Understanding the Order of Operations in Mathematics

Hi and welcome to Math Antics. Today we’re gonna talk about an important math concept called Order of Operations. Order of Operations is just a set of math rules that tell you which math operations (like addition or multiplication) to do first.

Now you might be wondering (especially if you’re a teenager), “Why do I need RULES to tell me which operations to do first? Can’t I just do them in any order I want?” Well, that’s a really good questions!

And to answer it, we’re going to give two TOTALLY different people the SAME math problem to solve. The problem is: 2 + 5 × 4 Hmmm… I like addition better than multiplication, so I’m gonna do that first!

Let’s see… 2 + 5 gives us 7. And then I just multiply that 7 by the 4 and I get 28. That was easy! But you’d better not copy my answer! Uh, don’t worry… I’m not gonna copy your answer, cuz I want the RIGHT answer!

And I prefer multiplying, so I’m going to do THAT first. Let’s see… 4 × 5 = 20, and THEN I’ll add the 2 which gives me 22 for a final answer. What makes you think THAT’S the right answer? All my calculations were correct.

I even checked it with a calculator! Pfffff - the only calculator I need is right up here! And the correct answer is… boop, boop, boop boop, boop…. 22!

Okay, so which one of these guys do you think is right? Neither one made any mistakes with the calculations. They just did the operations in a different order and got DIFFERENT answers! Well… since there were no mistakes, in a way, they were BOTH right!

But math would be a VERY confusing subject if there were different answers to the same problem. And that’s where Order of Operations can help us out. The Order of Operations Rules are a way for us all to agree on the order that you should do math operations in.

And if we always do operations in the same order, then we’ll always get the same answer. So now that you know WHY we need Order of Operations rules, let’s find out what those rules are. There’re basically four of them and they go something like this:

FIRST: do operation in parentheses and brackets NEXT: do exponents THEN: do multiplication and division

LAST: do addition and subtraction Let’s take a closer look at each one of these rules, and see some examples where they will help us. First on the list was: do operations in parentheses and brackets.

Now in case you haven’t seen parentheses or brackets used in math before, let me briefly explain how they work. Parentheses are just these symbols that curve forwards and backwards, and they’re used in pairs like this. And when we put numbers and operators in between them, it forms a group.

It’s almost like the parentheses form a package to hold whatever math stuff we put inside them. And brackets work exactly the same way as parentheses. They just have a different shape that looks a little more boxy, but they mean exactly the same thing.

So parentheses and brackets are used to group things together, and our rules tell us to do any operations inside these groups FIRST. For example, have a look at this problem: 10 × (4 + 5) It has 3 numbers and 2 operations: multiplication and addition.

But 2 of the numbers and the addition symbol are inside parentheses. That means that they form a group and we need to do that part of the problem first. 4 + 5 = 9, so the part inside the parentheses can just be replaced with the simplified value 9.

Oh… and once you do the math that’s inside parentheses and get a single number like this, you usually don’t need to show the parentheses anymore. Now that the parentheses are gone, we just have one operation left to do.

We multiply 10 × 9 and that gives us 90 as our final answer. So parentheses can really help you know what part of a problem you’re supposed to do first. But what if you get a problem that has more than one set of parentheses, like this: (5 − 3) + (6 × 2)

Fortunately, it doesn’t matter which set of parentheses you do first. You just need to do everything that’s inside the parentheses BEFORE you do anything that's NOT inside parentheses. In other words, we need to simplify BOTH of our parentheses groups before we can do this addition IN-BETWEEN them.

The first group (5 − 3) simplifies to 2, and the second group (6 × 2) simplifies to 12. Now we can do the last operation and add the values that we got from simplifying: 2 + 12 = 14 Okay, now that we know we always do operation in parentheses or brackets FIRST,

let’s take a closer look at the second rule that says the next thing we do is exponents. Now if you haven’t seen exponents before, they’re just a way of writing repeated multiplication. For example, the repeated multiplication 4 × 4 can be written in a shorter form as 4 multiplied twice.

And 4 × 4 × 4 can be written as 4 multiplied 3 times, and 4 × 4 × 4 × 4 can be written as 4 multiplied 4 times. …get the idea? This small number is called an exponent (or power).

It just tells you how many times to multiply the bigger number together. So after we take care of any parentheses, simplifying any exponents becomes the next highest priority. For example in this problem, we have to simplify the exponent BEFORE we can do the other multiplication.

The exponent is telling us to multiply 5 together twice. So 5 times 5 is 25. And after we do that, THEN we multiply the result by 3. So 25 times 3 is 75. Oh, and one thing I should point out…

sometimes you’ll get a problem that has exponents inside of parentheses, like this problem. And you may wonder, “How can I get rid of the parentheses BEFORE I do the exponent?” You might think that if you simplify the exponent first, you’re breaking the rules.

But the truth is that by doing whatever operations are inside the parentheses, you ARE doing the parentheses first. The parentheses really just tell you where to start. So in this problem, first we do 3 to the power of 2, which means 3 × 3 which is 9.

Then the part inside the parentheses is 9 × 4, which equals 36. And once the parentheses are gone, we add 36 + 6 and get 42 as our final answer. Alright, now we’re gonna look at the last two rules together.

These two rules are really important because they deal with the most common math operations: addition, subtraction, multiplication & division. And the rules tell us that we need to do multiplication and division BEFORE we do addition and subtraction.

To see how these rules work, lets look at a few quick examples that use those basic operations. First let’t try this one: 2 + 5 + 4 Ah Ha!… Does this look familiar? Yep, it’s the one we gave to my two friends earlier.

And now that we have our rules, we see that we have to do the multiplication before the addition. 5 × 4 = 20 and THEN we add the 2, which gives us 22… so the second guy WAS right! [Sarcastically] What a surprise!

Now let’s try this one: 3 × 5 − 1 Our rules tell us that multiplication is higher on the list than subtraction, so we do 3 × 5 first. That gives us 15, and THEN we subtract the ‘1’ which leaves 14 as our final answer.

Here’s one with division and subtraction: 20 − 10 ÷ 5 And since division has a higher priority, we do the 10 divided by 5 first, which equals 2. And then we subtract 2 from 20 and get 18 as our final answer.

And here’s another problem: 12 ÷ 6 + 5 Again, our rules say to do the division before addition, so 12 divided by 6 equals 2 and then we add the 5 to get 7. And here’s one last problem: 40 ÷ 4 × 5

Which do we do first?… the multiplication or the division? Our rules don’t tell us. Well, that’s because multiplication and division are TIED for priority or importance. So are addition and subtraction. And THAT’S the reason we need an extra part at the end of each of these rule that says “from left to right”

If you have a problem that has BOTH multiplication AND division, then you’re supposed to work it from left to right. That’s because in some cases, you can get a different answers if you go from right to left. For example, in this problem,

if you work from right to left (the wrong way) you would do the 4 × 5 first and get 20. And then 40 divided by 20 equals 2. But if you go from left to right, you would do 40 divided by 4 first, which is 10, and then 10 times 5 equals 50.

Wow! The direction we went made a BIG difference. So whenever you have a problem that has a mixture of multiplication and division OR a mixture of addition and subtraction, you know to do the operations in order from left to right.

Alright, we’re just about done, but let’s have one more look at all four of our rules before you start practicing with the exercises. The Order of Operations rules say: FIRST: do operation in parentheses and brackets

NEXT: do exponents THEN: do multiplication and division (from left to right) LAST: do addition and subtraction (from left to right)

Alright, that does it for this video. Good luck with the exercises and I’ll see ya next time. Learn more at www.mathantics.com