The Associative Property

What is the associative property and give an example?

It is a property that allows us to change the grouping of addition or multiplication operations with more than two terms, without changing the final result.

For example:

$5+3+8=5+3+8=5+\left(3+8\right)=16$

$5\times4\times2=5\times4\times2=5\times\left(4\times2\right)=40$

The name "the associative property" comes from the word associate, which means to group.

Using the associative property, we can associate (or group) several terms in an exercise as we see fit and in the order we choose, without changing the result.

The usual way of associating is by putting parentheses around certain terms, thus giving them priority in the order of operations.

In other words: this property allows us to isolate two of the terms, find the result of their addition or multiplication, and then add or multiply the third term to the previous result.

In addition expressions we have a few different options when it comes to association.

For example, we can start by grouping the firstand second addends, solve this sum and add the third addend to the result.

Another option is to first find the sum of the second and third addends and then add the first addend to the result.

One more option is to add the first and third addends and then add the second addend.

The more terms there are in the expression, the more options we have to associate the expression.

What's great about the associative property is that regardless of the order in which we choose to solve the exercise, the answer will always be the same.

Remember to place parentheses around the terms we want to add first.

Rule:

$a+b+c=(a+b)+c=a+(b+c)=b+(a+c)$

Note that the associative property of addition also works with algebraic expressions, although not with subtraction.

In multiplication expressions with three or more terms we can use the associative property to decide which factors we multiply first.

For example, in an expression with three factors we can:

1. Start by multiplying the first factor by the second factor and then multiply the product by the third factor.
2. Start by multiplying the second factor by the third factor and then multiply the product by the first factor.
3. Start by multiplying the first factor by the third factor and then multiply the product by the second factor.

The more terms there are, the more possibilities we have to cerate different groupings, or associations!

Regardless of the order, the result will always be the same.

Remember to place parentheses around the terms you want to multiply first.

Rule:

$a\times b\times c=(a\times b)\times c=a\times(b\times c)=b\times(a\times c)$

Note that the associative property of multiplication doesn't work in division expressions.

Let's take a step back. What is the associative property, and how can it help us?

As we mentioned earlier, the term "the associative property" comes from the word associate: to group. It tells us that we can group two terms together and multiply/ add them together before moving on to deal with the other terms.

At first it may sound a bit complicated, but the more you practice, the less you will even realize you're using it.

Sometimes we come across expressions with multiple terms that are arranged in an awkward way. At first glance, all the operations might seem overwhelming and messy. Thanks to the associative property, we can arrange the terms in a way that is easier to work with, and solve the exercise more easily.

The associative property works with addition and multiplication operations.

In addition exercises with three or more addends, we can use the associative property to group terms together and make the operation simpler.

Up until now we were taught to add the first addend to the second and then the third.

Naturally, that will give you the correct answer.

The associative property allows us to switch that order. For example, we can start by adding the second and third addends, and add the first addend.

Or, we can start by finding the sum of the first and third addends and then add the second addend to the result.

Note that we place parentheses around the addends that we want to calculate first. This helps us to clearly see the terms that we are starting with, and gives them priority in the order of operations.

Let's define the associative property of addition as:

$a+b+c=(a+b)+c=a+(b+c)=b+(a+c)$

Seem a little cofusing? Let's see it in an example:

According to the associative property of addition, we can take the first and second addends, add them together and then add the third addend to the result:

For example:

First, let's find the sum of the first and second addends. We will get $9$.

Then, we can add the third addend ($5$) to the previous total, i.e. to $9$.

We will solve in the following order:

$4+5=9$

$9+5=14$

So, how is this different than regular addition?

The associative property allows us to take whichever addends we want, regardless of their order in the expression, and add them, even if they're not in order.

Let's see it in the same expression:

Now we will add $5+5$ and to the total we will add the $4$.

$5+5=10$

$10+4=14$

As you see, we have arrived at the same answer, $14$.

The associative property can also help us in exercises with fractions:

Instead of starting by finding the common denominator right away, we can first simplify the expression by adding the first two terms, which gives us 1 in this case, and then add the third term.

$1+\frac{3}{5}=1\frac{3}{5}$

The associative property of addition also works with algebraic expressions.

Let's see it in the following example:

$4+5X+4X=$

As you can see, we can associate the terms $5X$ and $4X$ and add to the term $4$.

That is:

$4+5X+4X=9X+4$

Notice that even if the terms you want to associate are not one after the other, you could use the commutative property to change their location and then use the associative property.

Let's see it in an example:

$5X+4+4X=$

Using the commutative property, we can swap the 5X and the 4 and we get:

Then we can apply the associative property and arrive at:

$9X+4$

Now, let's put a number in place of the $X$, for example

$x=2$

and check if we get the same result:

$5\times2+4+4\times2=$

$10+4+8=22$

We will put $x=2$ in the equation and associate:

$9\times2+4=$

$18+4=22$

Indeed, we get the same result, $22$, in both expressions.

The associative property helped us to solve the exercise in a simpler way without changing the result.

It can be said that the expression

$5X+4+4X$

is equivalent to the expression

$9X+4$

Note: it is important to pay attention to the order of operations, especially when you replace $X$ with a number. Remember that if there is a number before the $X$ (called the coefficient), for example $4X$ the number that we put in place of the $X$ is multiplied by the coefficient.

Only after multiplying the $X$ we placed by its coefficient can we add the result to the other terms.

Note: The associative property of addition does not work with subtraction.

The associative property of multiplication

In multiplication exercise with three or more factors, we can use the associative property to group the terms together that will be easiest to multiply.

Up until now we multiplied the first factor by the second and then by the third.

The associative property allows us to change the order. We can start by multiplying the second by the third, the third by the first, and so on.

Let's define the associative property of multiplication as:

$a\times b\times c=(a\times b)\times c=a\times(a\times c)=(a\times c)\times b$

Remember that placing parentheses around the terms we want to multiply first will help us clearly see which terms we are dealing with, as well as give priority in the order of operations.

Let's look at the following example:

$7\times6\times\frac{3}{7}=$

At first glance, the exercise does not look simple to solve.

With the help of the commutative property and the associative property it can be solved fairly easily!

Let's start by applying the commutative property and changing the order of the terms:

$7\times\frac{3}{7}\times6=$

Now, by using the associative property of multiplication, we can start by multiplying the first two terms and then multiply the product by the third factor.

In other words:

We have chosen these two terms because we will be able to reduce the $7$ of the denominator and the numerator and stay with the whole number $3$.

We will get:

$3\times6=18$

Do you want to see if you got it right?

Try to solve the exercise as shown and see if you get the same result, $18$.

The associative property of multiplication also works with algebraic expressions.

For example:

$4\times X\times 5=$

By using the commutative property, we can change the place of the second and third terms and get:

$4\times 5\times X=$

Then, the associative property allows us to multiply the first two factors and, after that, multiply the product by the third factor.

That is to say: $20\times X=20X$

Now, let's put a number in place of the $X$, for example, $x=2$ and see if we arrive at the same result:

Solving the exercise:

$4\times 2\times 5=40$

Indeed, we got the same result: $40$.

Is this property starting to seem easy?

Later on you will use the associative property in almost any exercise without batting an eyelash, and you might be surprised to find that you don't even realize that you are using it.

Exercises:

Use the associative property to solve the following ten exercises without using a calculator:

$9\cdot4\cdot3=\left(9\cdot4\right)\cdot3=$

$15\cdot2\cdot9=15\cdot(2\cdot9)=$

$18\cdot5\cdot2=18\cdot\left(5\cdot2\right)=$

$27\cdot4\cdot25=27\cdot\left(4\cdot25\right)=$

$13+5+5=13+(5+5)=$

$8+2+7=(8+2)+7=$

$19\cdot2\cdot5=19\cdot(2\cdot5)=$

$38+2+8=38+(2+8)=$

$102\cdot10\cdot10=102\cdot(10\cdot10)=$

$13+7+100=\left(13+7\right)+100=$

$18\cdot1\cdot10=18\cdot(1\cdot10)=$

Solutions:

$9\cdot4\cdot3=\left(9\cdot4\right)\cdot3=108$

$15\cdot2\cdot9=15\cdot2\cdot9=270$

$18\cdot5\cdot2=18\cdot\left(5\cdot2\right)=180$

$27\cdot4\cdot25=27\cdot\left(4\cdot25\right)=2700$

$13+5+5=13+(5+5)=23$

$8+2+7=(8+2)+7=17$

$19\cdot2\cdot5=19\cdot(2\cdot5)=190$

$38+2+8=38+(2+8)=48$

$102\cdot10\cdot10=102\cdot(10\cdot10)=10200$

$13+7+100=\left(13+7\right)+100=120$

$18\cdot1\cdot10=18\cdot(1\cdot10)=180$

As we mentioned at the beginning of the article, we will briefly review a couple other properties that can make our lives easier: the distributive property and the commutative property.

The distributive property allows us to break down multiplication expressions into simpler addition and subtraction expressions. The ideas is to work with more comfortable numbers and make the exercise simpler.

Task:

$8\cdot28=8\cdot(20+8)=160+64=224$

$5\cdot93=5\cdot(90+3)=450+15=465$

$108:4=(100+8):4=100:4+8:4=25+2=27$

The distributive property can be expressed as follows:

$A\cdot(B+C)=AB+AC$

$A\cdot(B-C)=AB-AC$

If you need to, take a look at the article on the distributive property: the distributive property.

The commutative property allows us to change the order of the terms in an addition or multiplication expression without changing the result.

Here are some examples that demonstrate the use of this property:

$10+5=5+10=15$

$6\cdot7=7\cdot6=42$

$12+3+1=1+3+12=16$

$5\cdot4\cdot7=7\cdot4\cdot5=140$

If you need a refresher, take a look at the article about the commutative property: the commutative property.

Task:

$-5+3+4=$

Solution:

Using the associative property, we add parentheses to simplify the expression.

$-5+(3+4)=$

$-5+7=$

$-5+7=2$

Answer:

$2$

Task:

$3+2-11=$

Solution:

Using the associative property, we add parentheses to simplify the expression.

$(2+3)-11=$

$5-11=$

$5-11=-6$

Answer:

$-6$

Task:

$12:4−3+3×3=$

Solution:

Using the associative property, we add parentheses to simplify the expression.

$(12:4)-3+(3\times3)=$

$(3)-3+(9)=$

$(3)-3+(9)=9$

Answer:

$9$

Task:

$9:3-1.5\times2=$

Solution:

We add two sets of parentheses separate the equation into two expressions. After we solve the expressions separately, we subtract the results.

$(9:3)-(1.5\times2)=$

$(3)-(3)=$

$(3)-(3)=0$

Answer:

$0$

Task:

$1+2×3−7:4=$

Solution:

First, using the order of operations, we solve the multiplication and division from left to right.

$1+2\times3-7:4=$

$1+6-\frac{7}{4}=$

$1+6-7:4=$

Then, we solve addition and subtraction from left to right.

We group $(1+6)$

$(1+6)-\frac{7}{4}=$

$7-\frac{7}{4}=$

$7-1\frac{3}{4}=5\frac{1}{4}$

Answer:

$5\frac{1}{4}$

Task:

$3+10−2:4+1=$

Solution:

First, using the order of operations we solve multiplication and division from left to right.

$3+10-2:4+1=$

$3+10-\frac{1}{2}+1=$

Then, we solve addition and subtraction from left to right.

$(3+10+1)-\frac{1}{2}=$

$(14)-\frac{1}{2}=$

We group the numbers that share a common operation:

$14-\frac{1}{2}=13\frac{1}{2}$

Answer:

$13\frac{1}{2}$

If you found this article helpful, you might also be interested in the following:

Commutative property of addition

Commutative property of multiplication

Associative property of addition

Associative property of multiplication

The distributive property for students of 1st grade of ESO

Distributive property

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