what order do you solve a math problem

Make Math a Game

Everyone can do math, order of operations – what to do first in a math problem.

See Spot run.  Run see Spot.  Run Spot see.

If we use the same 3 words in a different order, they sound weird and make no sense – even if we understand each word by itself.

Just like our language, when we are writing an equation we need to understand the rules so that someone reading (or solving) the equation can get the intended meaning and can correctly find the answer. This order is part of the language of math. We follow rules naturally as we talk, read, and write, but our spoken language becomes second nature and we don’t have to think quite so much about the rules we are using while we speak. We can get to this point with math, with practice. Give yourself time to learn and apply the rules, and math will make more sense and it will be more fun.

To keep things straight in writing and solving equations, we have the  order of operations .  This is the set of rules that help everyone to solve a math problem the same way – it helps us know what to do first.  Why do need this?  Well, it matters – a lot! Take a look:

3-2+1 = ? If we do this problem from left to right, like reading, it is 3-2 = 1 , and then 1 +1 = 2

3-2+1 = ? If we choose to do this starting with the addition first, 2+1 = 3 , and so 3- 3 = 0

The two different approaches give us two different answers.  We need rules to help us know which is appropriate, so everyone can get the same result when given the same problem.

We use PEMDAS as the order of operations to solve problems.  What does PEMDAS mean?

PEMDAS is P arentheses, E xponents, M ultiplication, D ivision, A ddition, S ubtraction.

P is for Parentheses ( ). Parentheses are the first thing to look for in an equation. They are used to group parts of the equation together and indicate what needs to be done first.  Smaller parts of the equation can be “nested” inside of larger parts by using parentheses and should be worked from the inside out.

The E stands for Exponents.  Once the parentheses have helped you find where to start, do the exponents next.  The base number with an exponent attached cannot be multiplied, divided, added, or subtracted without first carrying out the exponent operation.

The M is for Multiplication, the D is for Division.  There are places where these can be switched, but it is easier to learn the order and stick to it.

The A is for Addition and the S is for Subtraction.  You saw in the first example above how switching these can really mess things up, even in simple equations.

Order of Operations Examples – using PEMDAS

Here are some examples that include combinations of operations so we can see how this works:

After our first operation should we have 12 x 7 or 14 – 14?  Using PEMDAS, Multiplication comes before Subtraction, so this equation should be solved by doing the 2 x 7 first, then the subtraction.  So, 14 – (2  x 7), where the parentheses are inserted to show “grouping” and indicate the first operation to be performed, becomes 14 – 14 = 0

NOTE: If you wanted to describe the equation to force the 14 – 2 to be done first, then use parentheses as in (14 – 2) x 7 which is then 12 x 7 = 84

The Parentheses really help break this into groups.  First, 8 + 8 = 16, hold on to that and go to the next group – start here with the Exponent, 2 squared is 4, so you now have 5 + 4 = 9.  This gives us (16) – (9) which equals 7!

This is definitely more complicated.  I wouldn’t really know where to start without the order of operations rules.  We have a couple of groupings to look at here, so we should start with the innermost Parentheses. 12 – 2 = 10, easy enough, the next group has an Exponent, so do that first – 9 x 9 is 81, which leads to 19 + 81 = 100.  We now have the inner groupings done and we can look at the next level out.  (4 x (10) x (100)), since they are all at the same level and all Multiplication, 4 x 10 x 100 = 4000.  We are now left with (4000) + 12/3, following PEMDAS, we do the Division before the Addition giving 4000 + 4 which is… 4004!

For fun, write some equations with several operations and see if you can solve them.   Then, give the equations to someone else and see if they can follow the same order of operations to get the same answer you did.

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General Education

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 "pem-dass," "pem-dozz," or "pem-doss."

Here's what each letter in PEMDAS stands for:

• P arentheses
• M ultiplication and D ivision
• A ddition and S ubtraction

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: P lease E xcuse M y D ear A unt S ally .

In the United Kingdom and other countries, students typically learn PEMDAS as BODMAS . The BODMAS meaning is the same as the PEMDAS meaning — it 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:

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

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!

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

11 − 8 + 5 × 6

8 ÷ 2 (2 + 2)

7 × 4 − 10 (5 − 3) ÷ 2²

√25 (4 + 2)² − 18 ÷ 3 (3 − 1) + 2³

Answer Explanations

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

#1 Answer Explanation

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

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:

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):

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 wrong — and clearly isn't using PEMDAS correctly!

#3 Answer Explanation

Things start to get a bit trickier now.

This math problem has parentheses, an exponent, multiplication, division, and subtraction. But don't get overwhelmed — let'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

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 exponents — this 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 the acronym SOHCAHTOAH means 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.

Need more help with this topic? Check out Tutorbase!

Our vetted tutor database includes a range of experienced educators who can help you polish an essay for English or explain how derivatives work for Calculus. You can use dozens of filters and search criteria to find the perfect person for your needs.

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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Order of Operations PEMDAS

"Operations" mean things like add, subtract, multiply, divide, squaring, etc. If it isn't a number it is probably an operation.

But, when you see something like ...

7 + (6 × 5 2 + 3)

... what part should you calculate first? Start at the left and go to the right? Or go from right to left?

Warning: Calculate them in the wrong order, and you can get a wrong answer !

So, long ago people agreed to follow rules when doing calculations, and they are:

Order of Operations

Do things in Parentheses First

Exponents (Powers, Roots) before Multiply, Divide, Add or Subtract

Multiply or Divide before you Add or Subtract

Otherwise just go left to right

How Do I Remember It All ... ? PEMDAS !

Divide and Multiply rank equally (and go left to right).

Add and Subtract rank equally (and go left to right)

So do it this way:

After you have done "P" and "E", just go from left to right doing any "M" or "D" as you find them.

Then go from left to right doing any "A" or "S" as you find them.

You can remember by saying " P lease E xcuse M y D ear A unt S ally".

You may prefer GEMS ( G rouping, E xponents, M ultiply or Divide, Add or S ubtract). In the UK they say BODMAS (Brackets, Orders, Divide, Multiply, Add, Subtract). In Canada they say BEDMAS (Brackets, Exponents, Divide, Multiply, Add, Subtract). It all means the same thing! It doesn't matter how you remember it, just so long as you get it right.

Example: How do you work out 3 + 6 × 2 ?

M ultiplication before A ddition:

First 6 × 2 = 12 , then 3 + 12 = 15

Example: How do you work out (3 + 6) × 2 ?

P arentheses first:

First (3 + 6) = 9 , then 9 × 2 = 18

Example: How do you work out 12 / 6 × 3 / 2 ?

M ultiplication and D ivision rank equally, so just go left to right:

First 12 / 6 = 2 , then 2 × 3 = 6 , then 6 / 2 = 3

A practical example:

Example: Sam threw a ball straight up at 20 meters per second, how far did it go in 2 seconds?

Sam uses this special formula that includes the effects of gravity:

height = velocity × time − (1/2) × 9.8 × time 2

Sam puts in the velocity of 20 meters per second and time of 2 seconds:

height = 20 × 2 − (1/2) × 9.8 × 2 2

Now for the calculations!

The ball reaches 20.4 meters after 2 seconds

Exponents of Exponents ...

What about this example?

Exponents are special: they go top-down (do the exponent at the top first). So we calculate this way:

So 4 3 2 = 4 (3 2 ) , not (4 3 ) 2

And finally, what about the example from the beginning?

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