PEMDAS is an acronym that stands for "Please excuse my dear aunt Sally," which is a mnemonic device intended to help with memorizing the order of operations.
What is order of operations
The order of operations is a set of rules used to determine which operations to perform first in order to evaluate a mathematical expression. If the correct order is not used, we will end up with an incorrect solution to an expression. PEMDAS is just one of the mnemonics used to help remember the order of operations. Different mnemonics are used in different countries. PEMDAS is commonly taught in the United States, while other mnemonics (described below) are more common in different parts of the world. The figure below depicts PEMDAS meaning:
The above figure tells us the order in which we need to perform the respective operations. Parentheses are evaluated first, followed by exponents. Then, any multiplication or division is performed, followed by addition or subtraction.
Why is PEMDAS important
PEMDAS is important because we need to know what order to perform operations in to get the right answer. If we were to just randomly perform the operations in a given expression, we could end up with many different, incorrect solutions. As a simple example, consider the expression 5 × 3 + 2. The correct order of operations is to multiply 5 and 3 first, then add 2:
5 × 3 + 2 = 15 + 2 = 17
If we added 3 + 2 before multiplying 5 and 3, we would instead get,
5 × 3 + 2 = 5 × 5 = 25
which is incorrect. In this case, there are only two ways to evaluate the expression, one of which gives us the wrong answer. In cases when more operations are involved, it is possible to end up with many more "solutions," only one of which is correct. Order of operations is what lets us correctly evaluate an expression.
When to use PEMDAS
You use PEMDAS when there are two or more operations in an expression. If there is only one operation, we do not need to use order of operations.
In most cases, order of operations is straightforward and we just follow the order in the figure above. However, there are some cases, particularly with multiplication and division, that can cause some complications. Generally, the order of operations rules are as follows:
- Parentheses - First, compute any operations that are within parentheses. If there are multiple parentheses, start your calculations from inside out. Once all parentheses have been taken care of, move on to exponents.
- Exponents - Calculate any exponents in the expression. If there are exponents raised to an exponent, (e.g. ), evaluate the exponents from top down such that the expression is read as: rather than .
- Multiplication and division - After handling parentheses and exponents, perform multiplication or division in the expression from left to right.
- Addition and subtraction - Finally, add or subtract from left to right.
Multiplication and division are grouped together because they are inverses, so they have the same priority; when multiplication and division occur together, assuming all higher priority operations have already been taken into account, compute the operations in order from left to right. The same process is used for addition and subtraction, which also have the same priority as each other. In some cases PEMDAS is written as PE(MD)(AS), to indicate this relationship.
Also, whenever a number or group of numbers is next to another number or group of numbers that are in parenthesis, if there is no explicit operation written between them, the operation is multiplication. Below are some pemdas examples:
|1.||5 ÷ 2(3+7)||=||5 ÷ 2 × (10)|
|=||2.5 × (10)|
This problem is slightly tricky because if we were to have multiplied the 2 × 10 instead of dividing the 5 ÷ 2, we would get an incorrect answer of 0.25.
|2.||=||12 – 16 + 4|
|=||27 – 3 + 9|
|=||-5 + 6|
How to practice order of operations
One way to practice order of operations is to construct a "maze" where progress through the maze is tied to successfully completing order of operations problems. There are many ways this can be done. Below is one example.
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 have 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.
PEMDAS vs BODMAS
PEMDAS is not the only mnemonic used to remember the order of operations. In other countries, BODMAS, BEDMAS, and BIDMAS are used. They are all essentially the same thing; they just use different terms to describe the order of operations. The following list shows the difference between PEMDAS, BODMAS, BEDMAS, and BIDMAS.
- PEMDAS - Parentheses, exponents, multiplication, division, addition, subtraction.
- BODMAS - Brackets, orders, division, multiplication, addition, subtraction.
- BEDMAS - Brackets, exponents, division, multiplication, addition, subtraction.
- BIDMAS - Brackets, indices, division, multiplication, addition, subtraction.
Notice that the other three mnemonics list division before multiplication. Recall that division and multiplication are inverse operations that have the same priority when performing order of operations, and that we just need to compute them from left to right. The rules for all of the mnemonics above are the same; they only differ in the terminology used to describe each operation.