I'm not sure I understand the answer to this problem from Cat 2, would someone mind clarifying it for me?

Recently Mary gave a birthday party for her daughter at which she served both chocolate and strawberry ice cream. There were 8 boys who had chocolate ice cream, and nine girls who had strawberry. Everybody there had some ice cream, but nobody tried both. What is the maximum possible number of girls who had some chocolate ice cream?

(1) Exactly thirty children attended the party.

(2) Fewer than half the children had strawberry ice cream.

The answer is said to be A, because you can create a matrix from the data with four variables that you can maximize and minimize in order to determine the amount of girls who had some chocolate ice cream. My question is, don't you have to at least know the ratio of girls to boys before you attempt to answer the question, a ratio that is not hinted at in the answers or question. Because if that were the case than the answer would be E. If you can make up the ratio of boys to girls at the party, how do I determine whether a question will allow me to do so?

Thanks a lot for your help in advance!

You would need a ratio if the question asked for the exact number of girls that had chocolate ice cream. However, it asks for the maximum. To get the maximum, I assume there are only 8 boys total and 9 girls that had strawberry. "A" tells us there are 30 kids total. Subtracting the 8 boys, leaves a maximum of 22 girls. The problem states that 9 of the girls had strawberry which leaves the 13 remaining girls the maximum number to have chocolate.

HTH

Nice explanation by HTH. I just want to make a general point:

In general, when you are asked to maximize one quantity in a problem, you will have to minimize everything else.

Had this been a problem solving question of this type, they would always give you a total, because max/min problems are all about the trade-off between variables. (e.g. If x + y = 100, [max x + min y = 100] and [min x + max y = 100].)

If you know the total is always required information, (1) might be more obviously sufficient.

4 types of children:
1. Strawberry-Boys-->minimize-->0
2. Strawberry-Girls = 9 (exact number known)
3. Chocolate-Boys = 8 (exact number known)
4. Chocolate-Girls-->maximize-->x

The trade-off is between children of type 1 and 4. We only need to know the total number of children because 30 (total) - 9 (of type 2) - 8 (of type 3) = 13 (of types 1 and 4 combined). If 13 = 0 + max x, max x = 13.

_________________
Emily Sledge
Instructor
ManhattanGMAT

An open question to the community.

If statement one remains unchanged but statement 2 is as follows:

2) More than half the children had strawberry ice cream.

What is the answer now? A followup question: If this changes the answer, does that mean that the original question's answer should now be changed?

If Statement (2) is changed to More than half the children had strawberry ice cream, then taking both conditions together gives the following distribution where we try to maximize Girls and Chocolate (there are others but they lead to the same result:

Boys and Chocolate: 8
Girls and Strawberry: 9
Boys and Strawberry: 7, 8, 9,10, 11, 12 or 13
Girls and Chocolate : 6, 5, 4, 3, 2, 1 or 0
where the numbers in the columns correspond to each other.

This means that the MAX number of girls who can try chocolate ice if more than half the children had strawberry ice cream AND the number of children is limited to 30 is 6. This is inconsistent with the answer we received from Statement 1 alone.

So the Answer here is C NOT A. [Oops! I corrected this statement now.]

I don't follow you Cosimo. Are you combining the original version of S2 with this new version of S2? If so, they contradict on their face, and so you could save quite a lot of trouble.

This new version of S2 works perfectly fine with S1, and in fact your account gives us the correct answer to the re-imagined question: 6.

By the way, as Emily pointed out, the way to maximize the value in one cell of the DSM is to minimize the other values. Careful, though. The four values that form the overlap of the first two rows and the first two columns are the relevant set here. It is these four that add up to 30. So to MAXIMIZE the number of girls with chocolate, MINIMIZE the other three cells in that set of four. Two of those are fixed at 8 and 9, so we need only MINIMIZE the number boys with strawberry. Since the total number of strawberry is boys with strawberry plus 9, we need to MINIMIZE the total number of strawberry. The least integer value greater than 15 is 16. 16-9 = 7. 30-(7+8+9)=6.

As it happens, it would have been more efficient in this case to set aside Emily's injunction, and MAXIMIZED the total chocolate, in order to MAXIMIZE total chocolate minus 8, but that would have required a bit of insight. Emily's approach is more reliable.

Oh, by the way, there will never be an actual question where the two statements contradict. The answer wouldn't be E. It just wouldn't happen.

I will try to clarify my concern Michael, I do admit that the point is rather subtle. Imagine the question to be as follows:

Recently Mary gave a birthday party for her daughter at which she served both chocolate and strawberry ice cream. There were 8 boys who had chocolate ice cream, and nine girls who had strawberry. Everybody there had some ice cream, but nobody tried both. What is the maximum possible number of girls who had some chocolate ice cream?

(1) Exactly thirty children attended the party.

(2) MORE than half the children had strawberry ice cream.

Looking at Statement One alone we arrive at the maximum number of girls who could have ate chocolate ice cream to be 13. I agree with this completely.

Looking at Statement Two by itself tells us nothing as there could be anywhere from 17 to an infinite number of children that attended the birthday party.

Combining the two statements gives us the maximum number of girls who could have tried chocolate ice cream to be six. You agreed with this in your earlier post so I will not explain the reasoning again.

My point is this. Statement One by itself gives an answer of 13. But if I combine both statements, the answer is 6. This is internally inconsistent. If statement one alone does not give the “correct” final answer, then it cannot be said to answer the question alone. Hence, the answer for the modified question should be C and not A.

I do apologize as I have just noticed that I posted E should be the answer in a previous post. I meant C and I hope that this has not caused too much confusion. The inconsistency is NOT present in the original question because both statement one and the combined statements give an answer of 13.

As I mentioned before, this is a subtle point and if I have not expressed myself clearly, I am more than willingly to let it go. I doubt the GMAT would pose such a question… and if they do I will do my best to answer come test day.

OK, I think that I understand. I should have understood earlier, because your concerns about the modified version of S2 mirrors my concern about the question portion here. Remember that the info in the question portion is never sufficient without at least one of the statements. (There is one exception in the GMAT Prep, just a badly written question). But if we apply your reasoning about about the modified version of S2 to this question portion, we get,

boys who had chocolate=8
girls who had strawberry=9
0<boys who had strawberry<boys in the world-8
0<girls who had chocolate<girls in the world-9
maximize girls who had chocolate

Well, there's your maximum right there.

I think that we were a little careless with our language, and I'll bounce this off the curriculum committee. I'd like to look at the actual GMAT questions that demand similar reasoning, to see whether they are similarly careless. I'd be surprised if they are.