Order of operations

Order of operations is an order, agreed upon by mathematicians, for performing operations to simplify expressions. This order is as follows:

  1. Perform all operations within parantheses or other grouping symbols
  2. Simplify expressions involving exponents
  3. Multiply and divide in order from left to right
  4. Add and subtract in order from left to right

Examples

Ex 1
102 ÷ (8 × 2 - 6) + 1  =
102 ÷ (16 - 6) + 1  =
102 ÷ 10 + 1  =
100 ÷ 10 + 1  =
10 + 1  =  11
Ex 2
 3 ÷ 3 + 3 × 3 - 3   =
1 + 9 - 3   =
10 - 3   =  7

Remembering order of operations

There are a two commonly taught acronyms for helping you remember the order of operations. The most common is PEMDAS, which you can remember using the mnemonic device "Please excuse my dear aunt Sally."



Another acronym that is used is BODMAS, which stands for brackets, order, division, multiplication, addition, subtraction. Notice that the position of division and multiplication are switched between PEMDAS and BODMAS. This is because division and multiplication have the same priority. Given that parentheses and exponents have already been accounted for, division and multiplication are typically addressed from left to right. The same is true of addition and subtraction. They have the same priority, and are addressed from left to right once all the higher priority operations have been accounted for.

Practicing order of operations

Learning order of operations simply requires practice. There are many available worksheets that can be used to practice order of operations. Below is a "maze" that can also be used for practice.

Solve the problem in the rectangle marked "Start," and follow the arrow for the solution that you get. If the solution that you get isn't available, that means that your solution is incorrect. However, just because the solution you acquire is available doesn't necessarily mean that it is correct. Solve your way through the maze until you arrive at the "Finish" rectangle. Once you do, check the solution below to see if the path you followed to get to the solution was the correct one.



It is also possible to construct more complicated mazes. The point is just to test your understanding of order of operations since some of the solutions in the maze are possible to acquire by making certain mistakes with the order of operations.

If you progressed through the maze correctly, you would've moved from rectangle one through two, four, five, six, and then to the finish. The solutions to each of the problems in the rectangles are listed below; the numbering in the list corresponds to those in the top of each rectangle.

  1. 46
  2. 35
  3. 74
  4. 8
  5. 59
  6. 7
  7. -1
  8. 1