Find the Quickest Way to Solve for X on the GMAT

I like to compare the GMAT to everyday things that hopefully resonate with people. To that end, I often like to use the analogy of routes to work to compare the different methods one can use to get the answer to a question. Invariably, there are multiple ways to get to the right answer on a math question, just as there are multiple ways to get to work. Some are just more direct than others. If I work on the island of Manhattan and live on the island of Manhattan, I can detour through The Bronx to get to work, but I’ll probably waste a lot of time. However, that doesn’t mean that I won’t get there, so it is an acceptable route for work. Of course, most of us are usually looking for the quickest way to get to work (for some reason my boss gets testy when I show up 3 hours late).

Your route to work might include highways, byways, under through-ways and over parkways, walking part way, taking the tramway, driving sideways on Broadway and generally finding some way to get there on your workday. You may notice there are many ways to work on any given day, but let’s get back to quant before we go too far astray.

Let’s look at a simple question that gives people fits and examine a few different ways to solve it.

48 + 48 + 48 + 48 = 4x. What is x?

(A)   4

(B)   8

(C)   9

(D)   16

(E)    32

The first thing you’ll notice is that the math is unwieldy to work out in your head. Since there’s no calculator on the GMAT, you may know that 48 is 65,536, but unless you have a degree in Computer Science, that is unlikely. 65,536 also doesn’t help you much here. So I’m adding 65,536 four times. My answer is 262,144. Does this necessarily help? (It was a rhetorical question for those of you nodding).

The easiest way to do this is through algebra, because exponents have very clear rules. However, exponents only have very clear rules in multiplication and division. All bets are off in addition and subtraction. Which means you must convert this into multiplication. Just as 4 + 4 becomes 2 x 4, this problem can be rewritten 4 x 48, which really is 41 x 48. When multiplying exponents with the same base, you just add the exponents on that base, so this is 41+8 = 49. Thus the answer is C. If this method makes sense to you, then by all means use it on test day, it will always work. If this isn’t obvious or you’d like to have a backup route to work in case the highway is jammed, what other approaches can we use?

How about if we just used the concept? You’re adding four numbers together. What if we made the numbers easier to grasp, like by replacing 48 by 41. What happens then? 41 + 41 + 41 +41 = 16, which is 42. This could mean that you increase the exponent by one (49, answer C) or it could mean we have to double the exponent (416, answer D). Let’s do another iteration to see the pattern more clearly. 42 + 42 + 42 + 42 = 16 + 16 + 16 + 16. That’s 64, which is 43. With two simple iterations we’ve seen that the exponent just increases by 1. The answer must thus be 49. A couple of simple parallel iterations with smaller numbers helps to unlock the concept and allows us to confidently pick answer choice C.

What if the highway is blocked and the service road is blocked, we can also try an alternate route: backsolving using the answer choices.  Look at the five choices again.

(A)   4

(B)   8

(C)   9

(D)   16

(E)    32

Could it really be A or B? I’m adding positive numbers (48 is necessarily even because the exponent is even) can I end up with something smaller than any of the numbers? If I have 5$, and then I add 5$ and then 5$ and 5$, I must certainly now have more than 5$! The answer is thus C, D or E. Looking further, can it be E? E is the most common trap answer, because if you multiplied 48 x 48 x 48 x 48 you’d get 432. Hopefully that realization is enough to easily see that addition is not the same as multiplication and lead you down the path of either algebra or concept to differentiate between C and D. If not you’ve at least got the answer choices narrowed down to a 50/50 just by eliminating impossible answer choices.

The final alternative route to work is picking numbers. There are no variables in this problem so it’s not a great strategy (sort of like taking the bridge to the Bronx) but it might help you understand the problem more deeply. Suppose I changed 48 to a different small number. Say 2: 2 + 2 + 2 + 2 = 8. 8 is 23. Okay, what about 3? 3 + 3 + 3 + 3 = 12, which is not 3anything.  How about 4? That’s what yielded 16, or 42. 5? 5 + 5 + 5 + 5 = 20, again not 5anything. Same thing for 6, 7, etc. It starts to become clear that this problem only works because there are 4 terms. The only other number that works is 2, which is itself a factor of 4. This breaks down to 4 x 48, which leads us right back to algebra (aka down the street from where I work).

The picking numbers answer doesn’t get you the answer very quickly, but it does make you realize that you can get anywhere from anywhere (well, maybe not to Atlantis). Any route to the answer is acceptable on the GMAT, as there are no part marks for work shown. Students often ask me “I got the right answer through a different way, is that okay?” My response is unflinching: “If it worked and it made sense to you, then go for it”. There are many other ways to get the correct answer, and if you know a few alternatives, you won’t get lost on the exam anymore than you would when going to work at rush hour.

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Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you occasional tips and tricks for success on your exam.  After graduating from McGill and receiving his MBA from Concordia, Ron started teaching GMAT prep and his Veritas Prep students have given him rave reviews ever since.

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