A survey of 280 culinary school graduates evaluated their expertise across three specializ… — GRE General Test

A survey of $280$ culinary school graduates evaluated their expertise across three specialized culinary disciplines: Molecular Gastronomy ( $M$ ), Pastry Arts ( $P$ ), and Sous-Vide Cooking ( $S$ ). The survey gathered the following data:
- $130$ graduates have expertise in Molecular Gastronomy.
- $140$ graduates have expertise in Pastry Arts.
- $120$ graduates have expertise in Sous-Vide Cooking.
- $50$ graduates have expertise in both Molecular Gastronomy and Pastry Arts.
- $45$ graduates have expertise in both Pastry Arts and Sous-Vide Cooking.
- $55$ graduates have expertise in both Molecular Gastronomy and Sous-Vide Cooking.
- $20$ graduates have expertise in all three disciplines.
- $20$ graduates have expertise in none of the three disciplines.

Which of the following statements MUST be true? Select all that apply.

Exactly $45$ graduates have expertise in Molecular Gastronomy only.Cevap
B
The total number of graduates with expertise in exactly two disciplines is $150$ .
The number of graduates who have expertise in Sous-Vide Cooking but not Molecular Gastronomy is $65$ .Cevap
D
More than $25\%$ of all surveyed graduates have expertise in Pastry Arts only.
E
The total number of graduates who have expertise in at least one of the three disciplines is $240$ .

Cevap

The statements confirming that exactly 45 graduates have expertise in Molecular Gastronomy only and that 65 graduates have expertise in Sous-Vide Cooking but not Molecular Gastronomy are correct.

The calculated region for Molecular Gastronomy only yields

130 - (30 + 35 + 20) = 45

, making the statement about Molecular Gastronomy only correct. Furthermore, the number of graduates with expertise in Sous-Vide Cooking but not Molecular Gastronomy equals the sum of the Sous-Vide only region (

40

) and the Pastry-Sous-Vide only region (

25

), which totals

65

, making that statement correct as well.

Adım Adım Çözüm

Determine the number of graduates in all three disciplines and in exactly two disciplines.

Triple intersection

|M \cap P \cap S| = 20

. Pairwise intersections excluding the triple intersection:

M \cap P \text{ only} = 50 - 20 = 30

P \cap S \text{ only} = 45 - 20 = 25

M \cap S \text{ only} = 55 - 20 = 35

. Total in exactly two disciplines =

30 + 25 + 35 = 90

Each given pairwise intersection contains the central triple intersection, so we must subtract 20 to find the exclusive double-overlap regions.

Calculate single-discipline exclusive counts.

Molecular Gastronomy only =

130 - (30 + 35 + 20) = 45

. Pastry Arts only =

140 - (30 + 25 + 20) = 65

. Sous-Vide Cooking only =

120 - (35 + 25 + 20) = 40

Subtracting all double-overlap and triple-overlap members from each set total yields the number of people belonging exclusively to that single set.

Evaluate the individual option statements.

Molecular Gastronomy only = 45 (True). Exactly two disciplines = 90 (Statement B claims 150, False). Sous-Vide Cooking without Molecular Gastronomy =

40 + 25 = 65

(True). Pastry Arts only percentage =

\frac{65}{280} \approx 23.21\% < 25\%

(Statement D claims

>25\%

, False). Total with at least one discipline =

280 - 20 = 260

(Statement E claims 240, False).

Comparing calculated region sizes against each option statement verifies which conditions must hold true.

Anahtar Kavram

3-Set Inclusion-Exclusion Principle and Venn Diagram Region Analysis

Tahmini Süre:2m 0s

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