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Luci

Post subject: Guests at a recent party ate a total of fifteen hamburgers. Posted: Tue Jul 24, 2007 7:17 am 


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 nonstudents, half the rate for nonvegetarians.
(2) 30% of the guests were vegetarian nonstudents.
The answer is A. (1) is sufficient. And here is the explanation:
For this overlapping set problem, we want to set up a twoset table to test our possibilities. Our first set is vegetarians vs. nonvegetarians; our second set is students vs. nonstudents.
/ VEG / NONVEG / TOTAL
_____________/______/__________/_____
STUDENT / / /
_____________/______/__________/_____
NONSTUDENT / / 15 /
_____________/______/__________/_____
TOTAL / x / x / ?
(It is dranw in a very rudimmentary way ;))
We are told that each nonvegetarian nonstudent 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 nonvegetarian nonstudent category. We are also told that the total number of vegetarians was equal to the total number of nonvegetarians; 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 bottomright 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 nonstudent"; 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 nonstudents. We're also told that this 2:3 rate is half the rate for nonvegetarians. In order to double a rate, we double the first number; the rate for nonvegetarians is 4:3 We can represent the actual numbers of nonvegetarians as 4a and 3a and add this to the chart below. Since we know that there were 15 nonvegetarian nonstudents, we know the missing common multiple, a, is 15/3 = 5. Therefore, there were (4)(5) = 20 nonvegetarian students and 20 + 15 = 35 total nonvegetarians (see the chart below). Since the same number of vegetarians and nonvegetarians attended the party, there were also 35 vegetarians, for a total of 70 guests.
/ VEG / NONVEG / TOTAL
_____________/______/__________/_____
STUDENT / / 4a or 20 /
_____________/______/__________/_____
NONSTUDENT / / 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 nonveg
/ VEG / NONVEG / TOTAL
_____________/______/__________/_____
STUDENT / 2a / /
_____________/______/__________/_____
NONSTUDENT / 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 / NONVEG / TOTAL
_____________/________/__________/_____
STUDENT / 2a or 10 / 4a or 20 /
_____________/________/__________/_____
NONSTUDENT / 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 nonveg.
IÃ‚Â´m probably missing something here, but can you explain what am I missing?
Thanks





Luci

Post subject: Tables Posted: Tue Jul 24, 2007 7:22 am 


Sorry but the tables donÃ‚Â´t appear as I drew them.
Anyway they are the regular simple table
1st line NOTHING VEGETARIANS NONVEGETARIANS TOTAL
2nd line STUDENTS CELL CELL CELL
3rd line NONSTUDENTS CELL CELL CELL
4th line TOTAL CELL CELL CELL
I guess everybody can imagine
Thanks





givemeanid

Post subject: Posted: Tue Jul 24, 2007 11:09 am 


Quote: 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 nonveg
You can. But if you are plugging it for Veg, then 3a is not equal to 15. That is because 3a is for nonstudent, nonveg. Here, you are plugging in for veg.
Your answer, however, is what I got too. (A).





SoniaTandon

Post subject: MGMAT. EX5. Quest 2 Posted: Tue Jul 24, 2007 11:43 am 


Dear Luci
Since, you've already calculated that the no. of Nonveg 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 nonstudents.
Veg. NonVeg
Students 14 20
Non Students 21 15
Hope this helps!!





Luci

Post subject: You are right Posted: Tue Jul 24, 2007 11:49 am 


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





jp.jprasanna

Post subject: Re: Guests at a recent party ate a total of fifteen hamburgers. Posted: Wed Aug 15, 2012 9:08 am 


Students 

Posts: 203

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

Post subject: Re: Guests at a recent party ate a total of fifteen hamburgers. Posted: Fri Aug 17, 2012 2:04 pm 


Forum Guests 

Posts: 125

I think this statement fits into the explanation that all the 15 hamburgers are eaten by the non vegetarian nonstudents 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

Post subject: Re: Guests at a recent party ate a total of fifteen hamburgers. Posted: Tue Aug 21, 2012 1:13 pm 


ManhattanGMAT Staff 

Posts: 4994 Location: Southwest Airlines, seat 21C

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





asharma8080

Post subject: Re: Guests at a recent party ate a total of fifteen hamburgers. Posted: Sat Nov 03, 2012 7:51 pm 


Course Students 

Posts: 23

I have the same question as the poster. I got this wrong as I am getting 25 veg and 35 nonveg.Does the fact that the # of veg = # of nonveg take precedence over ratio?? Is the "a" after the 2 different than the a after the 4? 2 a + 3a = 4a + 3a If a = 5, then 2a = 10 and 3a = 15, and # of veg = 25 But with the other method, we know 2 a + 3a = 4a + 3a 2 a + 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

Post subject: Re: Guests at a recent party ate a total of fifteen hamburgers. Posted: Mon Nov 05, 2012 11:20 pm 


ManhattanGMAT Staff 

Posts: 13070

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 nonvegetarians.
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.
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