Data Sufficiency

logitech wrote:Once again , if you see PRIME numbers in these kind of DS questions , beware of C trap! You might need only either A or B!

your heart's in the right place, but the issue here actually has nothing to do with prime numbers -- this is a fundamental issue with all of these problems, in which your answer is restricted to whole numbers.
if you have a linear equation(s) to which your answer must be in WHOLE NUMBERS, then the only reliable way to solve the equation is to TEST VALUES.
it makes absolutely no difference whether the coefficients are prime; there just isn't going to be any sort of simple way to predict the outcome from the coefficients of the equation.
for instance:
5x + 7y = 48 has only one whole-number solution (x = 4, y = 4).
5x + 7y = 47 has two whole-number solutions (x = 1, y = 6, and x = 8, y = 1).
still think primes tell you anything here?

--

you are correct, though, that there is a "c trap" here.

the reason that there is a "c trap", though, has nothing to do with primes, and everything to do with sucker-punching people who just use rules of thumb without really thinking.
i.e., the sole reason for the existence of this problem is to penalize people who just look at it and say, "oh hey, two equations, two unknowns, i'll pick c." bam! wrong!

lunarpower wrote: if you have a linear equation(s) to which your answer must be in WHOLE NUMBERS, then the only reliable way to solve the equation is to TEST VALUES.

There are always shortcuts in these questions that can save a ton of time. In the question above, the prices of the pencils are very close in value, which suggests that if we know the total cost, we'll be able to determine the total number of pencils. If Marta buys 5 pencils, the total cost is at most 5*0.23 = $1.15, and if she buys 7 pencils, the total cost is at least 7*0.21 = $1.47. So if she spent $1.30, she bought six pencils. If we know the total cost and the number of pencils, there can clearly be only one value for the number of 23 cent pencils, since the more 23 cent pencils she buys, the more she'll spend (alternatively, you have two distinct linear equations in two unknowns, so we can solve). So we don't actually need to test a single number here to see that the answer is B.

Q123 in the DS section of OG12, the 'stamps' question, is similar to the one in this thread, and involves a lot of number testing if you don't spot a shortcut. On Q123 you can use a divisibility shortcut, one that can also be applied to Q65 in the Problem Solving section, and which I described here (scroll down) :

https://www.beatthegmat.com/og-12-ps-65-t35672.html

Ian Stewart wrote:

lunarpower wrote: if you have a linear equation(s) to which your answer must be in WHOLE NUMBERS, then the only reliable way to solve the equation is to TEST VALUES.

There are always shortcuts in these questions that can save a ton of time. In the question above, the prices of the pencils are very close in value, which suggests that if we know the total cost, we'll be able to determine the total number of pencils. If Marta buys 5 pencils, the total cost is at most 5*0.23 = $1.15, and if she buys 7 pencils, the total cost is at least 7*0.21 = $1.47. So if she spent $1.30, she bought six pencils. If we know the total cost and the number of pencils, there can clearly be only one value for the number of 23 cent pencils, since the more 23 cent pencils she buys, the more she'll spend (alternatively, you have two distinct linear equations in two unknowns, so we can solve). So we don't actually need to test a single number here to see that the answer is B.

yeah, i thought about this sort of thing. and, yes, you are certainly correct -- with this reasoning, you can solve the whole problem without firing a single shot, so to speak.
there are also similar situations for other problems, on which clever estimation can eliminate the need for actual work (even if the problem doesn't involve whole numbers). for instance, on OG12 problem solving problem #126, you can realize that the combined time must be more than 1.5 hours (since that would be the time if both machines were as fast as the faster one) and also less than 2 hours (since that would be the time if both machines were as slow as the slower one). those boundaries -- more than 1.5 hours and less than 2 hours -- leave only the correct answer in play.

however, i have two big problems with advocating these sorts of shortcuts; i prefer simply to mention them, while continuing to emphasize that students should just plug in the numbers.
here are those two issues:
1) these shortcuts are almost never going to be consistent from problem to problem, while the general recommendation (plug in numbers) is highly generalizable.
remember that the student is never going to see this problem again, and so we should judge the different methods on the probability that the student could actually be able to use them again. (i.e., students already have tons of stuff to study, so i don't want to emphasize the "low-percentage shots")
2) to use this type of reasoning, on the fly, within the rather strict time limitations imposed by the gmat, requires an uncommonly high level of mathematical intuition (coupled with an uncommon lack of psychological pressure -- the intensity of the mental frame associated with this test makes protocol that much more important, since it's a known fact that creative problem-solving ability drops markedly under psychological pressure). i.e., it's cool to show this reasoning here, mostly in order to maximize the number of approaches that students can take to the problem, but many students just won't be able to come up with it.
i.e., your post seems to have a certain subtext that students should preferentially use methods like this one. for those blessed students to whom this sort of intuition comes naturally, i would agree, but, since 100% of students should be able to understand and execute the number plugging -- and because number plugging will be universally applicable on these problems -- that's the method we should put forward as #1.

in fact, just to underscore the dangers of recommending such shortcuts -- the above shortcut doesn't even cut it on my two previously cited examples:
* 5-cent pencils and 7-cent pencils totaling 48 cents: only one combination.
* 5-cent pencils and 7-cent pencils totaling 47 cents: two combinations.

in both of these problems, using the given "shortcut", you'd come up with a hypothesis of ambiguity:
-- 7 pencils could cost between 35 and 49 cents;
-- 8 pencils could cost between 40 and 56 cents;
-- 9 pencils could cost between 45 and 63 cents.
so, even though these situations are almost identical to the situation in the cited problem, the same reasoning is insufficient to settle the issue -- you'd have to go ahead and plug the numbers anyway, after you'd already spent time calculating these boundary values. so, ironically, the shortcut would become more of a long-cut, in this case.

on the other hand, good old-fashioned number plugging will yield the correct results (unique solution if it's 48 cents, but not if it's 47 cents) with boring but minimal effort.

lunarpower wrote:in fact, just to underscore the dangers of recommending such shortcuts -- the above shortcut doesn't even cut it on my two previously cited examples:

I think you're just explaining why your 5-cent/7-cent example couldn't appear on the GMAT. The test will never (or almost never) require you to test six or more different sets of numbers for a question, so there needs to be a shortcut available. On a question like this, one in which you get a linear equation with positive integer unknowns which has only one solution, there are only two shortcuts that could, in theory, be available; the one I described here and the one (using divisibility) that I described in the post I linked to.

lunarpower wrote: 1) these shortcuts are almost never going to be consistent from problem to problem, while the general recommendation (plug in numbers) is highly generalizable.
remember that the student is never going to see this problem again, and so we should judge the different methods on the probability that the student could actually be able to use them again. (i.e., students already have tons of stuff to study, so i don't want to emphasize the "low-percentage shots")

I certainly agree that there's no value in learning shortcuts which are unique to certain problem setups. If there's no useful general takeaway that might be applied to a variety of future problems, then there's no reason to bother learning a shortcut. There are even entire questions in the OG that I think are pointless to study since their setup is so idiosyncratic, the test taker is simply never going to see something similar again - the question late in the DS section about Carl's credit card balance is one example.

Both of the available shortcuts (estimation and divisibility) in the type of question in this post are, however, applicable to all kinds of questions, and are often specifically tested. The test taker needs to learn these things anyway. And, in the 29 cent/15 cent stamps question in the OG, if you don't spot a shortcut you could easily spend four minutes plugging in values, so spotting some kind of shortcut is almost a necessity.

lunarpower wrote: your post seems to have a certain subtext that students should preferentially use methods like this one. for those blessed students to whom this sort of intuition comes naturally, i would agree, but, since 100% of students should be able to understand and execute the number plugging -- and because number plugging will be universally applicable on these problems -- that's the method we should put forward as #1.

I've worked with a lot of people who are capable of at least sometimes seeing and applying shortcuts on questions like this one. The methods that should be 'put forward' really depend on the capabilities of the student. While I agree a certain mathematical sophistication is required to recognize when to use shortcuts, that sophistication certainly is not beyond the abilities of a 40-51 scoring test taker. Of course, the more time-consuming alternative methods - number plugging - should also be offered for those who won't see, under time pressure, a faster method; even a high level test taker won't see shortcuts much of the time. But if a test taker can spot a quick shortcut on even two or three questions during the test, he or she can save so much time that pacing may no longer be an issue. It's for that reason that I think these kinds of approaches are worthwhile for the higher level test taker - I teach methods like these not with the expectation that the test taker will see some super-fast technique for all 37 questions on the test and finish the Quant section in fifteen minutes, but rather with the goal of saving them 4-6 minutes on 2-3 questions somewhere in the test. That is, it's specifically because of the time pressure on the test that these shortcuts can be invaluable.