Guests at a recent party ate a total of fifteen hamburgers.

Question

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.

Bunuel · Accepted Answer

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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sriharimurthy · Answer

Good question.

Question Stem : 
 Total number of people attending the party , x = students + vegetarians + neither - both
Also, vegetarians = x/2 (this implies that non vegetarians also = x/2)
And, NVnonstudents = 15

St. (1) : Vstudents/ Vnonstudents = 2/3 ; NVstudents/NVnonstudents = 4/3
 The second ratio gives us NVstudents = 20
Therefore total Non Vegetarians = 20 + 15 = 35
This accounts for half the number of people at the party.
This total number of people = 70
Hence, Sufficient.

St. (2) : 30% were Vnonstudents.
 By itself, this statement gives us nothing.
Hence, Insufficient.

Answer : A

chaoswithin · Answer

First of all, we need to be able to find a solid number,the total number of guest, as an answer.

The only solid number we are given to work with is 15, the number of hamburgers eaten by the guests.

From the question we could see that the guests can be broken down into 4 categories.

VEGETARIAN STUDENT (V & S)
NON-VEGETARIAN STUDENT (NV & S)
VEGETARIAN NON-STUDENT (V & NS)
NON-VEGETARIAN NON-STUDENT (NV & NS)

Looking @ # of hamburgers eaten,

The question states that (NV & NS) ate exactly 1 hamburger and that 
no hamburger was eaten by any guest who was a student, hamburger eaten by (V & S) = (NV & S) = 0; a vegetarian, hamburger eaten by (V & S) = (V & NS) =0; or both, hamburger eaten by (V & S) = 0.

So from this we can conclude the # of (NV & NS) = 15.

The last piece of information given is that 1/2 Total = V, which also means 1/2 Total = NV, where NV + V = Total and NS + S = Total. Drawing a table can help understand this relationship.

Statement 1:

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

I think this should be reworded to  ratio  instead of rate.

Anyways this just means that for every 2 (V & S) there are 3 (V & NS), which is half the ratio of S to NS for NV. Therefore for every 4 (NV & S) there are 3 (NV & NS), which means  \frac{4}{3}=\frac{(NV & S)}{(NV & NS)}

Since we know that (NV & NS) = 15. We can solve for (NV & S) and Find Total because 1/2* Total = (NV & NS) + (NV & S)

Sufficient.

Statement 2:

30% of the guests were vegetarian non-students.

This give us no way to link 15 to the total number of guest. So insufficient.

This is more clear if you draw a table to help visualize things.

I'm sure someone will come up with a better explanation later, but I hope this can help till then.

KarishmaB · Answer

This is what I drew when I read the Question stem. Half of the guests were vegetarians so Total/2 stands for the complete vegetarian circle. All outside Vegetarian circle are Total/2. Ques.jpg Statement 1: Veg students : Veg non students = 2:3 Let me say they are 2x and 3x in number. Non veg students : Non veg non-students = 4:3 (Since veg's ratio is half of non veg's ratio) Let me say they are 4y and 3y in number. So now my diagram looks like this: Ques1.jpg 3y = 15 hence y = 5 Since 7y is half of the total, 35 is half of the total. So total number of students is 70. Sufficient. Statement 2: We get that 30% of the guests were veg non students and we already know that 50% of the guests are veg so 20% of the guests are veg students. Essentially, we have got the 3:2 ratio of above. But we do not have the 4:3 ratio of above hence we cannot equate 15 to anything. Therefore, statement 2 is not sufficient alone.

mbafall2011 · Answer

Good one! 
Stumbled in the test. I like the MGMAT solution.

Non veg - Non Students is 15.

Since Nonveg Student to non student ratio is 4:3, therefore non veg students will be 20.
4/3=x/15
Therefore x= 20. That makes 35 NV and 35 Veg so total 70

SOURH7WK · Answer

St 1: Sufficient: Veg attended in 2:3 so the Non veg attended in ratio 4:3. We now from Question stem Non veg - Non student ate 15 burgers. therefore Non veg attended the party in 20:15 (4:3). Total no of Non Veg = 35 nos. And total guests = 2*35=70 (from Question Stem).
St 2: Not sufficient: cant calculate other group of non-veg and only %age is provided.

Hence Answer A.

Krage17 · Answer

Bunuel, could you explain this statement: " --> so, non-vegetarian non-students compose 3/7 of all non vegetarians." Don't see how you get there. Thanks!

Bunuel · Answer

Since  non-vegetarian students  plus  non-vegetarian non-students  equals  all non-vegetarians , then  (non-vegetarian non-students/(total) = 3/(4 + 3) = 3/7 .

Hope it's clear.

Krage17 · Answer

It is. Took me a while to grasp it.

BrainLab · Answer

Hi Math Experts, I have one question regarding my approach for this question:

This is a rule From MGMAT

If a Data Sufficiency question asks for the concrete value of one element of a ratio, you will need
BOTH the concrete value of another element of the ratio AND the relative value of two elements
of the ratio

So, after reading Statement 1, I've marked Answer A, because Statement 1 gives us the relationship among all the groups + we have a concrete value from the question stem, hence, one can find the required value --> I've just pretty much followed the rule from MGMAT.... 
Do you think it's a valid approach for such kind of questions ?

manojach87 · Answer

I have outlined my approach in the attached file. Please share your thoughts about this approach. Thanks.

mbaapp1234 · Answer

The first step is to set up an overlapping sets table and insert the one known value: 15 non-vegetarian non-students. It's worded in a confusing way, but clearly the stimulus states that each relevant student at 1 hamburger each and only 15 hamburgers were eaten.

1) The ratio of vegetarian students to vegetarian non-students is 2:3, or half the ratio of non-vegetarians. So we know that non-vegetarians are in the ratio 4:3. 3x = 15, since we already know the number of non-vegetarian non-students, so x = 5. If x = 5, then the total number of vegetarians is 7(5) = 35. Since vegetarians are half of the guests, the total number of guests is 70. Sufficient.

2) This provides the relationship of the vegetarian non-student number to the total number of guests, but we don't know any other values. Not sufficient.

sahilvijay · Answer

main catch is 1 statement rate

Veg
2 student ---> 3 non student
1 s-------1.5 ns

half of it is for Non veg
1s-------1.5/2 ns

1s---------3/4 ns
4s => 3ns
4:3 ratio

=> 3x/7 = 15
x= 35
2x is total members => 70 is answer A.
since 2x are members x are veg and x are non veg
so took 4:3 s:ns for nonveg and since total number is x
so 3x/7 => x= 35 and 2x =70

and clearly 2 is insufficient

bumpbot · Answer

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Guests at a recent party ate a total of fifteen hamburgers.

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