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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.

Re: A certain fruit stand sells only apples for $0.26 each, bananas for $0 [#permalink]

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11 Feb 2017, 00:30

Bunuel 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.

Converting all the cost into cents for the ease of calculation. So cost of an Apple(A) - 26cents, a Banana(B)-24 cents, and a Cantaloupe(C) - 65 cents. Total amount Clark has is 500 cents.

(1) Clark does not like bananas - Assuming he has to buy Apples and Cantaloupes. nA+mC = 500, where n, and m are no. of Apples and Cantaloupes respectively. No combinations are possible so he cannot spend exactly 5$. So sufficient.

(2) Clark does not like cantaloupes - Assuming he has to buy Apples and Bananas. nA+mB=500. 10 Apples and 10 Bananas cost 260+240=500. So he can spend exactly 5$. - Sufficient

A certain fruit stand sells only apples for $0.26 each, bananas for $0 [#permalink]

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15 Feb 2017, 22:31

Bunuel 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.

Dear Bunuel,

This question is flawed. We can reach the answer from each statement alone but each statement contradicts each other. When you reveal the answer, can you post the official answer by Veritas?

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.

Dear Bunuel,

This question is flawed. We can reach the answer from each statement alone but each statement contradicts each other. When you reveal the answer, can you post the official answer by Veritas?

Thanks

Dear Mo2men,

I'm happy to help.

Here is my interpretation. Statement #1: Clark does not like bananas. Does he like cantaloupes? Maybe or maybe not: we don't know. Does he like apples? Maybe or maybe not: we don't know. If he likes just cantaloupes, or just apples, or both cantaloupes & apples, then in any of those combinations, it's impossible to spend exactly $5.00. You see, both 26¢ and 65¢ are multiples of 13, and any combination of something times the first plus something times the second will still have to be divisible by 13. Since 500 is NOT divisible by 13, there's no way to combine apples & cantaloupes, separately or together, to get $5.00. Of course, if he doesn't like any of the three fruits, then he can't possibly spend the money on fruit he likes! Every scenario under this statement gives a resounding "NO" to the prompt statement. Because we can give a definitive answer, this statement, alone and by itself, is sufficient.

Statement #2: Clark does not like cantaloupes. How could anyone not love cantaloupes? They're delicious! Does he like bananas? Maybe or maybe not: we don't know. Does he like apples? Maybe or maybe not: we don't know. Consider these two scenarios (a) Clark likes both apples & bananas. One apples + one bananas = 50¢. Multiply this by ten: 10 apples & 10 bananas = $5.00 exactly. This gives a "yes" answer to the prompt. (b) Clark doesn't like bananas; he only like apples. No multiple of 26¢ can equal exactly $5.00. This gives a "no" answer to the prompt. If the question were "does he spend exactly $5.00," the answer would be no. But the exact prompt is "can he spend exactly $5.00?" Yes, scenario (a) clearly provides a right by which he can spend $5.00. We can answer "yes" to the prompt question. Because we can give a definitive answer, this statement, alone and by itself, is sufficient.

Both statements sufficient. Answer = (D).

But, Mo2men, I believe you are right. Something is fishy here. The scenario that makes statement #2 work is explicitly prohibited by statement #1, so the statements together would not work. Even though we can get to an answer, this question doesn't have the deep logical coherence that an official question would have. It's always true in official DS questions, even questions that have answers of (A), (B), or (D), that all statements can be true as part of the answer scenario. I believe there is a design flaw in this question.

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.

Dear Bunuel,

This question is flawed. We can reach the answer from each statement alone but each statement contradicts each other. When you reveal the answer, can you post the official answer by Veritas?

Thanks

Dear Mo2men,

I'm happy to help.

Here is my interpretation. Statement #1: Clark does not like bananas. Does he like cantaloupes? Maybe or maybe not: we don't know. Does he like apples? Maybe or maybe not: we don't know. If he likes just cantaloupes, or just apples, or both cantaloupes & apples, then in any of those combinations, it's impossible to spend exactly $5.00. You see, both 26¢ and 65¢ are multiples of 13, and any combination of something times the first plus something times the second will still have to be divisible by 13. Since 500 is NOT divisible by 13, there's no way to combine apples & cantaloupes, separately or together, to get $5.00. Of course, if he doesn't like any of the three fruits, then he can't possibly spend the money on fruit he likes! Every scenario under this statement gives a resounding "NO" to the prompt statement. Because we can give a definitive answer, this statement, alone and by itself, is sufficient.

Statement #2: Clark does not like cantaloupes. How could anyone not love cantaloupes? They're delicious! Does he like bananas? Maybe or maybe not: we don't know. Does he like apples? Maybe or maybe not: we don't know. Consider these two scenarios (a) Clark likes both apples & bananas. One apples + one bananas = 50¢. Multiply this by ten: 10 apples & 10 bananas = $5.00 exactly. This gives a "yes" answer to the prompt. (b) Clark doesn't like bananas; he only like apples. No multiple of 26¢ can equal exactly $5.00. This gives a "no" answer to the prompt. If the question were "does he spend exactly $5.00," the answer would be no. But the exact prompt is "can he spend exactly $5.00?" Yes, scenario (a) clearly provides a right by which he can spend $5.00. We can answer "yes" to the prompt question. Because we can give a definitive answer, this statement, alone and by itself, is sufficient.

Both statements sufficient. Answer = (D).

But, Mo2men, I believe you are right. Something is fishy here. The scenario that makes statement #2 work is explicitly prohibited by statement #1, so the statements together would not work. Even though we can get to an answer, this question doesn't have the deep logical coherence that an official question would have. It's always true in official DS questions, even questions that have answers of (A), (B), or (D), that all statements can be true as part of the answer scenario. I believe there is a design flaw in this question.

Aw, man, Mike, you're really busting our balls today. We're gonna have to have a question-wording summit or something. (Feel free to PM me.)

I wrote this question, and I don't really agree with the reading you're advancing. After all, in statement 2, he can do it *if* he likes apples and bananas. That's a condition separate from the one we're considering, and it isn't answered anywhere. You sort of reinterpreted the question in your response. You said

Quote:

But the exact prompt is "can he spend exactly $5.00?"

but you left out the end of it. The full question reads "Can Clark spend exactly $5.00 at the fruit stand buying fruit that he likes?" [emphasis added]

I mean, I can squint and see the reading "it's possible, since it's possible he likes those fruits and it's possible to make that dollar amount with those fruits," but the layering of conditions makes it feel wrong. It would render basically any "can" statement immediately sufficient no matter what. And especially given that there's nothing inherently self-contradictory about the facts here, and given that (1) gives NO, it seems weird to stretch so far to make (2) from a "maybe" into a "yes." We worked a lot on this wording, and "does" isn't better since of course then it's just E (and quite boring) -- there's no information about what Clark actually does here.

Here's the official solution:

Quote:

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.

So, my personal take on the wording is that it's fine and GMAT-appropriate. But I'll certainly consider it further, discuss it with my boss, and see what other instructors think, and if it's unduly confusing we'll find a way to edit. We certainly don't want to mislead anyone with ambiguous wording!

Again, feel free to PM me to discuss.

P.s. The first definition of "can" in the dictionary is "to be able." If the question said "Is Clark able to...?" would that change your mind?

My brilliant colleague, see my private email for a long philosophical rant about this question.

With deep respect, Mike McGarry

Thanks Mike. You're quite the shameless flatterer. I'm discussing this wording with my boss, and I'm not sure whether we'll change it or not, but I'm definitely taking your interpretation into account. I'd also still love to see what Bunuel thinks. If we do change it, I'll update here. Regardless of the outcome, I really appreciate the time and insight you've devoted to this subject. I would have responded directly, by the way, but some of us don't accept private messages here.

A certain fruit stand sells only apples for $0.26 each, bananas for $0 [#permalink]

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17 Feb 2017, 06:59

Bunuel 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.

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 \(A\).

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Re: A certain fruit stand sells only apples for $0.26 each, bananas for $0 [#permalink]

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02 Nov 2017, 12:20

Bunuel 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.

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.