At an intellectual property law firm, a senior partner reviewed 180 patent applications ac… — GMAT

At an intellectual property law firm, a senior partner reviewed $180$ patent applications across three technical domains: Artificial Intelligence, Biotechnology, and Clean Energy. The audit revealed that $25$ applications belonged to none of these three domains. Furthermore, $90$ applications involved Artificial Intelligence, $85$ involved Biotechnology, and $80$ involved Clean Energy. If the number of applications that belonged to exactly two of these domains was three times the number of applications that belonged to all three domains, how many patent applications belonged to exactly one domain?

Cevap: 75 applications

Cevap

The total number of patent applications that belonged to exactly one domain is

75

The total number of applications in at least one domain is

180 - 25 = 155

. Let

E_1

be the number of applications in exactly one domain,

E_2

in exactly two domains, and

x

in all three domains. The total union is

E_1 + E_2 + x = 155

, while the sum of individual set sizes is

|A| + |B| + |C| = E_1 + 2E_2 + 3x = 90 + 85 + 80 = 255

. Subtracting the union equation from the sum equation gives

E_2 + 2x = 100

. Given

E_2 = 3x

, substituting gives

5x = 100 \implies x = 20

, which implies

E_2 = 60

. Finally, subtracting

E_2

and

x

from the total union yields

E_1 = 155 - 60 - 20 = 75

Adım Adım Çözüm

Calculate the total number of applications in at least one domain

|A \cup B \cup C| = 180 - 25 = 155

Subtracting the applications belonging to none of the domains from the overall total yields the total number of unique applications covered by the three domains combined.

Formulate regional Venn diagram equations

E_1 + E_2 + x = 155

and

E_1 + 2E_2 + 3x = 255

Summing individual set counts counts elements in exactly two domains twice and elements in all three domains three times.

Deduce the relationship between

E_2

and

x

E_2 + 2x = 100

Subtracting the equation for total union from the sum of individual sets isolates the overcounted regions.

Use the given proportion

E_2 = 3x

to solve for

x

and

E_2

x = 20

and

E_2 = 60

Substituting

E_2 = 3x

into

E_2 + 2x = 100

gives

5x = 100

, so

x = 20

and

E_2 = 60

Find the number of applications belonging to exactly one domain (

E_1

)

E_1 = 75

Subtracting the count of applications in exactly two domains (

60

) and all three domains (

20

) from the total union (

155

) gives

155 - 60 - 20 = 75

Anahtar Kavram

Three-Set Venn Diagram Region Partitioning

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