Order of operations

The order of operations is a very simple, but a very important rule. So important, in fact, that you will have to obide by it in every math problem you encounter – ever! It concernes the order in which these operations need to be performed. The order of operations lists mathematical operations in order in which they should be performed in a mathematical expression, from highest to lowest priority. Here is that list:

order of operations

The first letters of these operations (P-E-M-A) are often used as a mnemonic device to help you remember the order in which they should be performed. As you can see, operations inside parentheses have the highest priority, then exponents, then multiplication and, with the lowest priority, addition. To explain how this translates to problem solving, we will use this example:

(8 – 10)2 – (-9) * (-3)

This is a fairly complex example, but it contains all the elements needed to properly explain the order of operations . The first step we have to do is to get rid of the parentheses where it is possible. That can be done by solving the expressions inside the parentheses.

(-2)2 – (-9) * (-3)

Notice that we left -9 and -3 inside their parentheses. That is because there are no operations to perform there and we will get rid of them a bit later. We will leave them be to avoid clutter. Now we need to perform the exponentation, as it is second on the list of priorities. Since (-2)2 equals 4, we get:

4 – (-9) * (-3)

The next operation on the order of operations is multiplication, so let us do it. The result of multiplying (-9) and (-3) is 27, so we are left with:

4 – 27

The only operation left to perform now is subtraction, which has the lowest priority. The end result is:

-23

It is important to adhere to this order as it affects the end result. If we performed the operations in a different order, we could have gotten the wrong result.

If you wish to practice the order of operations, you can use the free worksheets below.

Order of operations exams for teachers

Exam Name File Size Downloads Upload date
Non positive numeric expression
Order of operations – non positive integers 725.4 kB 3604 October 13, 2012
Order of operations – non positive decimals 602.6 kB 1295 October 14, 2012
Order of operations – non positive fractions 901.4 kB 1435 October 13, 2012
Order of operations – non positive mixed numbers 933.8 kB 1222 October 13, 2012
Order of operations – non positive improper fractions 857.5 kB 1118 October 13, 2012
Positive numeric expression
Order of operations – positive integers 662.2 kB 4066 October 13, 2012
Order of operations – positive decimal 732.4 kB 1323 October 13, 2012
Order of operations – positive fractions 848.8 kB 1990 October 13, 2012
Order of operations – positive mixed numbers 878.5 kB 1403 October 13, 2012
Order of operations – positive improper fractions 794.9 kB 1146 October 13, 2012
Non positive algebraic expression
Order of operations – non positive integers 897.4 kB 1695 October 13, 2012
Order of operations – non positive decimals 897.4 kB 921 October 13, 2012
Order of operations – non positive fractions 1.2 MB 892 October 13, 2012
Order of operations – non positive mixed numbers 1 MB 996 October 13, 2012
Order of operations – non positive improper fractions 1020.2 kB 915 October 13, 2012
Positive algebraic expression
Order of operations – positive integers 865.7 kB 2121 October 13, 2012
Order of operations – positive decimals 982.5 kB 1047 October 13, 2012
Order of operations – positive fractions 1.2 MB 1195 October 13, 2012
Order of operations – mixed fractions 1 MB 1173 October 13, 2012
Order of operations – improper fractions 981.2 kB 1106 October 13, 2012


Order of operations worksheets for students

Worksheet Name File Size Downloads Upload date
Order of operations – Positive numeric expressions 781.9 kB 3702 October 14, 2012
Order of operations – Non positive numeric expressions 843.8 kB 2198 October 14, 2012
Order of operations – Positive algebraic expressions 924.8 kB 1917 October 14, 2012
Order of operations – Non positive algebraic expressions 1.4 MB 1706 October 14, 2012


Order of operations knowledge test