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

Math questions from any Manhattan Prep GMAT Computer Adaptive Test.
Luci

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

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
Luci

### Tables

Sorry but the tables donÂ´t appear as I drew them.
Anyway they are the regular simple table

1st line NOTHING VEGETARIANS NON-VEGETARIANS TOTAL
2nd line STUDENTS CELL CELL CELL
3rd line NON-STUDENTS CELL CELL CELL
4th line TOTAL CELL CELL CELL

I guess everybody can imagine

Thanks
givemeanid

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

You can. But if you are plugging it for Veg, then 3a is not equal to 15. That is because 3a is for non-student, non-veg. Here, you are plugging in for veg.

Your answer, however, is what I got too. (A).
SoniaTandon

### MGMAT. EX5. Quest 2

Dear Luci

Since, you've already calculated that the no. of Non-veg guests are 35, and we know from statement 1 that the no. of veg. guests = non veg. guests,

35 in the ratio of 2:3 (Ratio mentioned for veg students v/s non- students) means 14 veg. students and 21 veg non-students.

Veg. Non-Veg

Students 14 20
Non Students 21 15

Hope this helps!!
Luci

### You are right

You are absolutely right, I dunno what I was thinking about, :-)

Thanks
jp.jprasanna
Students

Posts: 200
Joined: Thu Nov 03, 2011 3:48 am

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

Hi - I understand the complete sol, but What i don't get is the significance of this statement "No hamburger was eaten by any guest who was a student, a vegetarian, or both"

where does this statement fit in the matrix? please help?

Cheers
krishnan.anju1987
Forum Guests

Posts: 125
Joined: Mon May 07, 2012 6:13 pm

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

I think this statement fits into the explanation that all the 15 hamburgers are eaten by the non vegetarian non-students and none were eaten by any other groups. That fact is what gives us 3a=15. If not for this fact, 3a could be any value less than 15
tim
ManhattanGMAT Staff

Posts: 5665
Joined: Tue Sep 11, 2007 9:08 am
Location: Southwest Airlines, seat 21C

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

thanks; let us know if there are any further questions on this one..
Tim Sanders
Manhattan GMAT Instructor

Follow this link for some important tips to get the most out of your forum experience:
https://www.manhattanprep.com/gmat/forums/a-few-tips-t31405.html
asharma8080
Course Students

Posts: 22
Joined: Wed Aug 17, 2011 7:36 am

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

I have the same question as the poster. I got this wrong as I am getting 25 veg and 35 non-veg.Does the fact that the # of veg = # of non-veg take precedence over ratio??

Is the "a" after the 2 different than the a after the 4?
2a + 3a = 4a + 3a

If a = 5, then 2a = 10 and 3a = 15, and # of veg = 25
But with the other method, we know
2a + 3a = 4a + 3a
2a + 3a = 35
3a = 15
so 2a = 20
Now, a is 10...? I am confused why a changes to be something else.

4a + 3a = 35
RonPurewal
Students

Posts: 19747
Joined: Tue Aug 14, 2007 8:23 am

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

asharma8080, the two ratios given in the problem (the ratios 2:3 and 4:3) are separate ratios, so you can't use the same coefficient letter "a" for both of them.
if you write the things in the 2:3 ratio as 2a and 3a, and you also write the things in the 4:3 ratio as 4a and 3a, then you are assuming -- incorrectly, as it turns out -- that all four quantities are in a fixed ratio of 2:3:4:3.
this is why you seem to be finding a contradiction here: the relationship that you've (accidentally) assumed, here, is impossible given that there are equal numbers of vegetarians and non-vegetarians.

instead, if you denote the things in the 2:3 ratio as 2a and 3a, then you should use a different letter for the things in the other ratio, e.g., 4b and 3b.