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Seventh grader omarion, from Georgia, asks, "Why can the “M” and the “D” switch places without changing the order of operations?"
Hi omarion! You've hit on one of the things that is most confusing about Order of Operations. It shouldn't be confusing, but I think sometimes teachers either don't teach Order of Operations correctly, or they teach it correctly but unclearly or incompletely.
You've probably been taught PEMDAS, which stands for: parentheses, exponents, multiplication, division, addition, and subtraction. And you may have been told you can remember PEMDAS by remembering the silly phrase, "Please Exuse My Dear Aunt Sally," in which the first letter of each word corresponds to the first letter of each word in the PEMDAS list.
If that's what you were taught, it's not wrong...exactly...but it's also not complete.
So let me tell it to you a different way.
- Grouping symbols (like parentheses) are most important.
- Exponents are the next most important operation.
- Multiplication and Division are equally important, and they are next after exponents.
- Addition and Subtraction are equally important, and they are after multiplication and division.
- If you have operations that are equally important, you do them in the order that you read the expression (from left to right!)
Multiplication and Division are equally important, so you need to do them from left to right. Similarly, you do addition and subtraction from left to right. So let's try an example problem.
3 · (3 + 5) - 2 ÷ (33 - 52) + 3 · 2
To help you understand how we do this, we're going to simplify it one step at a time. Ready?
Order of operations says that we do parentheses first. But there are two sets of parentheses, right? Which one has higher priority? That's where our last rule comes into play. The one on the left comes first. So first we evaluate the first set of parentheses:
3 · 8 - 2 ÷ (33 - 52) + 3 · 2
Then we evaulate the only other set of parentheses. But wait! the second set of parentheses has multiple operations! Two exponents, and one division! So we tackle this in the following order: first exponent, second exponent, then subtraction. Here it is:
3 · 8 - 2 ÷ (27 - 52) + 3 · 2
3 · 8 - 2 ÷ (27 - 25) + 3 · 2
3 · 8 - 2 ÷ 2 + 3 · 2
So what's next? Multiplication and division. Which is more important? Neither! They're equally important. So we do them from left to right. So follow each step, and see which operation I evaluate in each step.
24 - 2 ÷ 2 + 3 · 2
24 - 1 + 3 · 2
24 - 1 + 6
Now we only have addition and subtraction. Which is more important? Neither! So we go from left to right, which means doing the subtraction, and then the addition:
23 + 6
So, in answer to your question, why can we switch the "M" and the "D" in PEMDAS? It's because neither M nor D is more important than the other. Similarly, neither A nor S is more important than the other. So actually, if I wanted, instead of talking about PEMDAS, I suppose I could talk about...
Which, of course, stands for...
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