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?

(1) The vegetarians attended the party at a rate of 2 students to every 3 non-students, half the rate for non-vegetarians.

(2) 30% of the guests were vegetarian non-students.

The answer is A. (1) is sufficient. And here is the explanation:

For this overlapping set problem, we want to set up a two-set table to test our possibilities. Our first set is vegetarians vs. non-vegetarians; our second set is students vs. non-students.

/ VEG / NON-VEG / TOTAL

_____________/______/__________/_____

STUDENT / / /

_____________/______/__________/_____

NON-STUDENT / / 15 /

_____________/______/__________/_____

TOTAL / x / x / ?

(It is dranw in a very rudimmentary way ;-))

We are told that each non-vegetarian non-student ate exactly one of the 15 hamburgers, and that nobody else ate any of the 15 hamburgers. This means that there were exactly 15 people in the non-vegetarian non-student category. We are also told that the total number of vegetarians was equal to the total number of non-vegetarians; we represent this by putting the same variable in both boxes of the chart.

The question is asking us how many people attended the party; in other words, we are being asked for the number that belongs in the bottom-right box, where we have placed a question mark.

The second statement is easier than the first statement, so we'll start with statement (2).

(2) INSUFFICIENT: This statement gives us information only about the cell labeled "vegetarian non-student"; further it only tells us the number of these guests as a percentage of the total guests. The 30% figure does not allow us to calculate the actual number of any of the categories.

(1) SUFFICIENT: This statement provides two pieces of information. First, the vegetarians attended at the rate, or in the ratio, of 2:3 students to non-students. We're also told that this 2:3 rate is half the rate for non-vegetarians. In order to double a rate, we double the first number; the rate for non-vegetarians is 4:3 We can represent the actual numbers of non-vegetarians as 4a and 3a and add this to the chart below. Since we know that there were 15 non-vegetarian non-students, we know the missing common multiple, a, is 15/3 = 5. Therefore, there were (4)(5) = 20 non-vegetarian students and 20 + 15 = 35 total non-vegetarians (see the chart below). Since the same number of vegetarians and non-vegetarians attended the party, there were also 35 vegetarians, for a total of 70 guests.

/ VEG / NON-VEG / TOTAL

_____________/______/__________/_____

STUDENT / / 4a or 20 /

_____________/______/__________/_____

NON-STUDENT / / 3a or 15 /

_____________/______/__________/_____

TOTAL /x or 35 / x or 35 / ? or 70

The correct answer is A.

But in this explanation something does not fit, because as stated in (1) if vegetarians attended in the rate 2:3 we could similarly draw as we did for non-veg

/ VEG / NON-VEG / TOTAL

_____________/______/__________/_____

STUDENT / 2a / /

_____________/______/__________/_____

NON-STUDENT / 3a / /

_____________/______/_________/_____

TOTAL / / /

and so if we already know that 3a=15, then this will lead to 2a= 10 and 3a=15 that will give us a total of 25 and the final table will be:

/ VEG / NON-VEG / TOTAL

_____________/________/__________/_____

STUDENT / 2a or 10 / 4a or 20 /

_____________/________/__________/_____

NON-STUDENT / 3a or 15 / 3a or 15 /

_____________/________/__________/_____

TOTAL / x or 25 / x or 35 / ? or 60

But this will not be valid for the premise that half of the guest were vegetarians because here we have 25 veg and 35 non-veg.

IÂ´m probably missing something here, but can you explain what am I missing?

Thanks