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Lesson 53: Order of Operations (PEMDAS)
This is a very important topic that comes up constantly in all of our mathematical work. It is very important that you understand it fully.
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When doing calculations, normally we work from left to right, just the same way that we read. However, there are some special rules which force us to sometimes do calculations in a different order. This is called the order of operations.
The general idea of the rule is that multiplication and division are always done before addition and subtraction. This must be memorized.
For example, in the problem 2 + 3 x 4, we cannot do 2 + 3 to get 5, and then multiply by 4 to get 20. That would be wrong. We must first do 3 x 4 to get 12, and then add 2 to get 14.
If we have a combination of nothing but addition and subtraction, or, if we have a combination of nothing but multiplication and division, then we just work left to right. Important: It is not correct to say that we must always do addition before subtraction, because A comes before S in PEMDAS. When we have nothing left in our expression but addition and subtraction operations, we just work left to right. The same is true for when we have nothing left to do but multiplication and division problems.
For example: 4 - 1 + 7 - 2. We get 8 working left to right.
Another example: 5 x 6 ÷ 2 ÷ 5. We get 3 working left to right.
Remember, multiplication and division are always done before addition and subtraction, no matter where are they are located in the problem.
There is one exception to this rule. If you see something in parentheses, that calculation is always done first, no matter what.
For example: 4 x (2 + 3). Normally we would do the multiplication first, but since we have parentheses, we must do that first. That equals 5, and then we can multiply by 4 to get 20. As another example, (10 - 4) ÷ 2. Normally we would do the division first, but here we must first do 10-4 to get 6, and then divide by 2 to get 3.
There is a special way of remembering the order of operations. Remember the word PEMDAS, and if it helps you, remember "Please Excuse My Dear Aunt Sally" to help remember these initials. P stands for parentheses. E stands for exponents, and you'll learn about this shortly in an upcoming lesson M and D are multiplication, and A and S are addition and subtraction.
A student submitted some questions on this topic. The first one is: 6 + 3(7 - 2). First we must always do what is in parentheses. Don't even look at anything else yet. We now have 6 + 3(5). Sometimes in math, when we multiply, we leave out the x symbol. This is because later we'll actually use x for a different purpose, and we don't want to get confused. What we really have here is 6 plus 3 times 5. We always do multiplication before addition. It doesn't matter that the addition comes first in this problem. After multiplying 3 times 5, the problem is now 6 + 15. Now we just add to get 21.
A student submitted another order of operations problem: 60/12 x (4-1). Again, always do what is in parentheses first. We now have 60/12 x 3. Now we have a combination of division and multiplication operations, and no other operations.
In this case, just work left-to-right. This is a little bit confusing because when we use the PEMDAS rule, it tells us to do multiplication before division. That is a tiny bit of a fib. What it really means is that multiplication and division will always come before addition and subtraction. However, if we have nothing but multiplication and division, then just work left-to-right. First we'll do 60/12 to get 5, then we'll multiply by 3 to get 15.
Remember, if you have nothing but + and -, or if you have nothing but x and ÷, then you must work from left-to-right. PEMDAS just helps you remember the overall priority of operations relative to one another. You'll have more practice in later lessons.
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