The Associative Property of Multiplication

πŸ†Practice associative property

The associative property of multiplication allows us to multiply two factors and then multiply the product by the third factor, even if they're not in order from left to right.

A - The Associative Property of Multiplication

We can use this property in three ways:
1. We start by multiplying the first and the second factors, solving, and then multiplying the third factor by the result.
2. We start by multiplying the second and the third factors, solving, and then multiplying the first factor by the result.

  1. We start by multiplying the first and the third factors, solving, and then multiplying the second factor by the result.


We will place in parentheses around the factors that we want to group first in order to give them priority in the order of operations.

Note: The associative property of multiplication also works in algebraic expressions, but not in division expressions.


Let's define the associative property of multiplication as:

aΓ—bΓ—c=(aΓ—b)Γ—c=aΓ—(bΓ—c)=(aΓ—c)Γ—b a\times b\times c=(a\times b)\times c=a\times(b\times c)=(a\times c)\times b

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Test yourself on associative property!

einstein

\( 94+12+6= \)

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Let's see an example:

XΓ—5Γ—3=XΓ—15 X\times 5\times 3=X\times 15

We associate the second and third factors, multiply them, then multiply the product by the first factor.
We will get an expression equivalent to the first expression, since the associative property does not change the result.


Let's put a number in place of X to verify:
X=2X=2
2Γ—5Γ—3=2Γ—15 2\times5\times3=2\times15

30=3030=30
As we can see, after applying the associative property and multiplying the last two terms (the second and third factors), and then multiplying the product by the first term, the result is the same.


Let's take a look at the following examples:

4Γ—5Γ—2 4\times 5\times 2

It will be easier for us to solve the exercise if we first solve 5Γ—2 5\times2 and only then multiply the result by 4.
The associative property of multiplication allows us to do just that. We will get:

4Γ—5Γ—2=4Γ—10=40 4\times 5\times 2=4\times 10=40


Let's look at another example:

9Γ—25Γ—4 9\times 25\times 4

It will be easier for us to solve the exercise if we first solve 25Γ—4 25\times 4 and only then multiply the result by 9.
The associative property of multiplication allows us to do just that. Giving us:

9Γ—25Γ—4=9Γ—100=900 9\times 25\times 4=9\times 100=900

Practice this property over and over until it becomes second nature. It will come in hand down the road for many of the different types of mathematical exercises you will encounter.


Example exercises

Exercise 1

Task:

7Γ—5Γ—2= 7\times5\times2=

Solution:

7Γ—5Γ—2=7Γ—10=70 7\times5\times2=7\times10=70

Answer:

7070


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Exercise 2

Task:

3Γ—5Γ—5= 3\times5\times5=

Solution:

3Γ—5Γ—5=3Γ—25=753\times5\times5=3\times25=75

Answer:

7575


Exercise 3

Task:

35Γ—6Γ—2= 35\times6\times2=

Solution:

35Γ—6Γ—2= 35\times6\times2=

(30+5)Γ—6Γ—2= (30+5)\times6\times2=

=180+30Γ—2 =180+30\times2

=210Γ—2 =210\times2

=420 =420

Answer:

420 420


Do you know what the answer is?

Exercise 4

Task:

(8βˆ’3βˆ’1)Γ—4Γ—3= (8-3-1)\times4\times3=

Solution:

(8βˆ’3βˆ’1)Γ—4Γ—3=? (8-3-1)\times4\times3=\text{?}

(5βˆ’1)Γ—4Γ—3=? (5-1)\times4\times3=\text{?}

(5βˆ’1)Γ—4Γ—3=4Γ—4Γ—3 (5-1)\times4\times3=4\times4\times3

=16Γ—3=48 =16\times3=48

Answer:

48 48


Exercise 5

Task:

(7βˆ’4βˆ’3)(15βˆ’6βˆ’2)+(3Γ—5Γ—2)=? (7-4-3)(15-6-2)+(3\times5\times2)=?

Solution:

(7βˆ’4βˆ’3)(15βˆ’6βˆ’2)+(3Γ—5Γ—2) (7-4-3)(15-6-2)+(3\times5\times2)

(3βˆ’3)(9βˆ’2)+(3Γ—10) (3-3)(9-2)+(3\times10)

0Γ—7+30=0+30=30 0\times7+30=0+30=30

Answer:

3030


Check your understanding

Exercise 6

Task:

4.1Γ—1.6Γ—3.2+4.7=? 4.1\times1.6\times3.2+4.7=\text{?}

Solution:

4.1Γ—1.6Γ—3.2+4.7=? 4.1\times1.6\times3.2+4.7=\text{?}

=4110Γ—1610Γ—3210=4710=4\frac{1}{10}\times1\frac{6}{10}\times3\frac{2}{10}=4\frac{7}{10}

4110Γ—1610Γ—3210+4710=656100Γ—3210+4710 \frac{41}{10}\times\frac{16}{10}\times\frac{32}{10}+\frac{47}{10}=\frac{656}{100}\times\frac{32}{10}+\frac{47}{10}

41Γ—1610Γ—10=656100\frac{41\times16}{10\times10}=\frac{656}{100}

=656Γ—32100Γ—10+4710==\frac{656\times32}{100\times10}+\frac{47}{10} =

=209921000+47001000 =\frac{20992}{1000}+\frac{4700}{1000}

=256921000=25.692 =\frac{25692}{1000}=25.692

Answer:

25.692 25.692


Exercise 7

Task:

47Γ—32Γ—74=?\frac{4}{7}\times\frac{3}{2}\times\frac{7}{4}=\text{?}

Solution:

We start by associating the left side of the expression:

47Γ—32=4Γ—37Γ—2=1214=67\frac{4}{7}\times\frac{3}{2}=\frac{4\times3}{7\times2}=\frac{12}{14}=\frac{6}{7}

We continue with the whole expression:

47Γ—32Γ—74=67Γ—74=64=32=1.5\frac{4}{7}\times\frac{3}{2}\times\frac{7}{4}=\frac{6}{7}\times\frac{7}{4}=\frac{6}{4}=\frac{3}{2}=1.5

Answer:

32 \frac{3}{2}


Do you think you will be able to solve it?

Review questions

What is the associative property in multiplication?

The associative property of multiplication tells us that in a multiplication expression with three or more factors, we can multiply factors out of order without changing the result.

For example, one way to solve the following

6Γ—3Γ—5= 6\times3\times5=

is to associate the first two factors first

18Γ—5=90 18\times5=90

Or we can associate the second two factors first and the result is multiplied by the first factor.

6Γ—3Γ—5= 6\times3\times5=

6Γ—15=90 6\times15=90

As we can see, in both cases we will get the same answer.


How is the associative property used with addition and multiplication?

In addition

The associative property in addition is represented as follows: a+(b+c)=(a+b)+c=(a+c)+b a+\left(b+c\right)=\left(a+b\right)+c=\left(a+c\right)+b

This means that no matter how we order the addends, the result is the same.

Let's assign values to a=3 a=3 , b=8 b=8 and c=15 c=15

Then, applying the associative property, we will have:

3+(8+15)=3+23=26 3+\left (8+15\right)=3+23=26

(3+8)+15=11+15=26 \left (3+8\right)+15=11+15=26

(3+15)+8=18+8=26 \left (3+15\right)+8=18+8=26

We get the same answer in all three cases.

In multiplication

The associative property in multiplication can be represented as follows: aΓ—(bΓ—c)=(aΓ—b)Γ—c=(aΓ—c)Γ—b a\times\left(b\times c\right)=\left(a\times b\right)\times c=\left(a\times c\right)\times b

This means that no matter which numbers are multiplied first the result is the same.

Let's assign values to a=7 a=7 , b=3 b=3 and c=2 c=2

Then, applying the associative property, we will get:

(7Γ—3)Γ—2=21Γ—2=42 \left (7 \times3\right)\times2=21 \times2=42

7Γ—(3Γ—2)=7Γ—6=42 7\times \left (3\times2\right)=7 \times6=42

(7Γ—2)Γ—3=14Γ—3=42 \left (7\times 2\right) \times3=14 \times3=42

We can see that all three expressions gave us the same answer, even though the factors were in a different order.


What are the associative, commutative and distributive properties of multiplication?

Associative: We can regroup the factors of an expression without changing the result.aΓ—(bΓ—c)=(aΓ—b)Γ—c=(aΓ—c)Γ—b a\times\left(b\times c\right)=\left(a\times b\right)\times c=\left(a\times c\right)\times b

Commutative: We can reorder the factors in an expression without changing the result.

aΓ—b=bΓ—a a\times b=b\times a

Distributive: We can break down a multiplication operation into simpler expressions. aΓ—(b+c)=aΓ—b+aΓ—c a\times\left(b+c\right)=a\times b+a\times c


Test your knowledge

examples with solutions for the associative property of multiplication

Exercise #1

6:2+9βˆ’4= 6:2+9-4=

Video Solution

Step-by-Step Solution

According to the order of operations, we first solve the division exercise, and then the subtraction:

(6:2)+9βˆ’4= (6:2)+9-4=

6:2=3 6:2=3

Now we place the subtraction exercise in parentheses:

3+(9βˆ’4)= 3+(9-4)=

3+5=8 3+5=8

Answer

8 8

Exercise #2

3Γ—5Γ—4= 3\times5\times4=

Video Solution

Step-by-Step Solution

According to the order of operations, we must solve the exercise from left to right.

But, this can leave us with awkward or complicated numbers to calculate.

Since the entire exercise is a multiplication, you can use the associative property to reorganize the exercise:

3*5*4=

We will start by calculating the second exercise, so we will mark it with parentheses:

3*(5*4)=

3*(20)=

Now, we can easily solve the rest of the exercise:

3*20=60

Answer

60

Exercise #3

3+2βˆ’11= 3+2-11=

Video Solution

Step-by-Step Solution

According to the order of operations, we solve the exercise from left to right:

3+2=5 3+2=5

5βˆ’11=βˆ’6 5-11=-6

Answer

βˆ’6 -6

Exercise #4

4+5+1βˆ’3= 4+5+1-3=

Video Solution

Step-by-Step Solution

According to the order of operations, we solve the exercise from left to right:

4+5=9 4+5=9

9+1=10 9+1=10

10βˆ’3=7 10-3=7

Answer

7

Exercise #5

24:8:3= 24:8:3=

Video Solution

Step-by-Step Solution

According to the order of operations, we solve the exercise from left to right since the only operation in the exercise is division:

24:8=3 24:8=3

3:3=1 3:3=1

Answer

1 1

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