At a culinary institute, a cohort of 250 professional chefs was evaluated on three special… — GMAT

At a culinary institute, a cohort of $250$ professional chefs was evaluated on three specialized culinary skills: Molecular Gastronomy, Pastry Arts, and Artisanal Baking. Each chef in the cohort possessed proficiency in at least one of the three skills. According to the evaluation report:

- $150$ chefs were proficient in Molecular Gastronomy
- $140$ chefs were proficient in Pastry Arts
- $130$ chefs were proficient in Artisanal Baking
- The number of chefs proficient in all three skills was $30$

How many chefs in the cohort were proficient in EXACTLY ONE of the three skills?

A
80
B
90
110Answer
D
140
E
170

Answer

110 chefs were proficient in exactly one of the three skills.

The correct answer of 110 is derived by setting up two fundamental 3-set relationships. First, the total number of chefs is the sum of those proficient in exactly one skill (

x

), exactly two skills (

y

), and all three skills (

z = 30

), giving

x + y + 30 = 250

, so

x + y = 220

. Second, the sum of individual set sizes counts elements according to how many sets they belong to:

150 + 140 + 130 = x + 2y + 3(30)

, which simplifies to

x + 2y = 330

. Subtracting the first equation from the second yields

y = 110

, and substituting

y = 110

back into

x + y = 220

gives

x = 110

Step-by-Step Solution

Define variables for the regions of the 3-set Venn diagram

Let

x

be the number of chefs proficient in exactly 1 skill,

y

be the number proficient in exactly 2 skills, and

z

be the number proficient in all 3 skills (

z = 30

Categorizing elements by the exact number of sets they belong to simplifies the set inclusion-exclusion equations.

Set up the total population equation

x + y + z = 250 \implies x + y + 30 = 250 \implies x + y = 220

Since every chef is proficient in at least one skill, the sum of all distinct regions equals the total number of chefs (

250

Set up the sum of individual set sizes equation

150 + 140 + 130 = x + 2y + 3z \implies 420 = x + 2y + 3(30) \implies x + 2y = 330

Summing the individual set totals counts elements in exactly one set once, elements in exactly two sets twice, and elements in all three sets three times.

Solve the system of two linear equations for

x

and

y

Subtracting Equation 1 from Equation 2 gives

(x + 2y) - (x + y) = 330 - 220 \implies y = 110

. Substituting

y = 110

back into Equation 1 gives

x + 110 = 220 \implies x = 110

Finding

x

directly answers the question regarding chefs proficient in exactly one skill.

Key Concept

Overlapping Sets (3-Set Inclusion-Exclusion and Region Partitioning)

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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