order of operations problem solving examples

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Order of op

Order of operations

Here you will learn about order of operations , including what it means and how to calculate and solve order of operations problems using PEMDAS .

Students will first learn about order of operations as a part of operations and algebraic thinking in elementary school.

What is the order of operations?

Order of operations refers to the rule that explains the sequence of steps necessary for correctly evaluating a mathematical expression or math problem.

You will use the acronym PEMDAS to help recall the correct order, or priority, in which you complete mathematical operations.

Mathematical operations such as multiplication and addition have to be completed in a specific order. This sequence of steps helps us to evaluate any mathematical expression, both with numerical values and algebraic expressions.

To evaluate an expression using PEMDAS, you need to understand what PEMDAS represents and be able to apply the PEMDAS rule to any calculation.

PEMDAS stands for:

P arentheses, E xponents, M ultiplication and D ivision, A ddition and S ubtraction

Parentheses have a higher priority than exponents, so we calculate what is inside a pair of parentheses first. Exponents have a higher priority than division and multiplication, so any exponent that can be evaluated is calculated next, and so on.

The order can be remembered using the mnemonic device for PEMDAS, ‘Please Excuse My Dear Aunt Sally.’

It is important to note that multiplication and division are given equal priority , and addition and subtraction are given equal priority.

When completing calculations that involve multiplication and division or addition and subtraction, you work from left to right.

For example,

Consider the following expression 12-7+6.

12-7=5 and then 5+6=11

Consider the following expression 10 \div 5 \times 2.

10 \div 5=2 and then calculate 2 \times 2=4.

Visually you could represent PEMDAS as:

Common Core State Standards

How does this relate to 5th grade math and 6th grade math?

• Grade 5: Operations and Algebraic Thinking (5.OA.1) Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
• Grade 5: Operations and Algebraic Thinking (5.OA.2) Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.
• Grade 6: Expressions and Equations (6.EE.1) Write and evaluate numerical expressions involving whole-number exponents.

How to evaluate using order of operations

In order to evaluate expressions using order of operations , you would use PEMDAS :

Solve any calculations within parentheses.

Solve for any exponents.

Solve any division and multiplication calculations.

Solve any addition and subtraction calculations.

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Order of operations examples

Example 1: pemdas with addition and multiplication.

Evaluate 3+6\times{7}.

There are no parentheses.

2 Solve for any exponents.

There are no exponents.

3 Solve any division and multiplication calculations.

The multiplication that we need to calculate is 6\times{7}=42.

Replacing 6\times{7} with 42 gives us the calculation 3+42.

4 Solve any addition and subtraction calculations.

So 3+6\times{7}=45

Example 2: PEMDAS with division and subtraction

Evaluate 12-8\div{2}.

The division you need to calculate is 8\div{2}=4.

Replacing 8\div{2} with 4 gives us the calculation 12-4.

So 12-8\div{4}=8

Example 3: PEMDAS with parentheses and multiplication

Evaluate 3(2+4\times{6}-3).

You have a pair of parentheses and so you need to resolve what is inside the set of parentheses first. This is the calculation 2+4\times{6}-3.

Using PEMDAS, multiplication comes before addition and subtraction so you need to solve the multiplication 4\times{6}=24.

Replacing 4\times{6} with 24 in the calculation, you have 2+24-3.

Addition and subtraction should be completed from left to right and so you have 2+24=26 and 26-3=23. Therefore 2+24-3=23.

As 2+4\times{6}-3=23 and 3(2+4\times{6}-3) means 3\times(2+4\times{6}-3), you have the updated calculation 3\times{23}.

3\times{23}=69

There is nothing else to solve, so the final answer is:

3(2+4\times{6}-3)=69

Example 4: PEMDAS with an exponent and multiplication

Evaluate 4\times{3}^{2}.

Here you have to resolve 3^{2}=3\times{3}=9.

Replacing 3^{2} with 9 in the calculation, you now have:

The final step needed is to calculate 4\times{9}=36

4\times{3}^{2}=36

Example 5: PEMDAS with brackets, an exponent and division

Evaluate 3+(10\div{4}\times{20})^{2}.

Within the set of parentheses, you have the calculation 10\div{4}\times{20}.

You need to work from left to right.

Completing the division, you have 10\div{4}=2.5

Next, 2.5\times{20}=50

Therefore, 10\div{4}\times{20}=50.

Replacing 10\div{4}\times{20} with 50, you now have the updated calculation,

You now have to solve 50^{2}=50\times{50}=2500.

Updating the calculation, you now have 3+2500.

There are no divisions or multiplications to solve.

3+2500=2503 which gives us the final answer:

3+(10\div{4}\times{20})^{2}=2503

Example 6: PEMDAS with everything

Evaluate 4^{2}+2(14-8)\div{3}.

Within the brackets is the calculation 14-8=6.

Updating the calculation by changing the value in the bracket to 6, you have

4^{2}+2\times{6}\div{3}

As 4^2=4 \times 4=16, you now have

16+2\times{6}\div{3}

Here you have to calculate 2\times{6}\div{3}. Working from left to right, you calculate 2\times{6} and then divide the solution by 3.

2\times{6}=12

12\div{3}=4

Updating the calculation, you now have 16+4.

As 16+4=20, our final answer is:

4^{2}+2(14-8)\div{3}=20

Teaching tips for order of operations

• While PEMDAS is a great way for students to remember the order of operations, it’s important that the students understand how to solve for each of the operations required, not simply giving them the acronym to memorize.
• It is important to find a way for students to practice outside of order of operations worksheets. The more practice a student gets, the better. For example you can physically separate word problems from the answer choices in the classroom. Students would have to walk around to discuss and solve the problem and then find the answer that matches.
• Students could be given an expression without the symbols. They would have to then decide which operations/symbols would make the expression true.

Easy mistakes to make

• Multiplication/division and addition/subtraction in the incorrect order When you have a chain of multiplication and division calculations or a chain of addition and subtraction calculations, you work from left to right. For example, If you put parentheses into the calculation 12\div{4}\times{3} with the purpose of keeping the calculation exactly the same, you must place the parentheses around 12\div{4}. This is because 12\div{4}\times{3}=9, and (12\div{4})\times{3}=3\times{3}=9. If you placed parentheses around 4\times{3} you would have 12\div{(4\times{3})}=12\div{12}=1 which is a different answer that you would get without the parenthesis.
• Exponents as multiplying a number by 2 When introducing exponents, some students will interpret 4^2 as 4 \times 2 instead of 4 \times 4. The use of a number line to represent this expression or using expanded notation when introducing the concept can help students internalize the learning.
• Multiplying before exponents Take 2\times{3}^{4}. Because you would perform exponents before you perform multiplication within PEMDAS, you need to calculate the exponent term first, and then multiply the solution by 2. Inserting a pair of parentheses with the purpose of keeping the calculation exactly the same, 2\times{3}^{4}=2\times(3^{4})=162. A common misconception is that the multiplication is carried out first, and then the value is raised to the power; here this would look like (2\times{3})^{4} and is equal to 6^{4}=1296 which is the wrong answer.

Related properties of equality lessons

• Properties of equality
• Associative property
• Commutative property
• Distributive property

Practice order of operations questions

1. Calculate 6+5 \times 8.

To solve, follow the order of operations by using PEMDAS.

2. Calculate 16-15 \div 3.

To solve, follow the order of operations by using PEMDAS .

3. Calculate (8-2 \times 5+4) \div 2.

4. Calculate 2 \times 5^2.

5. Calculate 258-(10 \div 2 \times 3)^2.

6. Calculate 3(7-5)^3 \div 6+4.

Order of operations FAQs

In the order of operations, multiplication and division are seen as equal and would be performed as they are stated within the expression, starting from left to right. The same is true with addition and subtraction.

Both PEMDAS and BODMAS are correct. The acronym BODMAS is the UK version of the same rules. It can also be referred to as BIDMAS or BEDMAS . The acronym in the UK would read as: B rackets, I ndices, D ivision and M ultiplication, A ddition and S ubtraction.

An exponent refers to the number of times a number is multiplied by itself. For example 7^2 = 7 \times 7 or 4^3 = 4 \times 4 \times 4.

Parentheses are used within mathematical expressions to note a modification is the normal order of operations. In the expression, (3+4) \times 3, because the addition is within parentheses, it is solved before the multiplication.

The next lessons are

• Addition and subtraction
• Multiplication and division
• Types of numbers

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Practice With Order of Operations Problems and Answers

I see it happen all the time in my math classroom: students attempt math problems and get the wrong answer even though they are performing each math operation correctly. How can this be!? Isn’t math supposed to have a single correct answer?

Well, as it turns out, there is a set of rules that tells us the right order and wrong order that operations can be performed in a mathematical expression. And without knowing this very important set of rules, you can’t   be sure that you will always get the correct answer! Let’s take a look at this set of rules and dig into some practice with order of operations problems and answers so that you can make sure this doesn’t happen to you!

What is the Correct Order of Operations?

In math and algebra, the order of operations is an important set of rules that tell us the correct order that arithmetic operations should be performed in when working with a numerical expression. Performing operations in the right order using a standard method makes it so that two people will always get the same correct answer when solving a given problem.

In order to help remember the standard order of operations, we can use the acronym PEMDAS. 

The PEMDAS rule (sometimes known as BEDMAS or the BODMAS rule) works by matching the first letter of each operation to each of the mathematical operations. 

• P (Parentheses) – We start by performing the operations inside any set of parentheses first. It is important to start with any expressions within innermost parentheses and work outward.
• E (Exponents) – The next step is to evaluate or simplify any expressions involving exponents or powers. It is important to remember that this step also includes any square root.
• MD (Multiplication and Division) – It is important to perform multiplication and division from left to right in the order that they appear in the expression. This is a common place where students make mistakes!
• AS (Addition and Subtraction) – Finally, we perform addition and subtraction from left to right as they appear in the expression. Like multiplication and division, these operations have the same priority and are performed from left to right.

To help my students, I have told them to think of the acronym PEMDAS as standing for “Please Excuse My Dear Aunt Sally”. There are many different ways to remember the PEMDAS acronym, but I have found that this mnemonic device is a great way to help my students remember the order of the PEMDAS acronym.

If you follow the rules of the order of operations, you should find that arriving at the correct answer isn’t as hard as you once thought!

Why Does Order of Operations Matter?

In order to understand when order of operations matters, take a look at this simple 2-step order of operations problem. Consider the following expression:

$$2 + 4 \times 3$$

There are two approaches that you could take here, and only one of them will give you the correct answer! Which one do you think is correct?

As you can see, each strategy results in different answers. The strategy on the left adds 2 + 4 first, while the strategy on the right multiplies 4 x 3 first. Remember that we use the PEMDAS rule to help us identify the right order.

PEMDAS tells us that multiplication must be performed before addition. This tells us that the second solution is correct!

Order of Operations Problems and Answers

Let’s take a look at a few more examples of order of operations problems and answers! I’ll start by introducing you to some simpler problems with two basic operations, and we’ll work our way up to more complex 4-step order of operations problems! Just be sure to review the answer key for each problem to make sure you get the same answer!

2-Step Order of Operations Problems

Example #1:  \(5 – 3 \times 2\)

In this first example, following order of operations tells us to perform multiplication before subtraction. Taking a look at the given values, we know that this will result in:

\begin{split} &5 – 3 \times 2 \\ \\ =&5 – 6 \\ \\ =& -1 \end{split}

Remember, performing subtraction first is a common mistake that will prevent you from obtaining the correct answer!

Example #2:  \((4 \div 2) + 7\)

The first step of the PEMDAS rule is to tackle any math expressions inside parentheses. After that, we can add 7 to the result.

\begin{split} &(4 \div 2) + 7 \\ \\ =&2 + 7 \\ \\ =& 9 \end{split}

Example #3:  \(3^3 – 4\)

The first step in this example is to work out our exponent. After that, we can subtract 4 from the result.

\begin{split} &3^3 – 4 \\ \\ =&27 – 4 \\ \\ =&23 \end{split}

3-Step Order of Operations Problems

Example #4:  \(\sqrt{4} \times 3 – 5\)

Remember that the ‘E’ in the PEMDAS acronym also includes the square root operation. As such, we need to evaluate the square root of 4 before multiplying by 3 and subtracting 5.

\begin{split} &\sqrt{4} \times 3 – 5 \\ \\ =&2 \times 3 – 5 \\ \\ =&6 – 5 \\ \\ =&1 \end{split}

Example #5:  \((6 ÷ 2) + 3 × 2\)

In this example we must remember to tackle the parentheses first. While your instinct may be to add the 3 next, remember that you need to multiply 3 times 2 first!

\begin{split} &(6 ÷ 2) + 3 × 2 \\ \\ =&3 + 3 × 2 \\ \\ =&3 + 6\\ \\ =&9 \end{split}

Example #6:  \(7 – 2 × 3 ÷ 2\)

In this example, we multiply 2 by 3 first, then divide the result by 2. Remember that multiplication and division operations are performed in the order in which they appear from left to right.

\begin{split} &7 – 2 × 3 ÷ 2 \\ \\ =&7 – 6 ÷ 2 \\ \\ =&7 – 3  \\ \\ =&4 \end{split}

4-Step Order of Operations Problems

Example #7:  \(2 + 4 × 3 – 4 ÷ 2\)

\begin{split} &2 + 4 × 3 – 4 ÷ 2 \\ \\ =&2 + 12 – 4 ÷ 2 \\ \\ =&2 + 12 – 2  \\ \\ =&12 \end{split}

Example #8:  \(2^3 + (4 × 3) – 6 ÷ 2\)

\begin{split} &2^3 + (4 × 3) – 6 ÷ 2 \\ \\ =&2^3 + 12 – 6 ÷ 2 \\ \\ =&8 + 12 – 6 ÷ 2  \\ \\ =&8 + 12 – 3 \\ \\ =&17  \end{split}

Using Order of Operations to Solve Math Problems

When I teach order of operations in my classes, I always encourage my students to keep the PEMDAS rule handy for every problem. Sometimes these problems can seem very simple, but may actually require more thinking. In particular, problems with both multiplication and division operations tend to confuse students! 

Take the time that you need to fully understand this very important concept. You will find it comes up often in your studies of math, particularly when working with algebraic formulas !

Like all math concepts, mastering the use of order of operations takes practice and critical thinking. I am hopeful that these order of operations problems and answers have helped you feel more comfortable with this very important algebra skill!

Did you find these order of operations problems and answers helpful? Share this post and subscribe to Math By The Pixel on YouTube for more helpful mathematics content!

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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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Algebra Topics  - Order of Operations

Algebra topics  -, order of operations, algebra topics order of operations.

Algebra Topics: Order of Operations

Lesson 1: order of operations, introduction to the order of operations.

How would you solve this problem?

12 - 2 ⋅ 5 + 1

The answer you get will depend largely on the order in which you solve the problem. For example, if you work the problem from left to right —12-2, then 10⋅5, then add 1—you'll get 51 .

12 - 2 ⋅ 5 + 1 10 ⋅ 5 + 1 50 + 1 51

On the other hand, if you solve the problem in the opposite direction—from right to left —the answer will be 0 .

12 - 2 ⋅ 5 + 1 12 - 2 ⋅ 6 12 - 12 0

Finally, what if you did the math in a slightly different order? If you multiply first, then add , the answer is 3 .

12 - 2 ⋅ 5 + 1 12 - 10 + 1 2 + 1 3

It turns out that 3 actually is the correct answer because it's the answer you get when you follow the standard order of operations . The order of operations is a rule that tells you the right order in which to solve different parts of a math problem. ( Operation is just another way of saying calculation. Subtraction, multiplication, and division are all examples of operations.)

The order of operations is important because it guarantees that people can all read and solve a problem in the same way. Without a standard order of operations, formulas for real-world calculations in finance and science would be pretty useless—and it would be difficult to know if you were getting the right answer on a math test!

Using the order of operations

The standard order of operations is:

Parentheses

Multiplication and division, addition and subtraction.

In other words, in any math problem you must start by calculating the parentheses first, then the exponents , then multiplication and division , then addition and subtraction . For operations on the same level, solve from left to right . For instance, if your problem contains more than one exponent, you'd solve the leftmost one first, then work right.

Let's look at the order of operations more closely and try another problem. This one might look complicated, but it's mainly simple arithmetic. You can solve it using the order of operations and some skills you already have.

4 / 2 ⋅ 3 + (4 + 6 ⋅ 2) + 18 / 3 2 - 8

Always start with operations contained within parentheses. Parentheses are used to group parts of an expression.

If there is more than one set of parentheses, first solve for the ones on the left. In this problem, we only have one set:

In any parentheses, you follow the order of operations just like you do with any other part of a math problem.

Here, we have two operations: addition and multiplication . Because multiplication always comes first, we'll start by multiplying 6 ⋅ 2 .

4 / 2 ⋅ 3 + (4 + 6 ⋅ 2 ) + 18 / 3 2 - 8

6 ⋅2 is 12. Next, we'll add 4 .

4 / 2 ⋅ 3 + ( 4 + 12 ) + 18 / 3 2 - 8

4+12 is 16 . So we've simplified our parentheses to 16 . Since we just have a single number in the parentheses, we can get rid of them all together—they're not grouping together anything now.

4 / 2 ⋅ 3 + 16 + 18 / 3 2 - 8

Second, solve any exponents . Exponents are a way of multiplying a number by itself. For instance, 2 3 is 2 multiplied by itself three times, so you would solve it by multiplying 2 ⋅2 ⋅2 . (To learn more about exponents, review our lesson here ).

There's only one exponent in this problem : 3 2 . 3 2 is 3 multiplied by itself twice —in other words, 3 ⋅ 3 .

3 ⋅ 3 is 9 , so 3 2 can be simplified as 9 .

4 / 2 ⋅ 3 + 16 + 18 / 9 - 8

Next, look for any multiplication or division operations. Remember, multiplication doesn't necessarily come before division—instead, these operations are solved from left to right .

Starting from the left means that we need to solve 4 / 2 first.

4 divided by 2 is 2 . That makes our next problem 2 ⋅ 3 .

2 ⋅ 3 + 16 + 18 / 9 - 8

2 ⋅ 3 is 6 . Finally, there's only one multiplication or division problem left: 18 / 9 .

6 + 16 + 18 / 9 - 8

18 / 9 is 2 . There's nothing left to multiply or divide, so we can move on to the next and final part of the Order of Operations: addition and subtraction .

6 + 16 + 2 - 8

Our problem looks a lot simpler to solve now. All that's left is addition and subtraction.

Just like we did with multiplication and division, we'll add and subtract from left to right . That means that first we'll add 6 and 16 .

6 + 16 is 22 . Next, we need to add 22 to 2 .

22 + 2 is 24 . Only one operation left: 24 - 8 .

24-8 is 16 . That's it!

We're done! We've solved the entire problem, and the answer is 16 . In other words, 4 / 2 ⋅ 3 + ( 4 + 6 ⋅ 2 ) + 18 / 3 2 - 8 equals 16 .

4 / 2 ⋅ 3 + (4 + 6 ⋅ 2) + 18 / 3 2 - 8 = 16

Whew! That was a lot to say, but once we broke it down into the right order it really wasn't that complicated to solve. When you're first learning the order of operations, it might take you a while to solve a problem like this. With enough practice, though, you'll get used to solving problems in the right order.

Remembering the order of operations

If you use it a lot, you'll eventually get the hang of the order of operations. Until then, it can be helpful to use a word or phrase to remember it. Two popular ones are the nonsense word PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and the phrase Please Excuse My Dear Aunt Sally .

/en/algebra-topics/exponents/content/

Order of Operations Worksheets

Welcome to the order of operations worksheets page at Math-Drills.com where we definitely follow orders! This page includes Order of Operations worksheets using whole numbers, integers, decimals and fractions.

Elementary and middle school students generally use the acronyms PEMDAS or BEDMAS to help them remember the order in which they complete multi-operation questions. The 'P' or 'B' in the acronym stands for parentheses or brackets. All operations within parentheses get completed first. The 'E' refers to exponents; all exponents are calculated after the parentheses. The 'M' and 'D' are interchangeable as one completes the multiplication and division in the order that they appear from left to right. The fourth and final step is to solve for the addition and subtraction in the order that they appear from left to right.

More recently, students are being taught the acronym, PEMA, for order of operations, to avoid the confusion inherent in the other acronyms. For example, in PEMDAS, multiplication comes before division which some people incorrectly assumes means that multiplication must be done before division in an order of operations question. In fact, the two operations are completed in the order that they occur from left to right in the question. This is recognized in PEMA which more correctly shows that there are four levels to complete in an order of operations question.

Unless you want your students doing something different than the rest of the world, it would be a good idea to get them to understand these rules. There is no discovery or exploration needed here. These are rules that need to be learned and practiced and have been accepted as the standard approach to solving any multi-step mathematics problem.

Most Popular Order of Operations Worksheets this Week

Order of Operations With Whole Numbers and Integers

The worksheets in this section include questions with parentheses, addition, and multiplication. Exponents, subtraction, and division are excluded. The purpose of excluding some parts of PEMDAS is to ease students into how the order of operations works. To help students see a purpose for the order of operations, try to associate the expressions with related scenarios. For example, 2 + 7 × 3 could refer to the number of days in two days and three weeks. (9 + 2) × 15 could mean the total amount earned if someone worked 9 hours yesterday and 2 hours today for $15 an hour.

• Order of Operations With Whole Numbers (Addition and Multiplication Only) 2-Step Order of Operations with Whole Numbers ( Addition & Multiplication Only ) 3-Step Order of Operations with Whole Numbers ( Addition & Multiplication Only ) 4-Step Order of Operations with Whole Numbers ( Addition & Multiplication Only ) 5-Step Order of Operations with Whole Numbers ( Addition & Multiplication Only ) 6-Step Order of Operations with Whole Numbers ( Addition & Multiplication Only )

The worksheets in this section include questions with parentheses, addition, subtraction, and multiplication. Exponents and division are excluded. This section is similar to the previous one in that it is meant to help ease students into the order of operations without complicating things with exponents and division.

• Order of Operations With Whole Numbers (Addition, Subtraction and Multiplication Only) 2-Step Order of Operations with Whole Numbers ( Addition, Subtraction & Multiplication Only ) 3-Step Order of Operations with Whole Numbers ( Addition, Subtraction & Multiplication Only ) 4-Step Order of Operations with Whole Numbers ( Addition, Subtraction & Multiplication Only ) 5-Step Order of Operations with Whole Numbers ( Addition, Subtraction & Multiplication Only ) 6-Step Order of Operations with Whole Numbers ( Addition, Subtraction & Multiplication Only )

One last section to help ease students into the order of operations or simply for students who haven't learned about exponents yet. The questions on the worksheets in this section include parentheses and all four operations.

• Order of Operations With Whole Numbers (No Exponents) 2-Step Order of Operations with Whole Numbers ( No Exponents ) 3-Step Order of Operations with Whole Numbers ( No Exponents ) 4-Step Order of Operations with Whole Numbers ( No Exponents ) 5-Step Order of Operations with Whole Numbers ( No Exponents ) 6-Step Order of Operations with Whole Numbers ( No Exponents )

The worksheets in this section include questions with parentheses, exponents and all four operations.

• Order of Operations With Whole Numbers (All Operations, Parentheses and Exponents) 2-Step Order of Operations with Whole Numbers 3-Step Order of Operations with Whole Numbers 4-Step Order of Operations with Whole Numbers 5-Step Order of Operations with Whole Numbers 6-Step Order of Operations with Whole Numbers
• Order of Operations With Integers (No Exponents) 2-Step Order of Operations with Integers and No Exponents 3-Step Order of Operations with Integers and No Exponents 4-Step Order of Operations with Integers and No Exponents 5-Step Order of Operations with Integers and No Exponents 6-Step Order of Operations with Integers and No Exponents

The worksheets in this section include parentheses, exponents, and all four operations.

• Order of Operations With Integers (All Operations, Parentheses and Exponents) 2-Step Order of Operations with Integers 3-Step Order of Operations with Integers 4-Step Order of Operations with Integers 5-Step Order of Operations with Integers 6-Step Order of Operations with Integers

Order of Operations With Fractions and Decimals

As with other order of operation worksheets, the fractions order of operations worksheets require some prerequisite knowledge. If your students struggle with these questions, it probably has more to do with their ability to work with fractions than the questions themselves. Observe closely and try to pin point exactly what prerequisite knowledge is missing then spend some time going over those concepts/skills before proceeding. Otherwise, the worksheets below should have fairly straight-forward answers and shouldn't result in too much hair loss.

• Order of Operations with Positive Fractions (No Exponents) 2-Step Order of Operations with Positive Fractions (No Exponents) 3-Step Order of Operations with Positive Fractions (No Exponents) 4-Step Order of Operations with Positive Fractions (No Exponents) 5-Step Order of Operations with Positive Fractions (No Exponents) 6-Step Order of Operations with Positive Fractions (No Exponents)
• Order of Operations with Positive Fractions 2-Step Order of Operations with Positive Fractions 3-Step Order of Operations with Positive Fractions 4-Step Order of Operations with Positive Fractions 5-Step Order of Operations with Positive Fractions 6-Step Order of Operations with Positive Fractions
• Order of Operations with Positive and Negative Fractions 2-Step Order of Operations with Positive & Negative Fractions 3-Step Order of Operations with Positive & Negative Fractions 4-Step Order of Operations with Positive & Negative Fractions 5-Step Order of Operations with Positive & Negative Fractions 6-Step Order of Operations with Positive & Negative Fractions
• Order of Operations With Positive Decimals 2-Step Order of Operations with Positive Decimals 3-Step Order of Operations with Positive Decimals 4-Step Order of Operations with Positive Decimals 5-Step Order of Operations with Positive Decimals 6-Step Order of Operations with Positive Decimals
• Order of Operations With Positive and Negative Decimals 2-Step Order of Operations with Positive & Negative Decimals 3-Step Order of Operations with Positive & Negative Decimals 4-Step Order of Operations with Positive & Negative Decimals 5-Step Order of Operations with Positive & Negative Decimals 6-Step Order of Operations with Positive & Negative Decimals
• Order of Operations With Positive Decimals (European Format: Comma Decimal) 2-Step Order of Operations with Positive Decimals (Comma Decimal) 3-Step Order of Operations with Positive Decimals (Comma Decimal) 4-Step Order of Operations with Positive Decimals (Comma Decimal) 5-Step Order of Operations with Positive Decimals (Comma Decimal) 6-Step Order of Operations with Positive Decimals (Comma Decimal)
• Order of Operations With Positive and Negative Decimals (European Format: Comma Decimal) 2-Step Order of Operations with Positive & Negative Decimals (Comma Decimal) 3-Step Order of Operations with Positive & Negative Decimals (Comma Decimal) 4-Step Order of Operations with Positive & Negative Decimals (Comma Decimal) 5-Step Order of Operations with Positive & Negative Decimals (Comma Decimal) 6-Step Order of Operations with Positive & Negative Decimals (Comma Decimal)
• Order of Operations With Fractions and Decimals Mixed Order of Operations with Fractions & Decimals Mixed Order of Operations with Fractions & Decimals Mixed with some Negative Values

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Order of Operations

The order of operations is a set of rules that is to be followed in a particular sequence while solving an expression. In mathematics with the word operations we mean, the process of evaluating any mathematical expression, involving arithmetic operations such as division, multiplication, addition, and subtraction. Let us learn in detail about the order of operations rules and how well we can remember the rules using short tricks.

What is the Order of Operations?

The order of Operations is the rule in math that states we evaluate the parentheses/brackets first, the exponents/the orders second, division or multiplication third (from left to right, whichever comes first), and the addition or subtraction at the last (from left to right, whichever comes first). In math, there might be several operations to be done while evaluating an expression, and simplification at the end yields different results. However, we can only have one correct answer for any sort of expression. To identify the correct answer we simplify any given mathematical expression using a certain set of rules. These rules revolve around all the basic operators used in maths. Operators such as addition (+), subtraction (-), division (÷), and multiplication (×). Look at the given image to get a glimpse of how the order of operations exactly looks like.

Order of Operations Definition

As we discussed above Order of operations can be defined as, a set of basic rules of precedence we use while solving any mathematical expression, involving multiple operations. When a subexpression appears between two operators, the operator that comes first according to the list given below should be applied first. The order of operations, rules are expressed here:

• Brackets ( ), { }, [ ]
• Division (÷) and Multiplication (×)
• Addition (+) and Subtraction (-)

The above-mentioned set of rules always varies according to the respective given mathematical expressions.

Order of Operations Rules

While performing any sort of an operation on the respective numbers present in the expression we will follow the given basic rules in the particular sequence.

Order of Operations Rule 1: Observe the expression. The first rule is to solve the numbers present inside the parentheses or brackets. We solve inside to out grouping operations. Note the pattern of brackets present in the expression, there is a particular order to solve the parentheses, i.e., [ { ( ) } ]. First, solve the round brackets ( ) → curly brackets { } → box brackets [ ]. Inside the parantheses the order of operations are to be followed. Order of Operations Rule 2: After solving the numbers in the parentheses, look for any term present in the form of exponents and solve it. Order of Operations Rule 3: Now we are left with the basic four operators. Look for the numbers with the operation of multiplication or division, solve them from left to right. Order of Operations Rule 4: Lastly, look for the terms with addition or subtraction and solve them from left to right.

These rules have a specific acronym name. We call them PEMDAS or BODMAS . Let us learn now what exactly PEMDAS or BODMAS is.

Order of Operations - PEMDAS vs BODMAS

The PEMDAS or BODMAS is the two different acronym names given to learn the rules. These two names state the order in which the operations in an expression should be followed. Here is the detailed term for each letter used in the mentioned acronyms. First, we will discuss the PEMDAS.

Order of Operations PEMDAS

• P stands for Parentheses ( ), { }, [ ]
• E stands for Exponents (a 2 ) (For example, here, a is a number with exponent 2 )
• M stands for Multiplication (×)
• D stands for Division (÷)
• A stands for Addition (+)
• S stands for Subtraction (-)

Order of Operations BODMAS

• B stands for Brackets ( ), { }, [ ]
• O stands for Order

With the help of the above denotations, we can easily solve the mathematical expressions and get the correct answer.

How to Use Order of Operations?

Let us look at the different examples mentioned below to understand the accuracy of the rules used in order of operations.

1) For solving parentheses in order of operations :

Expression: 4 × (5 + 2) Solution: 4 × ( 7 ) = 28 (Correct (✔). This is a correct way to solve the parentheses) Let us look at another approach for the same expression. 4 × ( 5 + 2) = 20 + 2 = 22 (Incorrect (✘). This is an incorrect way to solve the parentheses)

2) For solving exponents in order of operations

Expression: 4 × (5 2 ) Solution: 4 × ( 25 ) = 100 (Correct (✔). This is a correct way to solve the exponents) Let us look at another approach for the same expression. 4 × ( 5 2 ) = 20 2 = 400 ((Incorrect (✘). This is an incorrect way to solve the exponents)

3) For multiplication or division and addition or subtraction

Expression: 3 + 5 × 2 Solution: 3 + 5 × 2 = 3 + 10 = 13 (Correct (✔). This is a correct way.) Let us look at another approach for the same expression. 3 + 5 × 2 = 8 × 2 = 16 (Incorrect (✘). This is an incorrect way.)

Expression: 3 - 6 ÷ 2 Solution: 3 - 6 ÷ 2 = 3 - 3 = 0 (Correct (✔). This is a correct way.) Let us look at another approach for the same expression. 3 - 6 ÷ 2 = (-3) ÷ 2 = -3/2 (Incorrect (✘). This is an incorrect way.)

Always remember while following the rules of order of operations do multiplication or division before addition or subtraction

Ways to Remember Order of Operations

We just read about the two different words PEMDAS and BODMAS. This is the best way to remember the order of operations. PEMDAS can be remembered by the phrase "Please Excuse My Dear Aunt Sally". In the order of operations, it means "Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction". Here multiplication and division, addition and subtraction are together. Similarly, we can remember the order of operations with the word BODMAS (Brackets, Orders, Division, Multiplication, Addition, and Subtraction.).

The easiest way to learn the order of operation is to perform the given steps:

• Start simplifying terms within the brackets first
• Solve the exponential terms.
• Perform division or multiplication.
• Perform addition or subtraction.

Note: While performing the order of operations on any given expression, we must observe the pattern of operators.

Real-Life Applications of Order of Operations

A lot of activities in our life require some sort of order of operation to perform it well. Let us take an everyday problem. Suppose you went to purchase five pepperoni pizzas that cost $20 each, and you want to split the total cost among 5 people evenly. To find out how much each person needs to pay let's use the order of operations here.

Total number of people = 5 Total number of pizzas = 5 Cost of one pizza = $20 Let us frame an expression using PEMDAS: Expression: (20 + 20 + 20 + 20 + 20) ÷ 5 or (5 × 20) ÷ 5 Solution: According to PEMDAS or BODMAS we will first solve the parentheses. (100) ÷ 5 = 20 According to the order of operations, each person needs to pay $20.

Similar to the above-mentioned problem, we have many day-to-day real-life instances where we use order of operations to deal with our problems.

☛Related Articles on Order of Operations

Check out the interesting articles below and learn more about the topic Order of Operations and its applications in detail.

• Subtraction
• Multiplication
• Order of Operations Worksheets 5th Grade

Order of Operations Examples

Example 1: Help Jack in solving the following problem with the help of order of operations rules. a) 18 ÷ (9 - 2 × 3)

Solution : Given expression: 18 ÷ (9 - 2 × 3) According to the order of operations rule, we have to solve parentheses first. Please note here inside the parentheses we have two operations present, multiplication, and subtraction. First, multiply 2 × 3 = 6 18 ÷ (9 - 6) Now subtract 6 from 9, 18 ÷ (3) Now divide 18 ÷ 3 = 6

Example 2: Simplify the given expression using the order of operations rules. ​(6 × 2 - 6 - 1) × 2 2

Solution: We know that the order of operation follows either PEMDAS or BODMAS . Let us follow the order of operations rules and simplify the given expression.

Step 1: First, we need to solve the numbers within the parentheses. Multiply 6 by 2 in the given expression, ​​​​​(6 × 2 - 6 - 1) × 2 2 , we get, (12 - 6 - 1) × 2 2 . Step 2- Now, we need to subtract 6 from 12 inside the bracket, so, we get, (6 - 1) × 2 2 . Step 3- Remove parentheses after subtracting 6 - 1, we get, 5 × 2 2 . Step 4- Solve exponent, i.e 2 2 = 4. Step 5- Multiply 5 by 4 to get the final answer, which is, 5 × 4 = 20. ∴ ​(6 × 2 - 6 - 1) × 2 2 = 20 .

Example 3: Evaluate the expression using the order of operations: (1 + 20 − 9 ÷ 3 2 ) ÷ ((2 + 1) 2 + 16 ÷ 2)

Solution: Let us see how we can apply the rules of the order of operations in solving the given expression. Step 1: First, we need to simplify the innermost bracket, (1 + 20 − 9 ÷ 3 2 ) ÷ (3 2 + 16 ÷ 2) Step 2: Now we have to evaluate exponents, (1 + 20 − 9 ÷ 9) ÷ (9 + 16 ÷ 2) Step 3: Now, we need to divide 9 by 9 and 16 by 2 inside the brackets, and we get, (1 + 20 − 1) ÷ (9 + 8) Step 4: Adding 1 and 20 we get 21. Now subtract 1 from 21 we get 20. Now, (20) ÷ (9 + 8) Step 5: Add 9 + 8 and divide the result by 20. 20 ÷ 17 = 20/17 Step 6: ∴ (1 + 20 − 9 ÷ 3 2 ) ÷ ((2 + 1) 2 + 16 ÷ 2) = 20/17

Example 4: Solve the statement problem using the order of operations. If 72 is divided by the sum of 4 and 5, then subtracted from 10, what will be the final answer?

Solution: Let us first write the given statement into mathematical form. 10 - [72 ÷ (4 + 5)] Using the order of operations rules this expression can be simplified as: = 10 - [72 ÷ (4 + 5)] = 10 - [72 ÷ 9] = 10 - 8 = 2 ∴ 10 - [72 ÷ (4 + 5)] = 2

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Practice Questions on Order of Operations

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FAQs on Order of Operations

What is the order of operations in math.

The order of operation in math is a set of rules revolving around 4 major operators. According to the order of operations, there is a particular sequence which we need to follow on each operator while solving the given mathematical expression.

How to Solve Order of Operations?

To solve the order of operations first observe the expression and note what pattern it exactly follows. Now start using either PEMDAS or BODMAS to solve the given expression. As per the rules of order of operations do look for parentheses first, then exponents, then move towards multiplication or division and addition or subtraction from left to right.

How to Do Order of Operations with Integers?

We know that integers are positive numbers and negative numbers. We can easily perform the order of operations with integers by following the given steps:

• Look for the integers present inside the parentheses or brackets and solve them.
• After solving the integers in the parentheses, look for any integer term present in the form of exponents and solve it.
• Now we are left with the basic four operators to be performed on integers. Look for the integers with the operation of multiplication or division and solve them from the left-hand side to the right-hand side.
• Lastly, look for the integers with addition or subtraction and solve them.
• In the case of integers, we need to make sure we are properly multiplying the signs. Such that, (-) × (-) = + and (+) × (-) = -

How to Remember Order of Operations?

To remember the order of operations we use two famous acronyms, i.e. PEMDAS and BODMAS. We use either of the two according to the rules of order of operations. PEMDAS or BODMAS helps in remembering the process of solving any order of operation for any n number of expressions.

How to do Order of Operations with Exponents?

According to the PEMDAS, the letter E stands for exponents which come as the second step in order of operations. Let us look at the given example to clearly understand how to do the order of operations with exponents. Expression: 7 × (2 2 ) Solution: 7 × ( 4 ) = 28 (Correct (✔). This is a correct way to solve the exponents) Let us look at another approach for the same expression. 7 × ( 2 2 ) = 14 2 = 196 ((Incorrect (✘). This is an incorrect way to do order of operations with exponents)

What is the Correct Order of Operations?

The correct order of operations can be easily expressed by using the word PEMDAS or BODMAS. The two words can be described as PEMDAS (Parentheses, Exponents, Multiplication or Division, and Addition or Subtraction). Similarly, for BODMAS (Brackets, Orders, Division, Multiplication, Addition, and Subtraction.)

What is the Order of Operations without Parentheses?

Going by the rules of the order of operations if we remove the parentheses, then we are left with EMDAS. EMDAS stands for (Exponents, Multiplication or Division, and Addition or Subtraction). If in the expression we don't have any exponential term then we need to perform multiplication or division first and moving forward we proceed with addition or subtraction. The situation may vary according to the operators present in the given expression

When Do We Use Order of Operations?

A lot of instances in our life pass through some sort of order of operations to perform it well. Every day we encounter such a scenario. For example, going to the grocery market and purchasing things we quickly perform the order of operations in our head. This helps us in reducing the turnaround time at the billing counter.

What Operation Is Completed First In the Order of Operations?

In the above sections, we read about two acronyms BODMAS and PEMDAS. According to both the acronyms, in order of operations, we simplify parentheses or the brackets first.

What is the Use of Order of Operation Calculator?

Order of operations calculator is an online tool and the fastest method with which we can evaluate any given numerical expression keeping the order of operations rules in mind. To use the order of operation calculator we need to enter the numerical expression in the correct format. Try Cuemath's order of operations calculator and solve the expressions quickly within a seconds.

☛Also Check:

For more practice try these:

• Order of Operations With Exponents Worksheets
• Advanced Order of Operations Worksheets
• Order of Operations Worksheets

What Are the 4 Order of Operations?

The 4 major order of operations are:

• Parentheses.
• An exponential term.
• Multiplication or division .
• At the end addition or subtraction.

The four order of operations can be easily recalled at any given point in time by learning the acronyms PEMD AS or B ODMAS .

Order Of Operations – Definition, Steps, FAQs, Examples

What is the order of operations in math.

• Order of Operations Steps

Solved Examples On Order Of Operations

Practice problems on order of operations, frequently asked questions on order of operations.

There are many operations in mathematics, such as addition , subtraction , multiplication , and division . They help us evaluate mathematical expressions.

Consider the following expression: 4+ 5 × 32 – 2

The expression consists of many operations. But which part do you calculate first?

You may start from the left and get one answer. But your friend may begin from the right and get a completely different answer!

Note: Both the methods given above are incorrect.

Hence, to avoid confusion, a standard rule was set to perform such calculations. This rule is known as the order of operations.

If you have an expression where all the operations are the same (example: only addition, only subtraction, only multiplication, or only division) then the correct way to solve it would be from left to right. But for expressions with multiple operations, we need to follow the order of operations.

The order of operations is the rule that tells us the sequence in which we should solve an expression with multiple operations.

A way to remember that order is PEMDAS. Each letter in PEMDAS stands for a mathematical operation. 

Order of Operations Steps:

Parentheses.

The first step is to solve the operation within parentheses or brackets . Parentheses are used to group things together. Work out all groupings from inside to out.

Work out the exponential expressions after the parentheses.

Multiplication and Division

Next, moving from left to right, multiply and/or divide, whichever comes first.

Addition and Subtraction

Lastly, moving from left to right, add and/or subtract, whichever comes first.

Related Worksheets

Why Follow the Order of Operations?

We follow the rules of the order of operations to solve expressions so that everyone arrives at the same answer. 

Here’s an example of how we can get different answers if the correct order of operations is NOT followed:

Example 1: Solve: 2 + 6 × (4 + 5) ÷ 3 – 5 using PEMDAS.

Step 1 – Parentheses : 2+6 × (4 + 5) ÷ 3 – 5 = 2 + 6 × 9 ÷ 3 – 5

Step 2 – Multiplication: 2 + 6 × 9 ÷ 3 – 5 = 2 + 54 ÷ 3 – 5

Step 3 – Division: 2 + 54 ÷ 3 – 5 = 2 + 18 – 5

Step 4 – Addition: 2 + 18 – 5 = 20 – 5

Step 5 – Subtraction: 20 – 5 = 15

Example 2: Solve 4 – 5 ÷ (8 – 3) × 2 + 5 using PEMDAS.

Step 1 – Parentheses: 4 – 5 ÷ (8 – 3) × 2 + 5 = 4 – 5 ÷ 5 × 2 + 5

Step 2 – Division: 4 – 5 ÷ 5 × 2 + 5 = 4 – 1 × 2 + 5

Step 3 – Multiplication:  4 – 1 × 2 + 5 = 4 – 2 + 5

Step 4 – Subtraction: 4 – 2 + 5 = 2 + 5

Step 5 – Addition: 2 + 5 = 7

Example 3: Solve 100 ÷ (6 + 7 × 2) – 5 using PEMDAS.

Step 1 – Multiplication inside parentheses: 100 ÷ (6 + 7 × 2 ) – 5= 100 ÷ (6 + 14) – 5

Step 2 – Addition inside parentheses: 100 ÷ (6 + 14) – 5 = 100 ÷ 20 – 5

Step 3 – Division: 100 ÷ 20 – 5 = 5 – 5 Step 4 – Subtraction: 5 – 5 = 0

Order Of Operations - Definition With Examples

Attend this Quiz & Test your knowledge.

Simplify 4+ (5 ×3² + 2) using PEMDAS.

Simplify 9 – 24 ÷ 8 × 2 + 3 using pemdas., simplify [(32 ÷ 4) + 3] × 2 using pemdas., simplify $(3 × 5² ÷ 5)$ – $(16 - 10)$ using pemdas..

What is the order of operations in math?

The order of operations are the rules that tell us the sequence in which we should solve an expression with multiple operations.

The order is PEMDAS: Parentheses, Exponents, Multiplication, and Division (from left to right), Addition and Subtraction (from left to right).

Is there a trick we can use to remember the order of operations?

Yes. You can use the phrase “Please Excuse My Dear Aunt Sally” to remember PEMDAS.

Can we perform subtraction before addition?

Yes, addition and subtraction are at the same level according to the PEMDAS rule. So, without brackets, we do the math from left to right if we are only dealing with addition and subtraction. For example, 9 – 6 + 3 = 3 + 3 = 6.

Can we perform division before multiplication?

Yes, multiplication and division are at the same level according to the PEMDAS rule. So, without brackets, we do the math from left to right if we are only dealing with multiplication and division. For example, 24 ÷ 8 × 2 = 3 × 2 = 6.

Are PEMDAS and BODMAS the same?

Yes. Both PEMDAS and BODMAS are acronyms for remembering the order of operations. They are different names for the same rule. What they call PEMDAS in the US is called BODMAS in the UK, Australia, India and various other countries.

NOTE – Related Readings:

• Multiplication

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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.

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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The Order of Operations: PEMDAS

PEMDAS Worked Examples Multiplication by Juxtaposition Fractions & Grouping

If you are asked to simplify something like " 4 + 2×3 ", the question that naturally arises is "Which way do I do this? Because there are two options!" I could add first:

4 + 2×3 = (4 + 2)×3 = 6×3 = 18

...or I could multiply first:

4 + 2×3 = 4 + (2×3) = 4 + 6 = 10

Which answer is the right one?

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The Order of Operations

It seems as though the answer depends on which way you look at the problem. But we can't have this kind of flexibility in mathematics; math won't work if you can't be sure of the answer, or if the exact same expression can be calculated so that you can arrive at two or more different answers. Mathematics doesn't do "loosey-goosey".

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To fix this "it depends on how you look at it" issue, mathematicians codified an ordering to the arithmetical operations of addition, subtraction, multiplication, division, repeated multiplication (that is, exponentiation), and grouping (that is, parentheticals). This codification of which comes before what is called "the order of operations".

What is the order of operations?

The order of operations is a listing of the basic mathematical operations according to precedence (that is, according to which operation is applied first, then which is applied second, etc). The ordering says that anything inside a parenthetical is simplified first; then exponents are applied; then multiplication and division are applied (going from left to right), and finally addition and subtraction are applied (again, going from left to right).

The order of operations was established informally at least as far back as the 1500s; by "informally", I mean that mathematicians just sort of agreed that the ordering made sense. The order of operations does not seem to have been formalized until the early 1900s. (For more on the history of the formalization of the order of operations, try TheMathDoctors .)

How can I remember the order of operations?

A common technique for remembering the order of operations is the abbreviation (or, more properly, the acronym) "PEMDAS", which has been turned into the mnemonic phrase "Please Excuse My Dear Aunt Sally". This phrase stands for, and helps one remember the order of:

• Parentheses,
• Multiplication and Division, and
• Addition and Subtraction

This listing tells you the ranks of the operations: Parentheses outrank exponents, which outrank multiplication and division (but multiplication and division are at the same rank), and multiplication and division outrank addition and subtraction (which are together on the bottom rank). In other words, the precedence is:

• Parentheses (simplify inside 'em)
• Exponents (apply them)
• Multiplication and Division (going from left to right)
• Addition and Subtraction (going from left to right)

When you have a bunch of operations of the same rank, you just operate from left to right. For instance:

15 ÷ 3 × 4

...is not 15 ÷ (3 × 4) = 15 ÷ 12 , but is actually:

(15 ÷ 3) × 4 = 5 × 4 = 20

...because, going from left to right, you get to the division sign first.

If you're not sure of this, test it in your calculator, which has been programmed with the Order-of-Operations hierarchy. For instance, typesetting the above expression into a graphing calculator, you will get:

Using the above hierarchy, we see that, in the " 4 + 2 × 3 " question at the beginning of this article, Choice 2 was the correct answer, because we have to do the multiplication before we do the addition.

What is BODMAS? or BEDMAS? or BIMDAS

Speakers of British English often instead use the acronym "BODMAS", rather than "PEMDAS". BODMAS stands for "Brackets, Orders, Division and Multiplication, and Addition and Subtraction". Since "brackets" are grouping symbols like parentheses, and since "orders" is another word for exponents, the two acronyms mean the same thing. (The I in BIMDAS stands for "indices", which is another word for exponents.) Also, note that the "M" and the "D" are reversed in the British-English version; this confirms that multiplication and division are at the same "rank" or "level".

Canadian English-speakers split the difference, using BEDMAS.

Why does "order" mean "exponent"?

In case you're wondering why British-English uses the word "orders" to stand for "exponents", it appears that nobody knows . The Wikipedia source for this equivalence is — I'm not making this up — the page you're reading right now. I can find no formal etymology for this usage. The canonical source for math-word origins only mentions order in the sense of the degree; its closest approach to orders as exponents is "parodick degrees", which refers to a polynomial having its terms in decreasing order. So your guess as to "why 'orders'?"is probably as good as anybody else's.

The order of operations was settled upon in order to prevent miscommunication, but PEMDAS can generate its own confusion. Many students using PEMDAS think that multiplication *must* be done before division; students using BODMAS think that addition *must* be done before subtraction. But multiplication and divison are on the same level, because division is just multiplication by the reciprocal; and addition and subtaction are on the same level, because subtraction is just addition of a negative. (The acronym could maybe be restated as BOPS , meaning "brackets, orders, products, and sums".)

Probably the best way to explain this is to work through some examples:

• Simplify 4 + 3 2 .

I need to simplify the term with the exponent before trying to add it to the 4:

• Simplify 4 + (2 + 1) 2 .

I have to simplify inside the parentheses before I can take the exponent through. Only then can I deal with the addition of the 4 .

4 + (2 + 1) 2

= 4 + (3) 2

• Simplify 4 + [−1(−2 − 1)] 2 .

I shouldn't try to do these nested parentheses from left to right; attempting to simplify that way is way too error-prone. Instead, I'll try to work from the inside out.

First I'll simplify inside the curvy parentheses, then I'll simplify inside the square brackets, and only then will I take care of the squaring. After that is done, then I can finally add in the 4 :

4 + [−1(−2 − 1)] 2

= 4 + [−1(−3)] 2

= 4 + [3] 2

How do I know what grouping symbols to use?

There is no particular significance in the use of square brackets (the "[" and "]" above) instead of parentheses. Brackets and curly-braces (the "{" and "}" characters) are often used when there are nested parentheses, as an aid to keeping track of which close-parentheses go with which open-parens. The different grouping characters are used for convenience only.

This is similar to what happens in a spreadsheet when you enter a formula using parentheses: each set of parentheses is color-coded, so you can tell the pairs:

I will simplify inside the parentheses first, and only then multiply by the 4 . Yes, this means that I'll be adding before doing the multiplication, but that's okay; the addition is inside grouping symbols, so it comes first.

URL: https://www.purplemath.com/modules/orderops.htm

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Course: 7th grade   >   Unit 5

Intro to order of operations.

• Worked example: Order of operations (PEMDAS)
• Order of operations example
• Order of operations with negative numbers
• Order of operations with rational numbers

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Video transcript