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Math Guide: Understanding the Order of Operations

By William Springer

Who is Aunt Sally, and what does she have to do with arithmetic? Read on to learn more about the Order of Operations and PEMDAS.

PEMDAS, or Please Excuse My Dear Aunt Sally

Last week, my entire family decided to go out for a nice dinner together, so we picked one of our favorite restaurants and headed down. I was hoping that all would go well and the worst would not happen. Unfortunately, after we started eating, I could see my Aunt Sally becoming uncomfortable. We were almost done and were about to leave when it happened - Aunt Sally let out this horrendous cloud of noxious, foul-smelling gas. It was so bad that we had to evacuate the restaurant!

As we were waiting to go back inside to pay for our meal, I went around to each of the employees to apologize. "Please," I said, "Please excuse my dear Aunt Sally!"

What is the point of this silly story? If you can remember the above sentence - please excuse my dear Aunt Sally - then you can remember the order of operations! Simply look at the first letter of each word: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction! A similar mnemonic, due to Danica McKellar, is "Pandas Eat: Mustard on Dumplings, and Apples with Spice".


When you begin evaluating an arithmetic expression, you always start by doing what's inside any parentheses; if there is morPublic domain image by Peter Valberg e than one set, you work from the inside out (and then left to right). For example, consider the following expression:

(5 x 3) + (3 x (4 - 2))

We start by evaluating the innermost set of parentheses; in this case, we'll replace the (4-2) with the result:

(5 x 3) + (3 x 2)

Evaluating the rest of the un-nested parentheses in order, we get 15 + 6 = 21.

When you're writing a problem and aren't sure how to figure out where to put the parentheses, the order of operations gives you the answer: just put them around whichever section of the problem you want to have evaluated first.


Once we've finished calculating all of the parentheses, it's time to do exponents! Notice that when you're evaluating the inside of a set of parentheses, you still follow the order of operations. So if I look inside a set of parentheses and see more parentheses, I evaluate those first. If I see some multiplication, some addition, and an exponent, I do the exponent first. You always follow the full order of operations.

When I'm inside the innermost set of parentheses, the first thing I do is check for any exponents. If there are any, I evaluate them first.

Multiplication and Division

Multiplication and division are listed together because they're really the same operation; dividing is just multiplying by the reciprocal. The same is true of addition and subtraction (subtracting is just adding a negative); as a result, the order of operations is often listed as P/E/MD/AS.

Once you've finished with your parentheses and exponents, do multiplication and division left to right. For example, suppose you have the problem 5 x 2 + 4 / 2 x 7. Going left to right and going each multiplication or division operation as we get to it, we get:

10 + 4 / 2 x 7

10 + 2 x 7

10 + 14 = 24

Addition and Subtraction

Like multiplication and division, addition and subtraction is done left to right; just like you did all of your arithmetic problems before you learned about the correct mathematical order of operations! (Technically, the "left to right" rule is not a law of mathematics, but it substitutes for an understanding of the commutative, associative, and distributive laws).

Now that you know what order you should do the operations in, can you work out the following problem?

3 x (4 - 2) + 3 x 7 - 23 = ?

(Hint: the answer is 19. Do you see why?)

For more practice, refer to our order of operations worksheets.