A certain fruit stand sells only apples for $0.26 each, bananas for $0

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 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:



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?

TRICKY QUESTION

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Official solution from Veritas Prep.

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 \(A\).

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?

Given, 26A + 24B + 65C = 500

Stat1: 1 Clark does not like bananas.
now, 26A + 65C = 500, 13* (2A+5C)= 500, but 500 is never divisible by 13. Hence, Clark will force to get some Bananas. Now, He has 3 fruits and he can decide, which one to like or not. Sufficient.

Stat2: Clark does not like cantaloupes.
26A + 24B = 500, Now, 13A + 12B = 250, so, 13*12= 156<250 (Linear Diophantine Equation), so there will one or two solutions for A and B. So, without cantaloupes, Clark will spend 500 cents. What if he likes cantaloupes or not. So, Clark must have all fruits available. In this case not. Not sufficient.

So, I think A.

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 AnthonyRitz,

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.

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.

I thought like this:

Must Clark buy any of the fruits?

Multiples of bananas : 24-48-72-96-120...240...360
Multiples of apples : 26-52-78-104-130...260...390
Multiples of cantaloupes: 65-130...260...290

The important multiples are 120, 130, 130, respectively. Clark obviously need to buy bananas to be able to reach exactly 250 (and so 5 dollars).

Statement (1) is therefore sufficient.
Statement (2) is therefore not sufficient. Maybe he buys cantaloupes to his mother, or maybe he doesnt.


Edit: Hmm, I see now why this is flawed. He must always buy bananas, and thats a fruit he does not like. So taken statement (2) we can still answer the question.

Correct answer must be D.