An environmental protection agency audited 240 coastal wetland sites to evaluate contamina… — GRE General Test

An environmental protection agency audited $240$ coastal wetland sites to evaluate contamination by three specific pollutants: Microplastics ( $M$ ), Heavy metals ( $H$ ), and Agricultural runoff ( $A$ ). The audit revealed the following findings:

- Exactly $30$ of the audited wetlands showed no contamination from any of the three pollutants.
- The total number of wetlands containing Microplastics, Heavy metals, and Agricultural runoff were $130$ , $110$ , and $100$ , respectively.
- The number of wetlands containing Microplastics and Heavy metals but NOT Agricultural runoff was $25$ .
- The number of wetlands containing Heavy metals and Agricultural runoff but NOT Microplastics was $35$ .
- The number of wetlands containing Microplastics and Agricultural runoff but NOT Heavy metals was $20$ .

Based on the audit data, which of the following statements must be true? Select all such statements.

Exactly $25$ wetlands contain all three pollutants.Cevap
The number of wetlands containing only Microplastics is $60$ .Cevap
C
The total number of wetlands containing exactly one pollutant is $85$ .
D
The percentage of all surveyed wetlands that contain at least two pollutants is $50\%$ .
The ratio of wetlands containing only Heavy metals to wetlands containing only Agricultural runoff is $5$ to $4$ .Cevap

Cevap

The statements confirming that exactly 25 wetlands contain all three pollutants, that 60 wetlands contain only Microplastics, and that the ratio of wetlands containing only Heavy metals to only Agricultural runoff is 5 to 4 are all correct.

The correct options are those stating that 25 wetlands contain all three pollutants, that 60 wetlands contain only Microplastics, and that the ratio of Heavy metals only to Agricultural runoff only is 5 to 4. Each of these follows directly from setting up the inclusion-exclusion equation for three sets and determining all eight mutually exclusive regions of the Venn diagram.

Adım Adım Çözüm

Determine the number of wetlands containing at least one pollutant.

Total in union

|M \cup H \cup A| = 240 - 30 = 210

The total audited set is 240, and 30 sites have no pollutants.

Apply the Principle of Inclusion-Exclusion for three sets.

|M \cup H \cup A| = |M| + |H| + |A| - (|M \cap H| + |H \cap A| + |A \cap M|) + |M \cap H \cap A|

, which simplifies to

210 = 130 + 110 + 100 - S_2 + x \implies S_2 - x = 130

, where

S_2

is the sum of pairwise intersections and

x

is the triple intersection.

This relates the total union to the individual set sizes and intersection regions.

Express

S_2

in terms of the given 'exactly two' regions and solve for

x

The number of wetlands with exactly two pollutants is

25 + 35 + 20 = 80

. Since

S_2 - 3x = 80

, substituting

S_2 = 130 + x

gives

(130 + x) - 3x = 80 \implies 2x = 50 \implies x = 25

The sum of pairwise overlaps counts the triple intersection three times, so subtracting

3x

yields the 'exactly two' region.

Calculate single-category region sizes.

Microplastics only =

130 - (25 + 20 + 25) = 60

; Heavy metals only =

110 - (25 + 35 + 25) = 25

; Agricultural runoff only =

100 - (20 + 35 + 25) = 20

. Total single-category =

60 + 25 + 20 = 105

Subtracting all dual and triple overlap counts from each set total isolates the exclusive membership.

Verify each choice statement against the calculated region counts.

Triple intersection is 25 (True). Microplastics only is 60 (True). Exactly one pollutant total is 105, not 85 (False). Percentage of total sample with at least two pollutants is

105 / 240 = 43.75\%

, not

50\%

(False). Ratio of Heavy metals only to Agricultural runoff only is

25:20 = 5:4

(True).

Validates exact values against option assertions.

Anahtar Kavram

Three-set Principle of Inclusion-Exclusion and Venn diagram region decomposition

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