As anyone who has taken or prepared for the GMAT realizes, there is a finite amount of general knowledge that we must do our best to master. There are however countless manifestations of questions that test this knowledge and therefore exposure to lots of questions from each general topic is highly beneficial to our preparedness for the as-of-yet-unseen manifestation we will surely encounter come test day.
In algebra, the more we practice with manipulating equations, simultaneous equations and quadratics, for example, the more likely we will be to recognize when the given information is sufficient to solve for x or not. In a geometry question, we are more apt to be able to solve for the area of a triangular region within a mixed shape if we’ve trained ourselves to spot vertical angles, similar triangles, the diagonals of squares or whatever the case may be for the particular scenario. And on the list goes. We also know that certain topics are tested more often that others and thus, though all topics matter, spending more time on the higher frequency areas gives the most payoff come test day. The two key words here are knowledge and recognition. Those two components allow us to execute most effectively.
When I took my most recent official exam in June 2013 (click here to read Amphibious Assault, a post about my water-logged testing experience), one question caught my attention. The question caught my attention not because I got it right or wrong, but because I knew I wasn’t answering it as effectively as possible. This problem is a classic example of how the GMAT is not only a challenge to your knowledge but also about how well you recognize when the knowledge you have is being tested. The lesson for you here is that even someone like me, who has been teaching GMAT non-stop for 4 years and scores in the high-700s, will get stumped on the occasional problem. The key is to not let one problem prevent you from getting your best score, and to always use your knowledge AND recognition in concert!
Here is a similar question to the one I saw on the test:
Take a moment and try it if you like. I’ll wait…
Okay, done? Good.
At first glance, I thought I’d FOIL (multiply first term of each, followed by the two outside terms, the two inside terms and lastly the second term of each). But alas, that second part is not in parentheses but under a radical. I rapped my fingers on the desk for a bit. Then I scratched my head. I rubbed my chin in ‘thinker’ fashion. More rapping of the fingers. No epiphanies to speak of. Um, FOIL? No, already ruled that out.
I was looking good on time, so I didn’t want to throw this one away, and I was feeling a wee bit stubborn so I did the only thing I could think of at the time – I used the brute force method.
Finally, multiplying the two terms yielded
3.2 × .9 ≈ 2.9
The problem with the method I used thus becomes apparent. None of the answer choices match up to the answer I found. If I use the closer approximation for of 2.236, then the second choice becomes 2.76. If I use the closer approximation for of 1.414, then the third choice is 2.83. But this is not confidence inspiring enough for me. The reason my answer was not closer to one of these is that I used the approximation of 2.2 for in the calculation. And the square root I took of the first term was thus an approximation of an approximation. Not very precise. And when the answer choices are this close, precision is the order of the day.
I chose the second answer. I didn’t feel great about it. But after what must have been at least five minutes, it was time to let go and move on.
Post-test, I contacted my trusted colleagues at the Bell Curves Command Center and pitched this question. When I was presented with the better approach, I had one of those ‘duh’ moments. Of course that’s how to do it! Well, hindsight is always 20-20 and much seems obvious in retrospect. The math was not dreadfully difficult. I just had to apply a simple manipulation.
Ready? Here it is…
Set the equation equal to y and square both sides.
Yup. That’s it. Let’s go through the steps:
Finally, we take the square root of both sides to answer the question asked:
So it looks like I answered this one incorrectly. I still got a high score on the section, in part because I didn’t let this one question impact my execution elsewhere, but it irked me nevertheless that I let this one get away. Why? Because it didn’t have to be that way. I should have heeded my own advice, advice that I give to my students and that served me so well throughout the rest of the test. Along those lines, there are some valuable insights to take away from this particular question:
Takeaway #1 – Approximation is approximate:
Well of course it is! But seriously, If you make an approximation, your answer will not be accurate. So if the answers are too close, this is not the best method to use.
Okay so we ruled out approximation. Then what? Are we supposed to remember the value of radicals out to four significant figures and take the square root of a decimal to a very exacting degree? No. The GMAT exam may be written by evil overlords, but that’s just not their style.
Takeaway #2 – Beware of seemingly intense calculations:
If we are going for an exact numerical answer and we find that the method for getting it involves some very grueling mathematical calculations, that is probably not the right method to use. Look for a different one.
Alright, so we’ve ruled out approximation AND using exact math based on the decimal values of radicals. So far, this only covers what not to do. But what DO we do? For starters, we should listen to this advice. While I was sitting there pondering this question, I should’ve forced myself to look for an alternative beyond grinding out the answer. I should have searched my database of knowledge points and recognition insights for alternative strategies. Instead, after I’d decided I could grind it out with the calculations I never looked back.
But, what would’ve been that alternative method? The alternative would’ve come from a basic math identity, one I’ve used countless times before but overlooked in search of something more “fancy”:
Takeaway #3 – if a = b, then a2 = b2
By extension, this means that if , then a = b. The challenge with this question was that the radical sign seemed to prohibit us from properly manipulating what was within it. But by applying this identity, we lose the big radical sign over the first term and become more empowered to solve. Therefore, using this identity would’ve allowed us to turn an expression into an equation. We can often manipulate equations much easier than expressions.
Note: DO NOT automatically assume that if a2 = b2 then a = b. Remember that any positive value has both positive and negative real roots and thus the sign of the intended root cannot be known for sure.
Okay, so now we’re cooking. We’ve got the squaring method in place. We then FOIL and get the exact answer in radical format and we’re good to go. I leave you with one last takeaway:
Takeaway #4 – You gotta know when to hold ‘em, Know when to fold ‘em… -Courtesy of Kenny Rogers
If you have the time to spare on a question AND you have a good feeling that you might be on the path to the right answer, go for it. If you are behind on your target pacing time OR if after giving it a minute or two you have absolutely no clue, let it go and move on.
Do note in my account that I mentioned having spent five minutes or more on this question. I spend a lot of time with students on the importance of proper pacing strategy, discipline, and the need to exercise caution before spending too much time on a question. My disproportionate attention to this question does not contradict this line of thinking because I
a) Was ahead of my allotted time based on the pacing strategy that I devised for myself on the basis of my expected test performance and
b) Believed, prior to ending up with an answer too close to two answer choices, that I had a reasonable shot at making my method work
Yes, on the second of these points I was wrong. And when you take a test, sometimes you ARE wrong. It happens. But it’s one thing to expend the effort when you think you might be on the right track and it’s an entirely different thing to expend the energy on a question just because you’re too stubborn to let it go and/or if you are in full knowledge that you really have no clue what you are supposed to do to solve it. My situation, at least initially, corresponded to the former. And yes, I probably spent another minute or two after discovering the difficulty of choosing between two close answers. But again, I had the time to spare.
In closing, remember that taking the test is not an exact science. There is more than one way to skin a CAT, so to speak (no I don’t skin cats, but it’s a fitting expression here). But the more we know and the more insights we are exposed to, the more equipped we are to make the best decision possible in the heat of the moment.
For information on Bell Curves courses or tutoring that could help you take the next step in your GMAT score, visit us at gmat.bellcurves.com.