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A certain fruit stand sells only........Veritas question

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A certain fruit stand sells only........Veritas question

by Mo2men » Sat Feb 11, 2017 2:46 am
A certain fruit stand sells only apples for $0.26 each, bananas for $0.24 each, and cantaloupes for $0.65 each. Can Clark spend exactly $5.00 at the fruit stand buying fruit that he likes?

(1) Clark does not like bananas.
(2) Clark does not like cantaloupes.

Source: Veritas
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Source: — Data Sufficiency |
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by GMATGuruNY » Sat Feb 11, 2017 4:25 am
Mo2men wrote:A certain fruit stand sells only apples for $0.26 each, bananas for $0.24 each, and cantaloupes for $0.65 each. Can Clark spend exactly $5.00 at the fruit stand buying fruit that he likes?

(1) Clark does not like bananas.
(2) Clark does not like cantaloupes.
Statement 1:
Since Clark may buy only apples and/or cantaloupes, we get:
26a + 65c = 500
13(2a + 5c) = 500
2a + 5c = 500/13.
If a and c are integers, then 2a + 5c cannot sum to a non-integer such as 500/13.
Implication:
a and c cannot both be integers.
Since the equation in red does not have an integral solution, the answer to the question stem is NO:
It is not possible for Clark to spend exactly 500 cents on the fruit he likes.
SUFFICIENT.

Statement 2:
Since Clark may buy only apples and/or bananas, we get:
26a + 24b = 500
13a + 12b = 250.
Since the right side is a multiple of 5, test multiples of 5 for the left side.
13a = 65, 130, 195
12b = 60, 120...
The values in blue sum to 250, implying that 13a + 12b = 250 has an integral solution.

Thus, if Clark likes apples and bananas, then the answer to the question stem is YES:
It is possible for him to spend exactly 500 cents on the fruit he likes.
But if Clark does NOT like apples and bananas, then the answer to the question stem is NO:
It is NOT possible for him to spend exactly 500 cents on the fruit he likes.

Since the answer could be YES or NO, INSUFFICIENT.

The correct answer is A.
Last edited by GMATGuruNY on Thu Feb 16, 2017 4:12 am, edited 2 times in total.
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by [email protected] » Sat Feb 11, 2017 11:22 am
Hi Mo2men,

Mitch's assessment of this question is spot-on, so I won't rehash any of that here. Does the source of this question include a correct answer and explanation (and if so, then what does the source offer?)?

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by Jay@ManhattanReview » Sun Feb 12, 2017 11:06 pm
Mo2men wrote:A certain fruit stand sells only apples for $0.26 each, bananas for $0.24 each, and cantaloupes for $0.65 each. Can Clark spend exactly $5.00 at the fruit stand buying fruit that he likes?

(1) Clark does not like bananas.
(2) Clark does not like cantaloupes.

Source: Veritas
Hi Mo2men,

I agree with Mitch and Rich. This question is either incorrectly copied or has a flaw.

One important aspect of DS question is that the prompt, statement 1, and statement 2 together present a holistic scenario. Even if one gets an answer to a 'What is the value?" DS question as D, one must ensure that answers from each statement return the same value.

Say, for example, if for a question, asking for the value x, the derived value from statement 1 is '2' and that from statement 2 is '3,' though each value is unique, one must check his/her work. x would either be 2 or 3 but not both. Same goes with Yes/No type of questions. If the answer is D, each statement should return either Yes/Yes or No/No.

As Mitch demonstrated that the answer from statement 1 is No and that from statement 2 is Yes, there is a flaw in the question.

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Hope this helps!

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by AnthonyRitz » Thu Feb 16, 2017 1:01 am
Hey guys! Note that Clark has to purchase fruit THAT HE LIKES. And we don't ever actually find out any fruits that he likes. So the only way to get sufficiency is with an answer of NO. Does that help?

I figure it's okay with Veritas if I post our official solution here, so here you go:

At the outset, we don't know what kinds of fruit Clark likes, so there's no way to determine whether he can spend $5.00 on fruit that he likes. And it's important to note that neither additional premise ever establishes any kind of fruit that Clark actually does like. It will not be possible at any point to prove that Clark *can* spend exactly $5.00 on fruit that he likes, so the only way in which sufficiency might be obtained is to prove that Clark in fact *cannot* spend exactly $5.00 on fruit that he likes.

If, as statement (1) establishes, Clark does not like bananas, then the only way that he can spend $5.00 on fruit that he likes is if he likes apples and/or cantaloupes and if the equation 26A + 65C = 500 has at least one solution for non-negative integers A and C. Note, however, that both 26 and 65 are multiples of 13. Factoring gives

13 * (2A + 5C) = 500

The left side is a multiple of 13, but the right side is not. But this is impossible for any non-negative integers A and C, so Clark definitely cannot spend exactly $5.00 buying fruit that he likes. Statement 1 is sufficient.

Statement 2 establishes that Clark does not like cantaloupes. In order for Clark to spend exactly $5.00 on fruit that he likes, he would have to like apples and/or bananas, and the equation 26A + 24B = 500 would have to have at least one solution for non-negative integers A and B. We can reduce the equation slightly by dividing through by 2. The result is

13A + 12B = 250

Since A, B, and C must be non-negative integers, this is what's known as a linear Diophantine equation. Students who have studied such equations may be aware that the linear Diophantine equation Mx + Ny = Q will always have at least one non-negative integer solution (x, y) provided that the greatest common divisor of M and N is 1 and that Q > MN - M - N. Both conditions are met here, so a solution exists.

Other students may wish to apply a bit of trial and error to actually determine such a solution. Note that 25 can be obtained as 13*1 + 12*1. Thus 250 can be expressed as 13*10 + 12*10, and one solution (in fact the only one) is for Clark to buy 10 apples and 10 bananas.

Either way, since Clark may or may not like apples and bananas, he may or may not be able to spend exactly $5.00 in this manner, and statement 2 is insufficient.

So, how does Clark like them apples? It doesn't really matter. The answer is A.

P.s. I wrote this question and solution, so feel free to yell at me if you hate it.
Last edited by AnthonyRitz on Thu Feb 16, 2017 10:37 am, edited 1 time in total.
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by GMATGuruNY » Thu Feb 16, 2017 3:54 am
AnthonyRitz wrote:Note that Clark has to purchase fruit THAT HE LIKES. And we don't ever actually find out any fruits that he likes.
Good point.
I've edited my response accordingly.
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by Mo2men » Thu Feb 16, 2017 1:44 pm
[email protected] wrote:Hi Mo2men,

Mitch's assessment of this question is spot-on, so I won't rehash any of that here. Does the source of this question include a correct answer and explanation (and if so, then what does the source offer?)?

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Hi Rich,

As requested, the official answer is posted above by the creator of the question.
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by Matt@VeritasPrep » Fri Mar 03, 2017 3:28 am
Jay@ManhattanReview wrote:As Mitch demonstrated that the answer from statement 1 is No and that from statement 2 is Yes, there is a flaw in the question.
But there isn't! "Clark does not like cantaloupes" ≠> "Clark DOES like apples and bananas". He might like both of these fruits, but we can't say for certain, meaning there's no definitive answer. (If he likes apples and bananas, the answer is yes, but if he only likes apples, the answer is no.)
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by AnthonyRitz » Fri Mar 03, 2017 9:41 pm
Matt,

Thanks, and I of course agree with you (it is, after all, my question). I don't think it's ambiguous, especially when you read the second statement in light of the existence of the first.

HOWEVER, while my goal was certainly to write a hard question, tricking/confusing multiple extremely bright GMAT instructors suggests that even if the question is valid it might nevertheless be harder than intended. I didn't intend to spark metaphysical discussions on the meaning of "can" and the nature of free will (believe it or not, I've gotten at least one of those because of this question so far).

In the interest of maximal clarity, we're looking at switching to the following question wording (which should not, in my view, change either the correct answer or its explanation):

"A certain fruit stand sells only apples for $0.26 each, bananas for $0.24 each, and cantaloupes for $0.65 each. If Clark spends exactly $5.00 at the fruit stand, does Clark like all of the fruit that he buys?

(1) Clark does not like bananas.
(2) Clark does not like cantaloupes."
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