# Property of Multiplication : Definition, Example & Problems

In this post we will discuss important properties of multiplication.

These properties are very basic and simple in nature but are very helpful to solve complex problem faster.

I strongly urge all the kids to learn the properties and practice the relevant question for your Math Examination.

## What are Properties of Multiplication?

### Commutative Property of Multiplication

It says that in multiplication operation even if you change the location of numbers, the end result will be the same.

The commutative property is expressed as:

Where A & B can be any possible number

Let us understand the property with examples:

Example 01
Let A = 2 and B = 3

Calculate A x B
⟹ A x B
⟹ 2 x 3
⟹ 6

Calculate B x A
⟹ B x A
⟹ 3 x 2
⟹ 6

Note that on changing the location of number A & B, we get the same result 6

Example 02
Let A = -7 and B = 6

Calculate A x B
⟹ A x B
⟹ -7 x 6
⟹ – 42

Calculate B x A
⟹ B x A
⟹ 6 x -7
⟹ – 42

Hence we get same result -42 for both the calculation

Conclusion: In Multiplication, changing the order of digit will not effect the end result

### Associative Property of Multiplication

It says that forming different groups in multiplication operation will not change the end result.

The associative property can be expressed as:

Where A, B & C can be any possible real number.

Example 01
Let A = 5, B = 2 & C = 4

Calculate (A x B) x C
⟹ (A x B) x C
⟹ (5 x 2) x 4
⟹ 10 x 4
⟹ 40

Now Calculate A x (B x C)
⟹ A x (B x C)
⟹ 5 x (2 x 4)
⟹ 5 x 8
⟹ 40

Note that both the calculation gives same result 40

Example 02
Let A = -11, B = 1 & C = 3

Calculate (A x B) x C
⟹ (A x B) x C
⟹ (-11 x 1) x 3
⟹ -11 x 3
⟹ -33

Now Calculate A x (B x C)
⟹ A x (B x C)
⟹ -11 x (1 x 3)
⟹ -11 x 3
⟹ -33

We get the same result -33 in both the calculation

Conclusion: In multiplication of numbers, forming different groups will not change the end result

### Distributive Property of Multiplication

There are two different part for this property:
(i) Distributive Property of Multiplication over addition
(ii) Distributive Property of Multiplication over subtraction

Both are very similar properties but we will learn both.

(i) Distributive property of multiplication over addition

It says that multiplication with sum of numbers is equal to sum of product of numbers.

The property can be expressed as:

(ii) Distributive Property of Multiplication over Subtraction

It says that multiplication with difference of numbers is equal to subtraction of product of numbers.

The property can be expressed as:

Given below are some examples of distributive property.

Example 01
A = 6, B = 2 & C = 10
Verify distributive property of multiplication over addition

Calculate A x (B +C)
⟹ 6 x ( 2 + 10)
⟹ 6 x 12
⟹ 72

Calculate A.B + A.C
⟹6.2 + 6.10
⟹ 12 + 60
⟹ 72

hence, both the calculation give the same result 72

Example 02
If A = 9, B = 1 & C = 3
Verify distributive property of multiplication over subtraction

Calculate A x (B – C)
⟹ 9 x ( 1 – 3)
⟹ 9 x -2
⟹ -18

Calculate A.B – A.C
⟹ 9.1 – 9.3
⟹ 9 -27
⟹ -18

Both the calculation give -18 as a result
hence, distributive property of multiplication is verified

### Identity Property of Multiplication

According to the property, if you multiply any number with 1 you will get same number as a result.

The property can be expressed as:

Where A can be any possible real number.

Why the property is called “Identity” Property?

The name “identity” is given as on multiplying number with 1, the identity of the number remain unchanged and we get the same number as a result

Given are some examples of identity property.

Example 01
Let A = 4

Calculate A x 1
⟹ 4 x 1
⟹ 4

Example 02
Let A = -17

Calculate A x 1
⟹ -17 x 1
⟹ -17

### Zero Property of Multiplication

According to the property, multiplication of any number with zero results in zero.

Given below are some examples

(a) 12 x 0 = 0

(b) – 6 x 0 = 0

(c) 99 x 0 = 0

(d) 0 x 0 = 0

(e) 1000 x 0 = 0

### Inverse property of Multiplication

According to the inverse property, if we multiply any number with its reciprocal we get number 1

The inverse property of multiplication is expressed as:

Where A can be any possible number and 1/A is its reciprocal

Here 1/A is also called Multiplicative Inverse

How to find the reciprocal of any number?

Just switch the numerator & denominator of any given number and you will get the reciprocal.

Given are some examples of reciprocal calculation

\mathtt{( i) \ 5\ =\ \frac{1}{5}}\\\ \\ \mathtt{( ii) \ \frac{2}{3} \ =\ \frac{3}{2}}\\\ \\ \mathtt{( iii) \ \frac{1}{7} \ =\ \frac{7}{1}}\\\ \\ \mathtt{( iv) \ \frac{1}{0} \ =\ not\ defined} \\\ \\

Note: keep two things in mind while finding reciprocal of a numbers:
(a) Reciprocal of number 0 is not defined
(b) Fraction with 0 in denominator has no reciprocal

### Multiplication Property of Equality

The property states that in a balanced equation if you multiply a number on both sides, the equation will remain balanced and valid.

Suppose the given equation is 2y + 4 = 6

If we multiply number 5 on both sides, the equation will still be valid

5. (2y + 4) = 6. 5

10y + 20 = 30

Hence, the equation 10y + 20 = 30 is correct and valid.

In General Math terms, the multiplication property of equality can be expressed as:

Note: Remember to multiply on both sides, otherwise the equation will be invalid

## Question on Property of Multiplication

(01) Find the property used in below expression
A + (B + C) = (A + B) + C

(a) Commutative Property
(b) Associative Property
(c) Distributive Property
(d) Inverse Property

Associative property is used in above expression
Option (b) is correct

(02) What is reciprocal of number 0?

(a) 1/0
(b) 0
(c) Not defined
(d) 0/1

Option (c) is correct

(03) According to commutative property, A x B = ?

(a) – ( B x A )
(b) 1 / ( B x A )
(c) – 1 / ( B x A )
(d) B x A

Option (d) is correct

The general equation for commutative property is :
A x B = B x A

(04) Which of the following is correct example of distribution property of multiplication over addition?

(a) 2 ( 6 – 3 ) = 2 . 6 – 2 . 3
(b) 2 (6 + 3 ) = 2 . 6 + 2 . 3
(c) 2 (6 + 3 ) = 2 . 6 – 6 . 3

Option (b) is correct

The general expression for distribution property of multiplication over addition is :
A x ( B + C ) = A. B + A.C

(05) Find multiplicative inverse of 1/6

(a) 6
(b) -6
(c) -1/6

Option (a) is correct

The reciprocal of the number is also called multiplicative inverse.

The reciprocal of 1/6 is 6

According to inverse property, multiplication of number with its inverse results in number 1.
1/6 x 6 = 1

(06) Find the name of property used by below expression
A x 1 = A

(a) Identity Property
(b) Inverse property
(c) Commutative Property