Blue Book, page 919 prob. 16. If xy=7 and x-y=5, then x^2y-xy^2 = ??? a. 2, b. 12, c. 24, d 35, e. 70.
x^2y-xy^2 factors xy (x-y), so substituting 7*5= 35.
If you try to solve the 2 equations and 2 variables, x=y+5, so (y+5)*y= 7, y^2+5y-7=0. By the quadratic formula, y = (-5 ± sqrt (53))/2, so x = (5± sqrt (53))/2. Then try substituting those values in, and it seems very complicated still.
Most problems can be solved by straight forward methods. Here, if you miss the trick, you could easily spend 5 minutes on the problem and not get it.
Someone is still confusing deeper math knowledge and mental acuity with … tricks. It’s not a math trap and it’s not absurd. It’s straightforward for the … right test taker!
Following the wrong approach leads to a time sink, and THAT is one of the purposes of the SAT as it rewards the people who do not solve problems with the “paint by the numbers” methods taught in high schools.
Recognizing patterns and recognizing formulas when applicable is what separates students from the ones who try to fit a square peg in a round hole. This problem is simple and elegant … if you can see it.
Yeah, this is not absurd or a trap. As MITer said, this is just factoring expressions. Anyone who has taken algebra knows how to solve this problem.
Whenever I see a problem like this, I automatically factor because I know they are always straight forward. It is testing your ability to see patterns, not your in depth knowledge of algebra.
I don’t know if I would say that anyone who has taken Algebra knows how to solve this problem. Every student I have worked with has taken Algebra and most of them would get this wrong or just wouldn’t be able to solve it in time (including people who have achieved near perfect scores on the Math section). That is because there are multiple ways to solve it (like most SAT questions) and as Xiggi said the reward is there for someone who thinks cleverly about how to solve the problem effectively instead of “knee-jerking” into a solution without first applying good problem solving methods. This is what is at the essence of the SAT Math section imo and this question is a great illustration of it (and its one of my favorite questions as a result).
I would actually go one step further and say that this is at its core a “solve for the expression” question where what I think they are really testing most of all is who will recognize that the question is asking you to solve for an expression and not for the variables themselves. That type of question appears on almost every SAT because again at its core it is a reasoning and problem solving issue as much or more than it is a Math issue.
@reasonsat Perhaps I should clarify, by anyone who has taken algebra “knows” how to solve this problem, I mean they have the tools to solve it. It may not register with them right away how to solve it, but they do know everything required to solve the problem.
Ok, got it. Yeah someone who has taken Algebra should have the tools to solve it, but yeah, the question requires you to really think about the most effective way to solve it and it also plays with people’s innate tendency to assume that they need to solve for the variables individually (since that is usually the point in Algebra class).
There are students who will try to apply what they learned and have become comfortable with, namely the quadratic formula. In the bee problem, chances are that a few people here, starting with MITer, will be familiar with http://mathworld.wolfram.com/Series.html but the reality remains the same. Almost every “puzzle” on the SAT could be solved within a few seconds. When someone embarks on a lengthy journey of seemingly complicated factors, chances are that he or she missed a key element in the question or the … proposed answers.
Being alert for possible time sinks is one of the tenets of the SAT.
Another question that I love and that plays with students’ tendency to apply what they have learned sort of automatically and uncritically is the following GMAT question. I have done this one with SAT/ACT/GMAT students alike and almost everyone begins by factoring the quadratic. It’s one of my favorite questions and it really teaches people to “think before they do.”
If (4-x) / (2+x) = x, what is the value of x^2 + 3x - 4?
And although the discussion about Math on the new SAT is in another post, I must also bemoan the fact that the above type of question will be less likely to appear on the new test. Boo hoo.
@xiggi yeah, I’ve seen that problem with the bee and the train. I gotta admit, I think I got tricked and used infinite geometric series the first time I saw that problem.
Another similar SAT-type problem where one should not just try the “obvious” method is something like:
Q: How many 4-digit positive integers greater than 2000 are there whose digits are strictly increasing?
(There are whole books/chapters of books devoted to counting by bijection, which I won’t get into here)
Coming from a math contest background, the most interesting math problems to me have some really creative solution that is harder to find (possibly in addition to an alternate, bashy solution). I feel that the new SAT doesn’t have as many creative problems.
Yeah answer is 0. I have done this question with literally hundreds of people and almost everyone starts by factoring the quadratic (and assuming that it equals 0 even though the quadratic is what they are asking for and not what they are giving as a fact). They then get x= -4 and x=1 and don’t know what to do with the 2 “answers.” At that point most people then realize that the question is asking for the quadratic not for x so they usually then proceed to plug -4 and 1 into the quadratic and with some relief then see that the output is 0 in both cases and then conclude that the answer is 0. It’s at this point that I usually point out to them that although they got lucky and got the question correct, the only reason that they got the quadratic to equal 0 is because they assumed it was 0 in the first place! Again it plays with people’s tendency to regurgitate what they learned in school and in most cases in school the quadratic is set to equal 0 and you are solving for x.
It’s a great teaching question because it illustrates so many things, but one very fundamental thing that it illustrates is the need to parse the question into the given facts on the one hand and the question that they are asking you to solve for on the other. That seems so basic but so many people fail to do it, especially on a question like this. Obviously the If (4-x) / (2+x) = x is the given fact so if you are trying to answer the question by completely ignoring the only given fact, then you are unlikely to be able to get to the answer.
The other great thing that it teaches is to pay attention to what the question is actually asking for. The other inefficient thing that many people do on this question is to manipulate the given fact (the (4-x) / (2+x) = x), get everything on one side of the equation, and then solve for x (by factoring the quadratic). They then plug the 2 values of x into what the question is asking for without ever realizing that they had x^2 + 3x - 4 = 0 in like the second step of the process. They were just so blind to what the question was asking for that they never noticed it and took the many extra steps of factoring the quadratic, solving for x, and then plugging the values of x back into that very same quadratic.
Great teaching question in my opinion. And very similar to the one that the OP was referencing.
@reasonsat that seems like a big problem…and it underlies the even bigger problem of learning algebra and all sorts of topics without really understanding them.
For example, here is a very simple variant of the same problem that should all be easy to us now:
Q: If (4-x)/(2+x) = x, what is the value of x^2 + 3x - 28?
The same student wouldn’t be so lucky here because x^2 + 3x - 28 factors to (x-4)(x+7), but neither x = 4 nor x = -7 satisfy the first equation.
Instead, the first expression is equivalent to x^2 + 3x + 4 = 0. Now, the trick of seeing “oh hey there’s an x^2 + 3x here and there” becomes more apparent.
My own take on this is that one is greatly helped by having his or her “SAT CAP” on when approaching the test. This includes the need to abandon what I would call the standard approaches described above and the usual … reflexes.
The key, IMHO, is to take a couple of seconds and look at the question and the suggested answers before marking “something down.” I think that the first entries on a paper are extremely important. If I look at mine in this example, my first entry served to eliminate the division.
(4-x) / (2+x) = x became 4-x = x^2+2x
Obviously that led to the next 0 = x^2 + 3x - 4 as I always like to test a zero answer.
As always, there are no hard rules. The type of “gymnastics” does, however, come from practicing, and often from repeating the problems one DID solve but only after wasting precious time.
It is important that when you review a test, you also review the ones you got right, especially if your path was long and twisty.
It is important to know the ecosystem. On the SAT (as it currently exists) when a question asks you to find some seemingly random expression involving x rather than solving for x itself, it is VERRRY likely that there is an easy way to get the value of the expression without ever finding the value of x. That is not a particularly important skill in the general math universe. But it sure is true in SAT land (again, for now).
@xiggi, I completely agree that that is the key on the SAT. You have to start by thinking about what you are going to do and why you are going to do it. It is a problem solving test more than it is a Math test IMO. I personally tell people not to write anything down when they first read a question because in my opinion that prevents people from thinking about the question before they “knee-jerk” into a particular approach.
I mean in above question they are giving you (4-x) / (2+x) = x and they are asking for x^2 + 3x - 4, so there must be some way to take the given information and solve for what they are asking for. And basically you can either manipulate that first equation and solve for x and then plug x into the quadratic or consider that since they are very suspiciously not asking for x but for the value of an expression (the quadratic) that maybe, just maybe, you could solve for that expression directly.
And here is another thing that I like. Even if you don’t know how to manipulate that first equation, you could just plug in values for x until you find one that works and then take that value of x and plug it into the quadratic. I have seen people who are not good at algebra do this and get the question right. Obviously you would be assuming that x will be a value that you can easily stumble upon, but usually on the SAT it is so it’s not bad idea to just plug in values like x = 1, 2, 3, 0, -1, -2, -3 and hope that you get something. If you know that you aren’t comfortable doing it algebraically then at least you have something. And of course x=1 works so you would quickly get that and then plug that value into the quadratic and have the right answer.
And @MITer94 I like that variety of the question a lot because a lot of people get the version that I posted correct simply because they get lucky and assume that the quadratic equals 0, but in your version that wouldn’t happen so people who didn’t really think carefully about what they were doing would probably not get the question right!
Maybe the recent popularity of the 2048 game would make that problem easier today than it was 2 years ago – students more likely to recognize powers of 2 when they see them.
