Art of Problem Solving: Associative Property of Addition
Question
The associative property of addition allows us to add the numbers in any order we want to get the same answer.
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Art of Problem Solving: Associative Property of Addition

The community property tells us that when we add two numbers, it doesn’t matter what order we add them in. But what if we have three numbers?

That’s where the Associative Property comes in.

Now once again I’ll start an example adding some hats. But unfortunately, I mean three types of hats and I only have two hands.

Got an another place to put some hats.

So start by putting some hats on my head, there we go, floppy hats on the head.

Looks good? Thank you.

And fuzzy caps here, and I got some ball caps.

So I got five ball caps, three fuzzy caps and two of these floppy hats. How many hats do I have?

Well you could start off by counting the hats in my hands, and adding other two hats on my head. That would sound like: 5 plus 3, and we put that in parentheses, the parentheses tell us we add those first, plus the two floppy hats on my head.

Of course that’s not the only way we could add them up. We could start, well, add the five caps over here and then three plus two, fuzzy hats and floppy hats over here. Now we look like this, five plus the combination of three and two. Again the parentheses tell us we add those two first.

Now of course it doesn’t matter what order we add these hats in, we’re gonna end up in the same number hats.

So these two have to be equal.

This is ridiculous. I’m gonna take these off. This one on, here we go.

Now it’s not special about five three and two. You can do this with any three numbers. We use variables to show that.

We take three numbers, a b and c. We add a and b first, and then c. We’re gonna get the same thing as we start off with a and added that to the total of b and c.

These two have to be the same.

And that’s what the Associative Property of Addition tells us.

It tells us it doesn’t matter if we add first two first, or last two first, you’re gonna get the same sum. What that means is you can get rid of these annoying parentheses, and just write, a plus b plus c because it doesn’t matter if we add the first two first or the last two first.

Then we combine it with community property which tells us we can reverse, we can flip the order of any two of these. We’re gonna write a plus b plus c as b plus a plus c. Alright we can flip these two and then a plus c plus b. This tells us that it doesn’t matter what order we add these three numbers in. We’re gonna get the same sum. And we can extend it to any number of numbers. Four numbers or five numbers and we use the associate property and community property to tell us we can add those numbers in whatever order we want.

And if you are anything like me, you like the one things anyway you want to.