Conceptual Thinking on the GMAT
A common complaint I hear from students is: “I’m not good at algebra.” Full disclosure, algebra is not my favorite topic either. Although algebra is a powerful tool for solving many questions on the GMAT, it is rarely the only means available to solve a given math problem.
The four most common strategies to get to the right answer are: algebra, conceptual thinking, picking numbers, and backsolving. Backsolving involves plugging in answer choices into the question to see whether it works. Picking numbers helps concretize abstract concepts with multiple variables. However the topic I’d like to focus on today is conceptual thinking.
The four strategies I mentioned above are not used arbitrarily in a vacuum; you can mix and match them to suit the problem at hand. Backsolving is often a good choice once you’ve done some algebra, or number picking can get you the right answer if you manipulate the equations provided. However, the strategy that will give you the greatest return is usually to start off with conceptual thinking. Look at the problem and think about what is being asked of you, as well as what information is provided that may guide your understanding. This tends to put you on the right path more often than not.
I often compare the GMAT to a game of Cranium. In Cranium, you must answer a question on a node, and then either take the fast track or the scenic path to the next node. As you can imagine, you can win the game even if you take the scenic path once or twice, but if you take it every time, you will assuredly run out of time. Similarly, on the GMAT, if you don’t know the best way to solve a question, it may take you 3 minutes to get a question right. This is fine if every second or third question is difficult, but if every question takes 3 minutes you’re only going to answer 25 questions out of 37.
Let’s look at a particularly challenging work/rate problem that is made much simpler by understanding what is going on:
Machine A takes 10 hours to complete a certain job and starts that job at 9AM. After one hour of working alone, machine A is joined by machine B and together they complete the job at 5PM. How long would it have taken machine B to complete the job if it had worked alone for the entire job?
(A) 15 hours
(B) 18 hours
(C) 20 hours
(D) 24 hours
(E) 35 hours
Using only algebra and the memorized formula for Work problems ( or ), we can break this problem into 3 parts: Machine A alone from 9 AM to 10 AM , Machines A and B from 10 AM to 5 PM, and then the hypothetical Machine B alone. Let’s set up these three steps to see how we can solve this on the scenic path.
Machine A takes 10 hours to do the job, so each hour it works finishes 1/10th of the total job. From 9 AM to 10 AM, Machine A works alone, so at 10 AM, machine B kicks in and 1/10th of the job is done. Ergo, 9/10th of the job is left to complete for both machines.
From 10 AM to 5 PM, 7 hours pass, during which 9/10th of the job gets completed. Thus we can calculate the rate of the machines working together: hours. .
Since we know that (i.e. rates are additive), we can leverage the fact that we know 2 of these 3 rates to find the third using basic fraction addition. . Putting them all on a common denominator: , so , or .
Now that we have B’s rate of , we can easily tell that it would take 35 hours to complete the entire job.
The correct answer is (E).
The algebraic solution works fine and gets you the right answer, but there are many moving parts to keep track of and many opportunities for mistakes. Can we get to the same answer but faster using conceptual understanding and avoid the scenic route entirely?
If machine A does 1/10th of the work in an hour, and it works from 9 to 5, then it works for 8 hours and accomplishes 80% of the job on its own. This means that machine B only accomplishes 20% of the job, and it does so in 7 hours (10 AM to 5 PM). If the machine does of the job in 7 hours, it will take () 35 hours to complete of the job. Answer choice E, using almost no math whatsoever, but rather by exploiting the logic of the question.
In general, if you see how to solve a problem via algebra and are confident you can solve it in 3 minutes or less, then by all means go for it. However, you can save some time if you really understand how questions are set up and what they are testing. It may not be possible to come up with a handy shortcut on test day because of nerves and stress, but during your preparation take a look at how problems are solved and see if you can find a more elegant solution. All roads lead to Rome, and the more routes you know, the less likely you are to get stuck in unfamiliar territory.
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