Add Faster and Multiply More Rapidly With Commutativity

October 13, 2008

Are you familiar with the concept of commutativity? You probably know the basics, even if the term is unfamiliar.

The idea is that, with operators such as addition and multiplication, it doesn't matter what order the numbers (technically, "operands") are in. For instance, 3 + 5 + 7 is the same as 3 + 7 + 5, which is the same as 7 + 5 + 3.

Not every operator is commutative. If you change the order of numbers around a subtraction sign, you can get a different answer: 7 - 3 doesn't equal 3 - 7. Same goes for division.

But with addition and multiplication (which are more common, anyhow), you can take advantage of the commutative law to save time on the GMAT. Just because a series of numbers are in a certain order doesn't mean you have to tabulate them that way.

Commutativity In Action

Here's one example. Say you had to add 36 + 59 + 64. No individual step is too hard, even if you go strictly left to right--most people would do (36 + 59) + 64, simplifying to 95 + 64 = 159.

It's faster if you recognize that 36 + 59 + 64 = (36 + 64) + 59, or 100 + 59 = 159. You won't always notice the pairs of numbers that add up to round figures, but if you look, you'll find some, and you'll get better with practice.

The same thing happens with multiplication. Perhaps you've gotten to the end of a problem and you're looking at an expression like this: (25)(37)(4). Rather than multiplying 25 by 37 (there's always a way around something like that on the GMAT!), recognize that 25 times 4 is familiar.

Rewrite (25)(37)(4) as (25)(4) x 37, or 100 x 37, or simply 3,700. There's no complicated arithmetic in those steps. Again, it will take some practice to notice when rearranging the terms will work in your favor, but if you keep a lookout for those opportunities, you'll find some of them.

Commutativity = Flexibility

I've said it many times on this site: The GMAT aims to measure your mental flexibility. Commutativity is a simple form of mental flexibility, but it fits into that general category. The exam isn't structured the way it is to test your arithmetic skills--it's designed to see if you can make analytical deductions.

Jeff Sackmann is a GMAT tutor based in New York City. He has created many resources for GMAT preparation, including the popular GMAT Math Bible and GMAT Verbal Bible, as well as 1,800 practice GMAT math questions.

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