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:

1. Parentheses
2. Exponents
3. Multiplication and division
4. 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

Parentheses

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:

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

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

Exponents

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 .

4 / 2 ⋅ 3 + 16 + 18 / 3 ² - 8

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

4 / 2 ⋅ 3 + 16 + 18 / 9 - 8

Multiplication and division

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 / 2 ⋅ 3 + 16 + 18 / 9 - 8

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

Addition and subtraction

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 + 2 - 8

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

22 + 2 - 8

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

24 - 8

24-8 is 16 . That's it!

16

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 .