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The PEMDAS Rule: Understanding Order of Operations

Posted by Hannah Muniz | Dec 2, 2019 2:00:00 PM

General Education

 

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Everyone who’s taken a math class in the US has heard the acronym "PEMDAS" before. But what does it mean exactly? Here, we will explain in detail the PEMDAS meaning and how it’s used before giving you some sample PEMDAS problems so you can practice what you’ve learned.

 

PEMDAS Meaning: What Does It Stand For?

PEMDAS is an acronym meant to help you remember the order of operations used to solve math problems. It’s typically pronounced either "pem-dozz" or "pem-doss."

Here’s what each letter in PEMDAS stands for:

  • Parentheses
  • Exponents
  • Multiplication and Division
  • Addition and Subtraction

The order of letters shows you the order you must solve different parts of a math problem, with expressions in parentheses coming first and addition and subtraction coming last.

Many students use this mnemonic device to help them remember each letter: Please Excuse My Dear Aunt Sally.

In the United Kingdom and other countries, students typically learn PEMDAS as BODMAS. The BODMAS meaning is the same as the PEMDAS meaningit just uses a couple different words. In this acronym, the B stands for "brackets" (what we in the US call parentheses) and the O stands for "orders" (or exponents).

Now, how exactly do you use the PEMDAS rule? Let’s take a look.

 

How Do You Use PEMDAS?

PEMDAS is an acronym used to remind people of the order of operations.

This means that you don’t just solve math problems from left to right; rather, you solve them in a predetermined order that’s given to you via the acronym PEMDAS. In other words, you'll start by simplifying any expressions in parentheses before simplifying any exponents and moving on to multiplication, etc.

But there’s more to it than this. Here's exactly what PEMDAS means for solving math problems:

  • Parentheses: Anything in parentheses must be simplified first

  • Exponents: Anything with an exponent (or square root) must be simplified after everything in parentheses has been simplified

  • Multiplication and Division: Once parentheses and exponents have been dealt with, solve any multiplication and division from left to right

  • Addition and Subtraction: Once parentheses, exponents, multiplication, and division have been dealt with, solve any addition and subtraction from left to right

If any of these elements are missing (e.g., you have a math problem without exponents), you can simply skip that step and move on to the next one.

Now, let’s look at a sample problem to help you understand the PEMDAS rule better:

4 (5 − 3)² − 10 ÷ 5 + 8

You might be tempted to solve this math problem left to right, but that would result in the wrong answer! So, instead, let’s use PEMDAS to help us approach it the correct way.

We know that parentheses must be dealt with first. This problem has one set of parentheses: (5 3). Simplifying this gives us 2, so now our equation looks like this:

4 (2)² − 10 ÷ 5 + 8

The next part of PEMDAS is exponents (and square roots). There is one exponent in this problem that squares the number 2 (i.e., what we found by simplifying the expression in the parentheses).

This gives us 2 × 2 = 4. So now our equation looks like this:

4 (4) − 10 ÷ 5 + 8    OR    4 × 4 − 10 ÷ 5 + 8

Next up is multiplication and division from left to right. Our problem contains both multiplication and division, which we’ll solve from left to right (so first 4 × 4 and then 10 ÷ 5). This simplifies our equation as follows:

16 − 2 + 8

Finally, all we need to do now is solve the remaining addition and subtraction from left to right:

16 − 2 + 8
14 + 8
= 22

The final answer is 22. Don’t believe me? Insert the whole equation into your calculator (written exactly as it is above) and you’ll get the same result!

 

body_math_homeworkDavid Goehring/Flickr

 

Sample Math Problems Using PEMDAS + Answers

See whether you can solve the following four problems correctly using the PEMDAS rule. We’ll go over the answers after.

 

Sample PEMDAS Problems

  1. 11 − 8 + 5 × 6
  2. 8 ÷ 2 (2 + 2)
  3. 7 × 4 − 10 (5 − 3) ÷ 2²
  4. √25 (4 + 2)² − 18 ÷ 3 (3 − 1) + 2³

 

Answers

  1. 33
  2. 16
  3. 23
  4. 176

 

Answer Explanations

Here, we go over each problem above and how you can use PEMDAS to get the correct answer.

 

#1 Answer Explanation

11 − 8 + 5 × 6

This math problem is a fairly straightforward example of PEMDAS that uses addition, subtraction, and multiplication only, so no having to worry about parentheses or exponents here.

We know that multiplication comes before addition and subtraction, so you’ll need to start by multiplying 5 by 6 to get 30:

11 − 8 + 30

Now, we can simply work left to right on the addition and subtraction:

11 − 8 + 30
3 + 30
= 33

This brings us to the correct answer, which is 33.

 

#2 Answer Explanation

8 ÷ 2 (2 + 2)

If this math problem looks familiar to you, that’s probably because it went viral in August 2019 due to its ambiguous setup. Many people argued over whether the correct answer was 1 or 16, but as we all know, with math there’s (almost always!) only one truly correct answer.

So which is it: 1 or 16? 

Let’s see how PEMDAS can give us the right answer. This problem has parentheses, division, and multiplication. So we'll start by simplifying the expression in the parentheses, per PEMDAS:

8 ÷ 2 (4)

While most people online agreed up until this point, many disagreed on what to do next: do you multiply 2 by 4, or divide 8 by 2?

PEMDAS can answer this question: when it comes to multiplication and division, you always work left to right. This means that you would indeed divide 8 by 2 before multiplying by 4. 

It might help to look at the problem this way instead, since people tend to get tripped up on the parentheses (remember that anything next to a parenthesis is being multiplied by whatever is in the parentheses):

8 ÷ 2 × 4

Now, we just solve the equation from left to right:

8 ÷ 2 × 4
4 × 4
= 16

The correct answer is 16. Anyone who argues it’s 1 is definitely wrongand clearly isn’t using PEMDAS correctly!

 

body_simple_math_problem_chalkboardIf only these sample PEMDAS problems were as easy as this ...

 

#3 Answer Explanation

7 × 4 − 10 (5 − 3) ÷ 2²

Things start to get a bit trickier now.

This math problem has parentheses, an exponent, multiplication, division, and subtraction. But don’t get overwhelmedlet’s work through the equation, one step at a time.

First, per the PEMDAS rule, we must simplify what’s in the parentheses:

7 × 4 − 10 (2) ÷ 2²

Easy peasy, right? Next, let’s simplify the exponent:

7 × 4 − 10 (2) ÷ 4

All that’s left now is multiplication, division, and subtraction. Remember that with multiplication and division, we simply work from left to right:

7 × 4 − 10 (2) ÷ 4
28 − 10 (2) ÷ 4
28 − 20 ÷ 4
28 − 5

Once you’ve multiplied and divided, you just need to do the subtraction to solve it:

28 − 5
= 23

This gives us the correct answer of 23.

 

#4 Answer Explanation

√25 (4 + 2)² − 18 ÷ 3 (3 − 1) + 2³

This problem might look scary, but I promise it’s not! As you long as you approach it one step at a time using the PEMDAS rule, you’ll be able to solve it in no time.

Right away we can see that this problem contains all components of PEMDAS: parentheses (two sets), exponents (two and a square root), multiplication, division, addition, and subtraction. But it’s really no different from any other math problem we’ve done.

First, we must simplify what’s in the two sets of parentheses:

√25 (6)² − 18 ÷ 3 (2) + 2³

Next, we must simplify all the exponentsthis includes square roots, too:

5 (36) − 18 ÷ 3 (2) + 8

Now, we must do the multiplication and division from left to right:

5 (36) − 18 ÷ 3 (2) + 8
180 − 18 ÷ 3 (2) + 8
180 − 6 (2) + 8
180 − 12 + 8

Finally, we solve the remaining addition and subtraction from left to right:

180 − 12 + 8
168 + 8
= 176

This leads us to the correct answer of 176.

 

What’s Next?

Another math acronym you should know is SOHCAHTOA. Our expert guide tells you what this acronym means (coming soon) and how you can use it to solve problems involving triangles.

Studying for the SAT or ACT Math section? Then you'll definitely want to check out our ultimate SAT Math guide/ACT Math guide, which gives you tons of tips and strategies for this tricky section.

Interested in really big numbers? Learn what a googol and googolplex are, as well as why it's impossible to write one of these numbers out.

 

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Hannah Muniz
About the Author

Hannah received her MA in Japanese Studies from the University of Michigan and holds a bachelor's degree from the University of Southern California. From 2013 to 2015, she taught English in Japan via the JET Program. She is passionate about education, writing, and travel.



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