Guests at a recent party ate a total of fifteen hamburgers. Each guest who was neither a student nor a vegetarian ate exactly one hamburger. No hamburger was eaten by any guest who was a student, a vegetarian, or both. If half of the guests were vegetarians, how many guests attended the party? We have 4 groups of guests:

1. Vegetarian students;

2. Vegetarian non-students;

3. Non-vegetarian students;

4. Non-vegetarian non-students.

Now, as guests ate a total of 15 hamburgers and each guest who was

neither a student nor a vegetarian (group #4) ate exactly one hamburger and also as

no hamburger was eaten by any guest who was a student, a vegetarian, or both (groups #1, #2 and #3) then this simply tells us that there were

15 non-vegetarian non-students at the party (group #4 = 15).

Make a matrix:

Note that we denoted total # of guests by \(x\) so both vegetarians and non-vegetarians equal to \(\frac{x}{2}\).

(1) The vegetarians attended the party at a rate of 2 students to every 3 non-students, half the rate for non-vegetarians --> \(\frac{vegetarian \ students}{vegetarian \ non-students}=\frac{2}{3}\) --> if the rate X (some fraction) is half of the rate Y (another fraction), then Y = 2*X --> \(\frac{non-vegetarian \ students}{non-vegetarian \ non-students}=2*\frac{2}{3}=\frac{4}{3}\) --> so,

non-vegetarian non-students compose 3/7 of all non vegetarians: \(non-vegetarian \ non-students = 15 = \frac{3}{7}*\frac{x}{2}\) --> \(x=70\). Sufficient.

(2) 30% of the guests were vegetarian non-students --> just says that # of \(vegetarian non-students\) equal to \(0.3x\) --> insufficeint, to calculate \(x\).

Answer: A.

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