What Your Math Teacher Didn’t Tell You About PEMDAS

by Chelsey Cooley May 25, 2017

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Here’s a phrase that might bring back some memories from middle school math class: Please Excuse My Dear Aunt Sally, or PEMDAS. (If you went to school outside of the U.S., you may have learned the acronym BEDMAS or BODMAS, instead.) You use this phrase to decide what order to do mathematical operations in: Parentheses first (from inside to outside), then Exponents, then Multiplication and Division (left to right), then Addition and Subtraction (also left to right).

PEMDAS isn’t terribly fancy stuff. It’s just a useful little tool that helps us communicate clearly—it’s what tells us, for instance, that “2x(3+4)” means something different from “2×3 + 4.” But if there’s one thing the GMAT loves, it’s making things look more complicated than they really are.

PEMDAS and Negative Numbers

Here’s an expression you could see on the GMAT: -2⁴. It looks simple, but there’s one small problem: there’s no “N,” for “negative numbers,” in PEMDAS. So, do you deal with the negative sign before or after you handle the exponent? If you handle it first, then you’d raise -2 to the fourth power, which would give you an answer of 16. But if you hold off on it, you’d raise 2 to the fourth power, then make the answer negative, which would give an answer of -16.

Here’s how to decide: a negative sign is the same thing as multiplying a number by -1. When you’re simplifying an expression, you handle the negative sign at the same time as you’d handle multiplication. That means it comes after the exponent, since M comes after E in the acronym. The right thing to do here is to raise 2 to the 4th power, then make the answer negative. The correct answer is -16.

You might also see negative numbers in parentheses: (-2)⁴. Parentheses come before exponents, so you handle the “multiplication” first. Then you raise -2 to the 4th power. The answer to this one is positive 16.

PEMDAS and Fractions

How do you simplify this fraction?

To simplify a complex fraction, you should pretend that there are three sets of invisible parentheses: a set around the top of the fraction, another set around the bottom of the fraction, and finally, a third set around the entire thing. (By the way, you identify the “top” and “bottom” of a multi-decker fraction on the GMAT by finding the longest, boldest line in the fraction.)

Here’s how this fraction really looks:

Now apply PEMDAS. Simplify everything inside each set of parentheses as much as you can. Note that it’s fine to leave the numerator as an improper fraction.

Now, divide 3/2 by 4, since that’s inside the outer set of parentheses:

Let’s put it all together. Here’s an expression that contains fractions, exponents, and negative numbers. Try simplifying it using PEMDAS.

Don’t get intimidated by the multi-decker fraction. Find the longest, boldest line first, and split the fraction there:

Then simplify what’s in the parentheses as much as you can. Let’s work on the numerator first. Be careful with the negatives:

Next, the denominator:

Here’s what the original fraction really says:

The whole thing simplifies to -1.

PEMDAS and Equations

Suppose you’re solving an equation like this one:

3x — 7 = 20

What do you do first? You add 7 to both sides. Next, you divide both sides by 3. Addition first, then division. But PEMDAS tells you to always divide before you add, right? Here’s why this happens.

Equations and expressions aren’t exactly the same thing. (-2)⁴ is an expression, and so is x³/x². On the other hand, 3x-7=20 is an equation. Remember this mnemonic: an equation contains an equals sign. PEMDAS can be useful when you solve equations, but it’s really meant for expressions. When you’re simplifying an expression, PEMDAS is a hard-and-fast rule. When you’re solving equations, you might do things in a different order, and that’s fine—as long as you understand the arithmetic you’re doing at each step.

You should learn PEMDAS so well that you rarely need to think about it. (For a lot more practice, check out Foundations of GMAT Math.) However, when you run into one of the special cases we’ve gone over in this article, slow down and think about what to do. The GMAT test writers want to overwhelm you and get you to make silly mistakes. But if you memorize some of the special cases here, you’ll greet them with confidence on test day. ?

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Chelsey Cooley is a Manhattan Prep instructor based in Seattle, Washington. Chelsey always followed her heart when it came to her education. Luckily, her heart led her straight to the perfect background for GMAT and GRE teaching: she has undergraduate degrees in mathematics and history, a master’s degree in linguistics, a 790 on the GMAT, and a perfect 170/170 on the GRE. Check out Chelsey’s upcoming GRE prep offerings here.