# Inverse Property of Multiplication

## What is Inverse Property of Multiplication?

The property says that when we multiply a number with its reciprocal we get number 1.

Th property is expressed as:

Where A can be any possible real number.

If the number is fraction then the inverse property is expressed as:

In the above expression A/B is the given fraction and B/A is its reciprocal.

When we multiply the fraction with its reciprocal we get number 1.

Note: Remember that the property will not work when either of A or B is equal to zero

### What is the reciprocal of number?

When you rotate the digit upside down you get reciprocal of the number.

In reciprocal, numerator becomes denominator and denominator becomes numerator.

Note: The reciprocal is also called multiplicative inverse.

Let us find reciprocal of number for our understanding.

\mathtt{( a) \ 3\ \Longrightarrow \ \frac{1}{3}}\\\ \\ \mathtt{( b) \ \frac{2}{5} \ \Longrightarrow \ \frac{5}{2}}\\\ \\ \mathtt{( c) \ \frac{-7}{9} \ \Longrightarrow \ \frac{-9}{7}}\\\ \\ \mathtt{( d) \ 1\ \Longrightarrow \ \frac{1}{1}}\\\ \\ \mathtt{( e) \ \ 0\ \Longrightarrow \ not\ defined}\\\ \\ \mathtt{( f) \ \frac{1}{0} \Longrightarrow \ not\ defined}\\\ \\

Note:
(i) There is no reciprocal for number 0 because when we inverse the digits, the number 0 gets into the denominator whose value is not defined in mathematics.

(ii) Similarly there is no reciprocal for number (1/0)

### Why the property is called Inverse property?

Because we multiply the number with its multiplicative inverse to get the number 1.

### Inverse Property of Multiplication Example

For your understanding we have provided some examples of inverse property

Example 01
Let A = 7
Then 1/A = 1/7

Calculating A x 1/A

\Longrightarrow \ 7\ \times \ \frac{1}{7}\\\ \\ \Longrightarrow \ \frac{7}{7}\\\ \\ \Longrightarrow \ 1

Example 02
Let A = 1/11
Then 1/A = 11

Calculating A x 1/A

\Longrightarrow \ \frac{1}{11} \ \times \ 11\\\ \\ \Longrightarrow \ \frac{11}{11}\\\ \\ \Longrightarrow \ 1

Example 03
Let A = -13
Then 1/A = -1/13

Calculating A x 1/A

\mathtt{\Longrightarrow \ -13\ \times \ \frac{-1}{13}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-13}{-13}}\\\ \\ \mathtt{\Longrightarrow \ 1}

Example 04
Let A = 8/9
Then inverse of A = 9/8

Calculating A x 1/A

\mathtt{\Longrightarrow \ \frac{8}{9} \times \ \frac{9}{8}}\\\ \\ \mathtt{\Longrightarrow \ \frac{8\times 9}{9\times 8}}\\\ \\ \Longrightarrow \ \frac{72}{72}\\\ \\ \mathtt{\Longrightarrow \ 1}

### Frequently asked Question – Inverse Property of Multiplication

#### How Is inverse property different from Identity Property?

In identity property, we multiply the given number with 1 to get the same number.

Where A can be any real number.

In inverse property, we multiply number with its reciprocal to get number 1

#### Can you name other important properties of multiplication?

Important properties of multiplication are given below:

(a) Commutative Property
(b) Associative Property
(c) Distributive Property
(d) Identity Property
(e) Inverse property
(f) Multiplication property of equality

## Solved Problems – Inverse Property of Multiplication

(01) What is the reciprocal of -1/2

(a) -1
(b) 1/2
(c) -2
(d) 2

Option (c) is correct

In reciprocal, we turn the number upside-down, i.e. numerator becomes denominator and denominator becomes numerator.

(02) if 3 x 1/y = 1
Find the value of y

(a) 1
(b) 1/3
(c) -3
(d) 3

Option (d) is correct

Explanation:
The expression is of inverse property of multiplication.
It can be written as:
3 x 1/3 = 1

On comparing, we get value of y = 3

(03) what is the reciprocal of number 1/0

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

Option (a) is correct

Explanation:
Any number with denominator 0 is not defined. So its reciprocal is also not defined.

(04) Find the value of below expression
(11/98) x (98/11) = ?

(a) 2
(b) 11
(c) 9
(d) 1

Option (d) is correct

Explanation:
The expression shows property of multiplicative inverse

Solving the given expression:
\mathtt{\Longrightarrow \ \frac{11}{98} \times \ \frac{98}{11}}\\\ \\ \mathtt{\Longrightarrow \ \frac{11\times 98}{98\times 11}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1078}{1078}}\\\ \\ \mathtt{\Longrightarrow \ 1}

(05) Find the reciprocal of number (6/17)

(a) -6/17
(b) 17/6
(c) -17/6
(d) 6/19

Option (b) is correct

Explanation:
Reciprocal can be found by inverting number upside down