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At a professional training conference, 44 attendees signed up for the data analytics workshop and 31 signed up for the public speaking workshop. Were at least 60 attendees at the conference?
(1) At least 16 attendees signed up for neither workshop.
(2) \(\frac{1}{4}\) of the attendees signed up for both workshops.
(1) At least 16 attendees signed up for neither workshop.
(2) \(\frac{1}{4}\) of the attendees signed up for both workshops.
30 May 2026, 09:52
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Official Solution:
At a professional training conference, 44 attendees signed up for the data analytics workshop and 31 signed up for the public speaking workshop. Were at least 60 attendees at the conference?
(1) At least 16 attendees signed up for neither workshop.
\(\{Total\} = \{data \ analytics\} + \{public \ speaking\} - \{both\} + \{neither\}\)
\(\{Total\} = 44 + 31 - \{both\} + \{neither\}\)
\(\{Total\} = 75 - \{both\} + \{neither\}\)
Since the greatest possible value of \(\{both\}\) is 31, \(75 - \{both\}\) is at least 44, and since \(\{neither\}\) is at least 16, we get:
\(\{Total\} = 75 - \{both\} + \{neither\} \geq 60\)
So the total number of attendees must be at least 60. Sufficient.
(2) \(\frac{1}{4}\) of the attendees signed up for both workshops.
This gives:
\(\{Total\} = \{data \ analytics\} + \{public \ speaking\} - \{both\} + \{neither\}\)
\(\{Total\} = 44 + 31 - \frac{\{Total\}}{4} + \{neither\}\)
\(\frac{5}{4} * \{Total\} = 75 + \{neither\}\)
Multiply by \(\frac{4}{5}\):
\(\{Total\} = 60 + \frac{4}{5} * \{neither\}\)
Since \(\{neither\}\) cannot be negative, the total number of attendees must be at least 60. Sufficient.
Answer: D
At a professional training conference, 44 attendees signed up for the data analytics workshop and 31 signed up for the public speaking workshop. Were at least 60 attendees at the conference?
(1) At least 16 attendees signed up for neither workshop.
\(\{Total\} = \{data \ analytics\} + \{public \ speaking\} - \{both\} + \{neither\}\)
\(\{Total\} = 44 + 31 - \{both\} + \{neither\}\)
\(\{Total\} = 75 - \{both\} + \{neither\}\)
Since the greatest possible value of \(\{both\}\) is 31, \(75 - \{both\}\) is at least 44, and since \(\{neither\}\) is at least 16, we get:
\(\{Total\} = 75 - \{both\} + \{neither\} \geq 60\)
So the total number of attendees must be at least 60. Sufficient.
(2) \(\frac{1}{4}\) of the attendees signed up for both workshops.
This gives:
\(\{Total\} = \{data \ analytics\} + \{public \ speaking\} - \{both\} + \{neither\}\)
\(\{Total\} = 44 + 31 - \frac{\{Total\}}{4} + \{neither\}\)
\(\frac{5}{4} * \{Total\} = 75 + \{neither\}\)
Multiply by \(\frac{4}{5}\):
\(\{Total\} = 60 + \frac{4}{5} * \{neither\}\)
Since \(\{neither\}\) cannot be negative, the total number of attendees must be at least 60. Sufficient.
Answer: D
