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Sixth grader Elise asks, "I don't get BODMAS. Can you help me?"

Well, Elise, this is one of my favorite questions, and I get asked this a lot. But I've never written it up on the "Ask Professor Puzzler" blog, so here we go!

Before I get started explaining BODMAS, I need to mention that in different parts of the world, this "rule" is known by different names. You call it BODMAS, but some people call it PEMDAS. So when I explain what it means, in parentheses I'll explain what it means to people who call it something different.

The six letters each stand for something you can do to combine numbers in mathematics:

B = Brackets (P = Parentheses)
O = Order (E = Exponents)
D = Division
M = Multiplication
A = Addition
S = Subtraction

These six letters indicate the order in which you do the operations in a mathematical expression. For example, if you see the following:

2 - (3 + 2),

You notice that "3 + 2" is in brackets (parentheses), so you do that FIRST: 3 + 2 = 5

Now you have 

2 - 5 = -3

Why does it matter which order you do things? It matters because you would get a different answer if you did the subtraction first:

2 - 3 = -1

-1 + 2 = 1

Uh oh! One way we get -3, and the other way we get 1!

Here's another example. Suppose you have 1 + 2^3 (1 + 2 cubed). If you did that from left to right, you would add 1 + 2 and get 3. Then you would cube that and get 27.

But what you're SUPPOSED to do, is evaluate the exponent first: 2 cubed is 8. Then you add 1 + 8, and get 9. It all depends on what order you do things, so you have to get the order right!

You see, we have to have an order of operations, or nobody would ever calculate expressions the same way. Order of Operations is a rule that helps to make sure EVERYONE evaluates the same expression in exactly the same way.  If we didn't have order of operations, people would get different answers for the same problem, and that would be horrible - nothing would ever get done, and none of our technology would work right because teams of engineers would always be fighting over how to evaluate the equations and formulas they work with, and if they didn't use proper order of operations, not only would things not work right, you could end up with some pretty horrible catastrophes (imagine engineers using heat formulas in nuclear reactors or power plants, and not calculating correctly how much cooling they need!).

Now here's the tricky part (and I've even had emails from MATH TEACHERS who don't understand this part!): you DON'T do all your multiplication before all your division, and you DON'T do all your addition before all your subtraction. Multiplication and division are on the same order of priority, and addition and subtraction are on the same order of priority. If a problem has both multiplication and division operations, you do them from left to right. If a problem has both addition and subtraction, you do them from left to right as well.

So really, it should be written BO (DM) (AS)* to remind you that division and multiplication go together, and addition and subtraction go together.

* Or PE (MD) (AS)

If you want some practice using BODMAS (PEMDAS), you can try our One To Ten Game, which challenges you to put toghether expressions that add up to all the numbers from one to ten.

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