An environmental auditing agency surveyed 250 manufacturing plants regarding their complia… — GRE General Test

An environmental auditing agency surveyed $250$ manufacturing plants regarding their compliance with three environmental standards: Air Quality ( $A$ ), Water Discharge ( $W$ ), and Waste Management ( $M$ ). The survey yielded the following data:

- $130$ plants meet Air Quality standards ( $A$ ).
- $140$ plants meet Water Discharge standards ( $W$ ).
- $120$ plants meet Waste Management standards ( $M$ ).
- $40$ plants meet all three standards.
- $20$ plants meet none of the three standards.
- The number of plants meeting both Air Quality and Water Discharge standards is equal to the number of plants meeting both Water Discharge and Waste Management standards.
- The number of plants meeting both Air Quality and Waste Management standards is $10$ fewer than the number meeting both Air Quality and Water Discharge standards.

How many of the surveyed plants meet exactly one of the three environmental standards?

Answer: 110

Answer

110

By setting up the 3-set inclusion-exclusion equation, the unknown pairwise intersections are found to be 70, 70, and 60. Subtracting the 40 plants that meet all three standards gives the exclusive double-overlap regions (30, 30, and 20). Subtracting these along with the central intersection from each single set yields 40 plants meeting only Air Quality, 40 meeting only Water Discharge, and 30 meeting only Waste Management, totaling 110 plants.

Step-by-Step Solution

Find total number of plants meeting at least one standard

|A ∪ W ∪ M| = 250 - 20 = 230

Subtracting plants that meet no standards from the total surveyed gives the union of all three sets.

Set up algebraic expressions for pairwise intersections

|A ∩ M| = k, |A ∩ W| = k + 10, |W ∩ M| = k + 10

Define the smallest pairwise intersection as k and express the other two based on the given relationships.

Apply the Principle of Inclusion-Exclusion (PIE) for three sets to solve for k

230 = 130 + 140 + 120 - (k + 10 + k + 10 + k) + 40 => k = 60

Substitute set sizes and the triple intersection into the 3-set inclusion-exclusion formula.

Calculate the number of plants in each exclusive region

Only (A ∩ W) = 30, Only (W ∩ M) = 30, Only (A ∩ M) = 20

Subtract the triple intersection (40) from each pairwise intersection.

Calculate plants meeting exactly one standard and sum them

Only A = 40, Only W = 40, Only M = 30; Total = 40 + 40 + 30 = 110

Subtract all double-overlap and triple-overlap regions from each individual set total.

Key Concept

Three-Set Principle of Inclusion-Exclusion and Venn Diagram Region Decomposition

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