ôô

# What is the Associative Property? Learn what the associative properties of addition and multiplication are, and how they can help speed up arithmetic.

By
Jason Marshall, PhD
Episode #60

In past articles, we’ve talked about the two arithmetic properties known as the commutative property of addition and the distributive property. In one way, these properties are just fancy names for things you already know how to do. But the ideas behind them really are important, and they can help you to not just understand math better, but also to solve problems faster!

The two properties that we’ve looked at before are not the only arithmetic properties. So, to round out our knowledge, today we’re talking about a third type of property called the associative property.

## What is the Associative Property of Addition?

The commutative property that we talked about before says that it’s okay to swap the order of two numbers when adding or multiplying them. But what if you’re dealing with more than two numbers? For example, what if you’re adding or multiplying three numbers? Or four numbers? Does the order in which you do things matter then?

Well, that’s precisely where the associative property comes into the picture. The associative property of addition simply says that the way in which you group three or more numbers when adding them up does not affect the sum. In other words, we can solve the problem 4 + 6 + 8 either by adding the first two numbers, 4 + 6 = 10, and then adding this sum to the last number 10 + 8 = 18, or by first adding the last two numbers, 6 + 8 = 14, and then adding this sum to the first number 14 + 4 = 18.

## Parentheses and the Associative Property

When writing this, we show the difference using parentheses. The first way is written (4 + 6) + 8, with the 4 + 6 part in parenthesis to show that it happens first, and the second way is written 4 + (6 + 8) to show the opposite ordering. Of course, no matter which way you solve this problem the answer is always 18. Although one way does make things a little easier than the other…which we’ll come back to in a minute.

## What is the Associative Property of Multiplication?

Similar to how it works for addition, the associative property of multiplication says that the way in which you group three or more numbers together when multiplying them doesn’t matter. In other words, in the problem 2 x 3 x 4, you can either multiply the first two numbers, 2 x 3 = 6, and then multiply this by the last number 6 x 4 = 24—written as (2 x 3) x 4—or you can multiply the last two numbers, 3 x 4 = 12, and then multiply this by the first number 12 x 2 = 24—written as 2 x (3 x 4). The answer is 24 either way.

## Real World Examples of the Associative Property

Up until now we’ve been talking about the associative property in an abstract way. So let’s take a minute to think about putting a more real world spin on things to help you get a better feeling for what associativity really means.

You can visualize the associative property in terms of making concrete from a combination of three ingredients: cement, gravel, and water. If you first pour a bag of cement into a bucket along with some gravel, then add water to this mix and stir, everything will work out fine. But if you instead pour cement and water into your bucket, stir, and wait a while, you’ll find that you can’t mix in the gravel because the cement has already set rock hard! In other words, adding together cement, gravel, and water to make concrete is not an associative process!

## How to Use the Associative Property to Speed Up Arithmetic

Okay, now that you have a real world picture of what it means, it’s reasonable for you to wonder why you should care about the associative property?

[[AdMiddle]Back when we first learned about the commutative property of addition, we found that we could use the idea of rearranging the order of the numbers in a sum to help us perform lightning fast addition in our heads. And, if you think about it, you’ll see that not only were we relying on the commutative property to do this, but we were also making use of the associative property. So you have good reason to be thankful for it!

Just as with addition, you can also use the commutative and associative properties of multiplication to help you multiply numbers faster. The idea is that instead of multiplying a list of numbers in the order they’re written from left-to-right, you can multiply them in any order you want. For example, in a problem like 3 x 4 x 10 x 5, you’d be wise to first multiply 4 x 5 to get 20 (the reason this is wise is that it gives you an even power of 10), then multiply 20 x 10 to get 200, and finally multiply 200 x 3 to get 600.

Why is that a good idea? Because it’s a lot less work than going from left-to-right like this: 3 x 4 = 12, 12 x 10 = 120, 120 x 5 is equal to…um, hang on a second I need to think about that…okay, it’s 600. See what I mean? I definitely prefer doing the problem the first way and working with even powers of 10 as much as possible!

## Wrap Up 