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At a conference, every attendee chose exactly one of three tracks: research, policy, or practice. Exactly 35 percent of the attendees chose research and exactly 24 percent chose policy. How many attendees chose practice?
(1) Fewer than 250 people attended the conference.
(2) More than 180 people attended the conference.
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Correct answer
C
In plain English first: you can't have a fractional person. So '35 percent chose research' only makes sense if 35 percent of the headcount lands on a whole number, and likewise for the 24 percent, and that quietly restricts how big the conference can be. That single observation turns the total from 'any number' into 'one of a short list,' which is what makes two range statements enough to pin it down.
Now the algebra. Let T be the total. Research = 0.35T = 7T/20 must be a whole count, so 20 divides T; policy = 0.24T = 6T/25 must be whole, so 25 divides T. Hence T is a multiple of 100: T ∈ {100, 200, 300...}. Practice = T − 0.35T − 0.24T = 0.41T, determined once T is, so 'how many chose practice?' is the same as 'what is T?'
Statement (1): T < 250 ⇒ T ∈ {100, 200} ⇒ practice = 41 or 82. not sufficient.
Statement (2): T > 180 ⇒ T ∈ {200, 300...}. not sufficient.
Together: 180 < T < 250 with 100 dividing T ⇒ T = 200 only ⇒ practice = 82. Unique; neither alone.
The continuous-space (E) trap fires on solvers who miss that the percentages force 100 to divide T; once seen, the two bounds bracket from opposite ends to a single value. Choosing (A) forgets the lower bound, (B) the upper bound.
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