Why Logic is More Important Than Algebra on the GMAT

QuestioningOne common complaint I get from students is that their algebra skills aren’t where they need to be to excel on the GMAT. This complaint, invariably, is followed by a request for additional algebra drills.

If you’ve followed this blog for any length of time, you know that one of the themes we stress is that Quantitative Reasoning is not, primarily, a math test. Though math is certainly involved – How could it not be? – logic and reasoning are far more important factors than conventional mathematical facility. I stress this in every class I teach. So why the misconception that we need to hone our algebra chops?

I suspect that the culprit here is the explanations that often accompany official GMAC questions. On the whole, they tend to be biased in favor of purely algebraic solutions.  They’re always technically correct, but often suboptimal for the test-taker who needs to arrive at a solution within two minutes. Consequently, many students, after reviewing these solutions and arriving at the conclusion that they would not have been capable of the hairy algebra proffered in the official solution, think they need to work on this aspect of their prep. And for the most part it isn’t true.

Here’s a good example:

If x, y, and k are positive numbers such that [x/(x+y)]*10 + [y/(x+y)]*20 = k and if x < y, which of the following could be the value of k? 

A) 10
B) 12
C) 15
D) 18
E) 30

A large percentage of test-takers see this question, rub their hands together, and dive into the algebra. The solution offered in the Official Guide does the same – it is about fifteen steps, few of them intuitive. If you were fortunate enough to possess the algebraic virtuosity to solve the question in this manner, you’d likely chew up 5 or 6 minutes, a disastrous scenario on a test that requires you to average 2 minutes per problem.

The upshot is that it’s important for test-takers, when they peruse the official solution, not to arrive at the conclusion that they need to solve this question the same way the solution-writer did. Instead, we can use the same simple strategies we’re always preaching on this blog: pick some simple numbers.

We’re told that x<y, but for my first set of numbers, I like to make x and y the same value – this way, I can see what effect the restriction has on the problem. So let’s say x = 1 and y = 1. Plugging those values into the equation, we get:

(1/2) * 10 + (1/2) * 20  = k

5 + 10 = k

15 = k

Well, we know this isn’t the answer, because x should be less than y. So scratch off C. And now let’s see what the effect is when x is, in fact, less than y. Say x = 1 and y = 2. Now we get:

(1/3) * 10 + (2/3) * 20  = k

10/3 + 40/3 = k

50/3 = k

50/3 is about 17. So when we honor the restriction, k becomes larger than 15. The answer therefore must be D or E. Now we could pick another set of numbers and pay attention to the trend, or we can employ a bit of logic and common sense. The first term in the equation x/(x+y)*10 is some fraction multiplied by 10. So this term, logically, is some value that’s less than 10.

The second term in the equation is y/(x+y)*20, is some fraction multiplied by 20, this term must be less than 20. If we add a number that’s less than 10 to a number that’s less than 20, we’re pretty clearly not going to get a sum of 30. That leaves us with an answer of 18, or D.

(Note that if you’re really savvy, you’ll recognize that the equation is a weighted average. The coefficients in the weighted average are 10 and 20. If x and y were equal, we’d end up at the midway point, 15. Because 20 is multiplied by y, and y is greater than x, we’ll be pulled towards the high end of the range, leading to a k that must fall between 15 and 20 – only 18 is in that range.)

Takeaway: Never take a formal solution to a problem at face value. All you’re seeing is one way to solve a given question. If that approach doesn’t resonate for you, or seems so challenging that your conclusion is that you must purchase a host of textbooks in order to improve your formal math skills, then you haven’t absorbed what the GMAT is really about. Often, the relevant question isn’t, “Can you do the math?” It’s, “Can you reason your way to the answer without actually doing the math?”

*Official Guide question courtesy of the Graduate Management Admissions Council.

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By David Goldstein, a Veritas Prep GMAT instructor based in Boston. You can find more articles by him here.

GMAT Tip of the Week

A Logical Approach To Sentence Correction

(This is one of a series of GMAT tips that we offer on our blog.)

There is probably no more illogical discipline in academia than English grammar, in which seemingly the only rule is that there are generally several exceptions to each rule. Alas, as English has become a primary language of the business world, the GMAT requires examinees to be proficient in a number of grammatical devices through the Sentence Correction style of question, which comprises 1/3 of the Verbal section of the exam. Fear not (or, maybe fear slightly), however, as the GMAT focuses on a limited scope of the English language, and the Veritas Prep verbal syllabus teaches those particular skills thoroughly.

Because of the seemingly-illogical nature of the English language, it would seem illogical that some of the more-difficult GMAT Sentence Correction questions can be answered using logic. Say, for example, that you narrowed your answer choices down to the following:

B) Gary mistook the car, which was backfiring, for a gunshot.

D) Gary mistook the sound of a backfiring car for a gunshot.

While neither may appear to have a distinct grammatical error, answer choice B is simply illogical – a gunshot is a sound (or event) while a car is an object, and therefore one could not logically mistake the car for the gunshot. D, which correctly equates the sound of the car with the sound of the gunshot, is an accurate sentence, and is thus correct.

How can you use this to your advantage? Consider this — it doesn’t behoove business schools to admit students based upon their knowledge of obscure and complicated English language idioms, but it does make sense for schools to admit students who can think logically. While certain elements of grammar are distinctly testable, and you need to understand them thoroughly, if you don’t see an opportunity to employ one of the major error types, your time may be better spent by considering the logic of the remaining options than by worrying over tiny constructions of idiomatic grammar.

For more GMAT prep tips and resources, visit Veritas Prep.