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

[Luci]

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.

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

Thanks

[Luci]

Sorry but the tables don´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]

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

Thanks

[jp.jprasanna]

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]

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]

thanks; let us know if there are any further questions on this one..

[asharma8080]

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]

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.