Learn about Order of operations
Khan Academy Presents: Using the order of operations to evaluate expressions
Learn about Order of operations
Welcome to my presentation on Order of Operations. So what we’re going to do now is give you a structure of how you should tackle a mathematical expression when it’s maybe not initially clear on I guess what you should do first. Let me give you an example. If I were to tell you—if I had 7 plus 2 times 3, how do you interpret that?
One person might say, okay, I’m going to add 7 plus 2 and get—so let’s see this is one way I can interpret it. I could say this is 7 plus 2 so that would be 9 times 3 and so that would be 27. But another person could say, no, no, no I think we should do the multiplication first, we should do the 2 times 3 first and 2 times 3 is 6 of course then we get 7 plus 6 and 7 plus 6 is 13.
So, who did the right way? They did just go straight left to right or for some reason you do multiplication first I mean we can’t do it both ways it could clearly—we don’t like to get two different answers for the same problem because otherwise how can we mark people’s question is wrong because we always want to be consistent with each other.
But anyway, what’s the correct way of tackling this problem? And this is what the Order of Operations is all about. And I will now tell you what they are. So the first thing you should worry about when you look—when you do Order of Operations is--parenthesis is first. Parenthesis, whenever you see a parenthesis that tells you to do this first.
Then, exponents and if you don’t know what an exponent is don’t worry I’ll show you some examples probably not on this video but in the next one after this because this one is for more general or younger crowd. Then you do multiplication and division and then you do addition and subtraction. So let’s do recall so parenthesis, exponents, multiplication, division, addition, subtraction. ― in parenthesis I think it almost makes sense to you they force you to do that part first and then you kind of do the faster operations first.
Exponents will increase the number very, very quickly while—anyway and exponents are kind of very fast operations that numbers get big very fast. Multiplication, division they’re a little less fast and addition and subtraction are kind of like the slowest way of operating on numbers, that’s how I think about I apologize if that happens to confuse you.
So let’s do some examples. This is all about practice, so watch this video and then do a ton of practice problems preferably on Khan Academy and it should make some sense. So let me see—let me see new tools according to this video so it’s a—here we go. So let’s say I had 3 plus 5 minus 3 times 6 divided by 2―that’s divided by― I can write the division sign there too.
So, what do I do first? I have all these numbers, there are bunch of different ways I could tackle. Well, if you remember what I just done, parenthesis always first. So let’s tackle the parenthesis, so we tackle this part first. That’s fine but not even within the parenthesis I’m not sure what I do first. So I’m in the parenthesis and then within the parenthesis, well let’s see I could do subtraction first, I could do multiplication first. And if you remember what I just wrote, you do the multiplication first and if you remember what I just wrote we’ll do the multiplication first. So this is what we will tackle first.
The 3 times 6 so 3 times 6 is obviously 18, so then the expression becomes 3 plus 5 minus 18 divided by 2—that’s a percent sign divided I write divided by like that we could be that slash too I guess I’m always programming computer that’s why I’m used to just dividing― but anyway, this is a division just in case you don’t know. So back to what we’ve done before, we still have fairly complicated expressions plus because I wrote it so messily over here. So what do I do next?
Well, once again parenthesis first. So we go into this parenthesis and there’s only thing I can do within this parenthesis is I can do 5 minus 18. Well, what is 5 minus 18? It is negative 13. So now, I have 3 plus minus 13 divided by 2. And our next—and I could even—I’ll leave the parenthesis there but I’m just having parenthesis there but you know I'm just having parenthesis around numbers.
So the next thing I do—what I do next? Do I do the addition or do I do the division first? And actually what I was just thinking about is I was tempted to do this but no. Division comes ahead of addition, division and multiplication are kind of the same speed that’s how I view them and addition and subtraction are slower so we do division first. So we do the negative 13 divided by 2 so we get 3 plus—this goes to this and now we want to do the 13 divided by 2.
What’s negative 13 divided by 2? It’s minus 6.5. That became minus 6.5 and now there’s nothing left to do we have 3 plus minus 6.5 well that is minus 3.5 and we’re done. It does look complicated but all you have to remember we did the parenthesis first and then we went within the parenthesis we did, we said, are there any other parenthesis no, but there’s multiplication so we do the multiplication we got here then we just did the rest of the parenthesis, we got the negative 13 then we did the division before the addition we got here then we just solve the problem.
Let’s do another one because I want your brain to hurt—if I can just learn how to use this tool I’m using. What if I had 3 plus 5 times 2 minus 3 times 6? Well, the first thing we always look for is the parenthesis and so we go on to this parenthesis which is a big parenthesis and then within that parenthesis there are more parenthesis so we keep kind of focusing in and we see this parenthesis right here and let’s—.
So this is the parenthesis within the parenthesis so that’s what we always want to do first. And so the value of 3 plus 5—oh that’s easy that’s 8 so then we get 8 times 2 minus 3 times 6 because we just took this 3 plus 5 has became this 8. And what we’re going to do next once again we have parenthesis so we do whatever inside the parenthesis.
First, we do the 8 times 2 so the 8 times 2 is 16. So now we have 16 minus 3 times 6. Now, what do we do next? We do the subtraction first or do we do the multiplication first? Well, we do the multiplication because multiplication comes ahead of subtraction. Remember, parenthesis first then multiplication, division then addition and subtraction. So let’s do this next. So we have 16 minus 3 times 6 that’s 16 minus 18, 16 minus 18 is minus 2. So we did it.
So let me ask you a question. I’m going to ask you something that should hopefully be bigly thought provoking. If I have 2 times 3 times 5. Does it matter whether I do 2 times 3 first of 3 times 5 first? Let’s test this out. If we did 2 times 3 first we get 6 then that times 5. If we do the 3 times 5 first, we have 2 times 15 and then 6 times 5 is 30 and 2 times 15 is also 30. It didn’t matter and you might say, “Well, maybe it only works for this Sal you fix the numbers.”
You pick exact numbers that wouldn’t matter which way I did it. And what I’m going to leave you is an exercise and to try it with any group of random numbers then you will see that when you’re doing only things of one level if this was all multiplication and division, it would matter what order you did them in left or right. And that’s why an order of operations you put multiplication and division at the same level the same thing as true of addition and subtraction.
If I had 1 plus 2 minus 3 plus 5, it doesn’t matter what order I do this in as long as I do the operations and I’ll get to the same answer. The Order of Operations only matter when you start mixing the multiplication and the subtraction and the parenthesis together. That’s all I have time for right now. Have fun, do some practice and I might record another video soon.
Learn about Order of operations
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