food?"
This represents the region \((D \cup R) \cap C'\).
We sum the disjoint regions that are in \(D\) or \(R\), but completely outside \(C\):

• Only washed dishes: \(n(D \text{ only}) = 2\)
• Only operated register: \(n(R \text{ only}) = 10\)
• Washed dishes and operated register, but did not cook: \(n(D \cap R \cap C') = 5\)

Summing these values:

$$ 2 + 10 + 5 = 17 $$

Answer:

e) How many of the employees washed dishes or operated the cash register but did not cook food?
<blank>17</blank> (Simplify your answer.)