Never Do Long Division Again

March 17, 2010

When I was 19 years old and started my test-preparation career working with middle schoolers on a high-school admissions exam, I got a nasty jolt: I had completely forgotten how to do long division. It was understandable: I hadn't taken a math class in years, and when I did have to divide, a calculator was always at hand.

Of course, you can't use a calculator on the GMAT. You will, however, find yourself in several situations where you want to set up a long-division problem, despite the fact that it's not a skill you've practiced very much since you were 12 or 13 years old, and one that, even at its best, is mighty-time consuming.

Three problems with long division on the GMAT

1. It invites mistakes. Everybody wants to avoid careless errors, yet we all seem to make them. It may feel like there's no rhyme or reason behind the mistakes you make, but there is definitely a way to cut down on them: do less tedious math. Long division is the worst form of tedious calculation. Instead of just one or two steps (like, say, multiplying two-digit numbers), long division can include several steps, each of which is a chance to make a careless error.

2. You're probably not that good at it. My apologies if you're a long-division whiz, but the vast majority of students I've worked with are about as comfortable as I was when I started teaching middle-school kids. Most people have divided with a calculator for their entire adult life.

3. It takes too long. You don't need me to tell you that time management is key on the GMAT. Even if you make it through the several steps of a single long-division problem without a careless mistake, you've probably spent 20 seconds or longer doing it--a big chunk of the time you should allot for that question. If there's a faster way, you should be using it.

Know the testmaker

You're not going to business school so that you can become a clerk for an 19th-century accounting firm. Luckily, the writers of the GMAT realize that too. Business schools aren't looking for calculators, they're looking for students who can solve problems effectively while under pressure. This obviously applies to much more than just long division, but it's particularly important to remember here. If you're doing a lot of tedious work on a single problem, you're almost certainly missing out on a more efficient method.

How to avoid long division

The first, simplest way to avoid long division on the GMAT is this: Don't calculate until you have to. If, after the first step of a problem, you come out with a fraction, say 135/7, it may be tempting to convert that to a decimal or compound number. Avoid the temptation until there's no alternative.

For example, let's say you're working a problem about a factory producing widgets, and you deduce that it's producing 135/7 widgets per day. If you had a calculator, it'd be easy to convert that to 19 2/7, but on the GMAT it's not worth the time. Perhaps the question ends up by asking, "if the factory produces widgets at a constant rate seven days per week, how many widgets does it produce in two weeks?"

In other words, you're asked to multiply the day rate by 14. You probably see where I'm going with this: 7 is a factor of 14, so rather than multiplying 135 by 14, eliminate the denominator and multiply 135 by 2. In that case, not only did you save yourself the trouble of long division, you also made the second step easier! 135 times 2 is much simpler than 19 2/7 times 14.

Another way: Factoring

My example above may seem artificial, and it's true, I did just invent it for the purpose of this tip. But the end result--a denominator canceling out--is a very common occurrence in actual GMAT questions. However, it doesn't happen all the time.

The second method is: factor, factor, factor. Perhaps you have to simplify 84/16. Instead of doing long division and having to solve to two decimal places, simplify the fraction one factor at a time. Dividing the top and bottom by two gives you 42/8; by another two results in 21/4. You may, at that point, have to resort to long division to find out what 21/4 is, but it's far more likely that the question is asking for a fraction, or that you can approximate it yourself.

Factoring is one of the best secrets of effective GMAT test-taking; it often amazes my students just how often factorization holds the key to a quicker solution to a problem. I'll be writing much more about the uses and benefits of factorization in the coming weeks.

Still another way: Approximation

Many students assiduously learn approximation methods, then only apply them on questions that explicitly ask for an approximation. It's a start, but those strategies are widely applicable. The GMAT often will not pack answer choices in a very small range: instead of 3.1, 3.2, and 3.3, it's more likely (especially on word problems) for the answers to be much more widely dispersed.

Here's how it works for you. If, at the end of a word problem, you come up with an answer of 25/7 and the answer choices are in decimal form, first check and see how widely dispersed the choices are. 25/7 is greater than 3 (21/7) and less than 4 (28/7), and a little closer to the latter. If you approximated 3.6, you'd be safe: to five demical places, the answer is 3.57143. If the answer choices are 3.1, 3.6, 4.1, 4.6, and 5.1, there's no need to do calculate any further: you've got the answer; you can move on.

Making these methods work

When you first apply these techniques, you might find yourself cursing me, or at least wondering why I'm advocating something that slows you down. In the beginning, you might occasionally take longer to complete problems while factoring. But with sufficient practice, I guarantee that using these methods will take less time and help you avoid careless mistakes.

Further, practicing these techniques---basically, thinking of numbers in terms of their factors--is a key part of many types of GMAT math problems, not just ones in which you might be tempted to use long division. The better you become at doing quick, simple calculations in your head, the more successful you will be on the test.

About the author: Jeff Sackmann has written many GMAT preparation books, including the popular Total GMAT Math, Total GMAT Verbal, and GMAT 111. He has also created explanations for problems in The Official Guide, as well as 1,800 practice GMAT math questions.

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