# Associative Property of Multiplication

Associative Property of Multiplication
Go back to  'What are Numbers'

Don’t you think it’s easier to calculate $$5 \times (10 \times 10),$$ than $$10 \times (5 \times 10)?$$

In this mini-lesson, we will figure out why it is so and explore the world of the associative property of multiplication. The journey will start with the definition of the associative property and will lead us to interesting facts and exciting examples.

You can check out the interactive calculator to know more about the lesson and try your hand at solving a few real-life practice questions at the end of the page.

What are we waiting for, let’s get started!

## What Is the Associative Property of Multiplication?

The Associative property tells us that we can add/multiply the numbers in an equation irrespective of the grouping of those numbers.

Grouping is mainly done using parenthesis.

Thus, associativity helps us in solving these equations regardless of the way they are put in parenthesis.

## Associative Property of Multiplication

We have already learned about Associative Property. So, let's take a step forward and understand the associative property of multiplication definition.

Before we get into the actual definition of the associative property of multiplication, let us take any general function (F) of multiplication as an example.

$$F = a \times b \times c$$

But, Associative Property of Multiplication tells us that

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

Take a look at how this property works from the example given below.

Now try putting values of any 3 numbers here and see how Associative property for Multiplication works for other functions and numbers!

From the above example and simulation, we can say that the associative property of multiplication is defined as the property of multiplication where the product of three or more numbers remains the same regardless of how the numbers are grouped.

Important Notes
1. Associative Property helps us in solving equations regardless of the way they are put in parenthesis
2. Associative Property of Multiplication states that
$$(a \times b) \times c = a \times (b \times c)$$

## Solved Examples

 Example 1

Rachel was leaving early from school when her teacher told her that $$(x \times y) \times z = w$$.

Her teacher then asked her the value of $$x \times (y \times z)$$.

Rachel was asked to solve the question before leaving school.

Can you help Rachel solve this question so that she can leave early?

Solution

According to the Associative Property of Multiplication,

$$(x \times y) \times z = x \times (y \times z)$$

So, the answer Rachel is looking for has already been given to her by her teacher!

 $$\therefore$$ The answer is $$w$$
 Example 2

Vinay knows that $$3 \times (2 \times 9) = 54$$

Now, his father asked him the value of $$(3 \times 2) \times 9$$

What should Vinay reply to his father?

Solution

As Vinay knows that $$3 \times (2 \times 9) = 54$$, he can apply the Associative Property of Multiplication on it.

Hence, we get,

$$3 \times (2 \times 9) = (3 \times 2) \times 9$$

Both the results are the same.

 $$\therefore$$ The answer is 54
 Example 3

Sameer knows that $$5 \times 2 =10$$

His teacher asks him the value of $$5 \times 2 \times 3$$

Can you help Sameer find the right answer?

Solution

Sameer knows that
$$5 \times 2 =10$$
Now, we know from the Associative property of Multiplication that
$$5 \times 2 \times 3 = (5 \times 2) \times 3$$
From the information available to Sameer, we can say that
$$(5 \times 2) \times 3 = 10 \times 3$$
Hence, the right answer will be
$$10 \times 3 = 30$$

 $$\therefore$$ The answer is 30
 Example 4

Jia's mother told her that the value of $$7 \times 3 = 21$$

Now, she asks her the value of $$(2 \times 7) \times 3$$

Can you help Jia find the right answer?

Solution

We will first try to arrange the brackets in the form of the value that we already know (that is - the values told by Jia's mother).

Using Associative property of Multiplication, we can say that
$$(2 \times 7) \times 3 = 2 \times (7 \times 3)$$
Now, we know that $$7 \times 3 = 21$$. Hence, we can say that
$$2 \times (7 \times 3) = 2 \times 21$$
$$2 \times 21 = 42$$

 $$\therefore$$ The answer is 42
 Example 5

Rahul knows that $$2 \times 4 = 8$$ and $$5 \times 8 = 40$$

Now, Rahul wants to know the value of $$5 \times 2 \times 4$$

Can you help Rahul figure out the answer?

Solution

We will first try to arrange the brackets in the form of the value that we already know (that is - the values Rahul is aware of).

Using Associative property of Multiplication, we can say that

$$5 \times 2 \times 4 = 5 \times (2 \times 4)$$

Now, Rahul knows that $$2 \times 4 = 8$$, so we get

$$5 \times (2 \times 4) = 5 \times 8$$

Rahul also knows that $$5 \times 8 = 40$$

Substituting these values, we get

$$5 \times 8 = 40$$

 $$\therefore$$ The answer is 40

Challenging Questions
1. Megha knows that $$p \times (q \times r) = z$$. Now, she is trying to figure out the value of $$(p \times r) \times q$$. Think about all the different properties you might have to use in order to solve this equation.
2. Will 15% of 60 be the same as 60% of 15? Is this an example of Associative Property on Multiplication?

## Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

## Let's Summarize

We hope you enjoyed learning about the Associative Property of Multiplication with simulations and practice questions. Now you will be able to understand associative property multiplication, associative property of multiplication definition, multiplication associative, associative property definition math, and associative multiplication in detail.

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

## 1. What is an example of associative property?

An example of associative property is $$(2+3)+4=2+(3+4)$$

## 2. How do you find the associative property of multiplication?

The associative property of multiplication is a postulate. Thus, it cannot be found. However, we can use it in any operations involving multiplication and brackets.

More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus