A market research firm surveyed 200 smartphone owners regarding two security features enab… — GMAT

A market research firm surveyed $200$ smartphone owners regarding two security features enabled on their devices: Facial Recognition and Fingerprint Scanning. Exactly $140$ owners enabled Facial Recognition, $110$ owners enabled Fingerprint Scanning, and $30$ owners enabled neither feature. How many of the surveyed smartphone owners enabled Facial Recognition ONLY?

A
30
60Answer
C
80
D
90
E
140

Answer

60 smartphone owners enabled Facial Recognition only.

Out of

200

owners,

30

used neither feature, leaving

170

owners who enabled at least one feature. Using the overlapping set formula

\text{At Least One} = \text{Facial} + \text{Fingerprint} - \text{Both}

, we get

170 = 140 + 110 - \text{Both}

, so

\text{Both} = 80

. To find the number of owners who enabled Facial Recognition ONLY, we subtract the overlap from the total Facial Recognition count:

140 - 80 = 60

Step-by-Step Solution

Calculate the total number of owners who enabled at least one security feature.

200 - 30 = 170

owners.

Owners who used neither feature must be subtracted from the total universe of surveyed owners.

Apply the Overlapping Sets formula to find the number of owners who enabled both features.

\text{Total At Least One} = \text{Facial} + \text{Fingerprint} - \text{Both}

170 = 140 + 110 - \text{Both} \implies \text{Both} = 250 - 170 = 80

Summing individual sets double-counts the intersection area.

Subtract the intersection from the total Facial Recognition set to find Facial Recognition only.

140 - 80 = 60

owners.

The Facial Recognition group consists of those who enabled Facial Recognition only plus those who enabled both features.

Key Concept

Two-Set Overlapping Sets Formula: Total = Set A + Set B - Both + Neither

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