Multiplying Two-Digit Numbers

April 13, 2010

As regular readers know, if there's a way to avoid long, drawn-out arithmetic, I'll use it. Given something like 21 times 29 or 35 squared, I'm looking for a shortcut.

Sure enough, there's a better way, and it uses a technique that you probably already know: the "FOIL" method.

Given an expression like (x + 3)(x - 4), you apply the FOIL method, with results of x^2 - 4x + 3x - 12, or x^2 - x - 12. Easy, right?

You can do the same thing with two-digit numbers.

Algebra Meets Arithmetic

Break each number down into two components, representing each digit. For instance, 21 = 20 + 1, or 97 = 90 + 7. (This is a useful way of thinking about two-digit and larger numbers in some other situations, as well.)

Let's apply the FOIL method to help us find the square of 21, a calculation that comes up occasionally on GMAT questions.

21^2
= (20 + 1)(20 + 1)
= 400 + 20 + 20 + 1
= 441

The number of steps involved probably isn't any fewer than it would take to square 21 the old-fashioned way. But there are several reasons why it's preferable.

First, there's no "carrying." Carrying leads to arithmetic errors, and the pressure of a high-stakes test makes it worse. This is a big reason why I never do long division.

Second, it's very intuitive. Once you learn to think about two-digit numbers as the sum of a multiple of ten and a smaller number, it will make much more sense why certain multiplication problems come out the way they do.

Third, along the same lines, it goes hand-in-hand with approximation. Notice that as you apply the FOIL method in a case like this, each step gets you an additional level of precision. First, 20^2 = 400. For many problems, that's good enough. If it's not, you can do (20)(1) twice. 440 is another approximation, and at this point, you might not need to keep going. Finally, if you need to be perfect, you can add the (1)(1).

Don't Forget Subtraction

This approach tends to be the speediest when the units digits are small--calculations like 31 times 43, for instance. Believe it or not, you can take advantage of small units digits even when the units digits aren't small!

Let's try 29 times 29. If we do it the way I've described, here's the process:

= (20 + 9)(20 + 9)
= 400 + 180 + 180 + 81
= 841

It works, but it's not much of a shortcut. It certainly doesn't give the "approximation" benefit. You have to go through the entire process before you get very close to 841.

Instead, use subtraction! 29 equals 20 plus 9, but it also equals 30 minus 1. We like small units digits, so try it the other way:

= (30 - 1)(30 - 1)
= 900 - 30 - 30 + 1
= 841

Much better! It's faster, there's less room for error, and as soon as you figure 30 squared, you're quite close to the answer. You can use subtraction any time you're working with a number ending in 7, 8, or 9, and you can combine addition and subtraction, as well.

Put It All Together

Let's try one more: 28 times 82. Applying everything we've learned, we think of these numbers as (30 - 2) and (80 + 2), respectively:

= (30 - 2)(80 + 2)
= 2400 + 60 - 160 - 4
= 2296

It's not magic, but like all mental math techniques, it will save you time and get you thinking more intuitively about numbers. It's no accident that those skills translate directly into greater success on the GMAT.

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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