Data Sufficiency

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.

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.

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.

[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?)?

GMAT assassins aren't born, they're made,
Rich

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.