A tech hub tracked 150 start-up companies to determine their use of three cloud services: … — GMAT

A tech hub tracked $150$ start-up companies to determine their use of three cloud services: Storage ( $S$ ), Analytics ( $A$ ), and Machine Learning ( $M$ ). Every company uses at least one of the three services. Exactly $95$ companies use Storage, $80$ use Analytics, and $65$ use Machine Learning. If exactly $20$ companies use all three services, and $45$ companies use exactly two of the services, how many companies use exactly one of the three cloud services?

A
$65$
$75$ Cevap
C
$85$
D
$95$
E
$105$

Cevap

The number of companies that use exactly one of the three cloud services is

75

Using the region-based formula for three overlapping sets, the total number of elements is equal to the sum of elements in exactly one set, elements in exactly two sets, elements in all three sets, and elements in none of the sets. Here, Total

= 150

, Exactly Two

= 45

, Exactly Three

= 20

, and Neither

= 0

. Substituting these values gives

150 = \text{Exactly One} + 45 + 20 + 0

, which simplifies to

\text{Exactly One} = 75

Adım Adım Çözüm

State the standard 3-set overlapping sets formula in terms of exact regions.

Total

= (\text{Exactly 1}) + (\text{Exactly 2}) + (\text{Exactly 3}) + (\text{Neither})

Dividing the Venn diagram into mutually exclusive regions simplifies calculating specific counts.

Substitute the known values into the regional total equation.

150 = (\text{Exactly 1}) + 45 + 20 + 0

Since every company uses at least one service,

\text{Neither} = 0

Solve for the number of companies using exactly one service.

\text{Exactly 1} = 150 - 65 = 75

Subtracting the sum of companies using exactly two services (

45

) and all three services (

20

) from the total (

150

) yields

75

Anahtar Kavram

Three-Set Overlapping Sets Regional Decomposition

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