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

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Guests at a recent party ate a total of fifteen hamburgers. [#permalink]  18 Nov 2009, 20:47
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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.
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Re: Guests at a recent party ate a total of fifteen hamburgers. [#permalink]  19 Nov 2009, 04:46
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kairoshan wrote:
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.

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.

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Re: data suffeciency [#permalink]  09 Nov 2010, 11:15
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.

Last edited by chaoswithin on 09 Nov 2010, 11:24, edited 2 times in total.
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Re: data suffeciency [#permalink]  09 Nov 2010, 11:19
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mrinal2100 wrote:
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.

can someone explain in detail

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:
Attachment:

Stem.PNG [ 2.33 KiB | Viewed 12452 times ]
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.
Attachment:

1.PNG [ 2.59 KiB | Viewed 12574 times ]

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

2.PNG [ 2.38 KiB | Viewed 12449 times ]

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Re: data suffeciency [#permalink]  10 Nov 2010, 18:16
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mrinal2100 wrote:
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.

can someone explain in detail

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.
Attachment:

Ques.jpg [ 17.29 KiB | Viewed 12375 times ]

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:
Attachment:

Ques1.jpg [ 18.33 KiB | Viewed 12380 times ]

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.
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Re: data suffeciency [#permalink]  25 Feb 2011, 22:19
I got this on the mgmat test and boy was this hard! is there a thread for similar questions?
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Re: Guests at a recent party ate a total of fifteen hamburgers. [#permalink]  27 Feb 2011, 18:24
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
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Re: Guests at a recent party ate a totaal of [#permalink]  20 Oct 2012, 22:59
nitzz wrote:
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.

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.

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Last edited by SOURH7WK on 21 Oct 2012, 08:36, edited 1 time in total.
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Re: Guests at a recent party ate a total of fifteen hamburgers. [#permalink]  22 Jul 2013, 23:17
Expert's post
From 100 hardest questions.

Bumping for review and further discussion.
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Guests at a recent party ate a total of fifteen hamburgers.. [#permalink]  05 Sep 2013, 08:52
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 other guests ate hamburgers. 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.

Can anyone please elaborate any method for solving the problem?
Unable to find an easy approach.
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Re: Guests at a recent party ate a total of fifteen hamburgers.. [#permalink]  05 Sep 2013, 08:55
Expert's post
sapto123 wrote:
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 other guests ate hamburgers. 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.

Can anyone please elaborate any method for solving the problem?
Unable to find an easy approach.

Merging similar topics. Please refer to the solutions above.
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Re: Guests at a recent party ate a total of fifteen hamburgers. [#permalink]  14 Sep 2014, 14:54
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Guests at a recent party ate a total of fifteen hamburgers. [#permalink]  01 Dec 2014, 04:54
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!

Last edited by Krage17 on 01 Dec 2014, 06:34, edited 1 time in total.
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Guests at a recent party ate a total of fifteen hamburgers. [#permalink]  01 Dec 2014, 05:14
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Krage17 wrote:
Bunuel, could you explain this statement: " --> so, non-vegetarian non-students compose 3/7 of all non vegetarians." Don't see how you there. Thanks!

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.
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Re: Guests at a recent party ate a total of fifteen hamburgers. [#permalink]  01 Dec 2014, 08:35
It is. Took me a while to grasp it.

Bunuel wrote:
Krage17 wrote:
Bunuel, could you explain this statement: " --> so, non-vegetarian non-students compose 3/7 of all non vegetarians." Don't see how you there. Thanks!

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.
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OVERLAPPING SETS [#permalink]  28 Jun 2015, 21:00
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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 vegetarian, a student, 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.
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Re: OVERLAPPING SETS [#permalink]  28 Jun 2015, 22:33
campus1995 wrote:
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 vegetarian, a student, 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.

Hello campus1995

This topic is discussed here
guests-at-a-recent-party-ate-a-total-of-fifteen-hamburgers-104547.html

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Re: OVERLAPPING SETS   [#permalink] 28 Jun 2015, 22:33
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