Subscribe to our YouTube channel for the latest videos, updates, and tips.


Properties of Multiplication of Rational Numbers

We will learn the properties of multiplication of rational numbers i.e. closure property, commutative property, associative property, existence of multiplicative identity property, existence of multiplicative inverse property, distributive property of multiplication over addition and multiplicative property of 0.


Closure property of multiplication of rational numbers:

The product of two rational numbers is always a rational number. 

If a/b and c/d are any two rational numbers then (a/b × c/d) is also a rational number.

For example:

(i) Consider the rational numbers 1/2 and 5/7. Then, 

(1/2 × 5/7) = (1 × 5)/(2 × 7) = 5/14, is a rational number . 

(ii) Consider the rational numbers -3/7 and 5/14. Then 

(-3/7 × 5/14) = {(-3) × 5}/(7 × 14) = -15/98, is a rational number . 

(iii) Consider the rational numbers -4/5 and -7/3. Then 

(-4/5 × -7/3) = {(-4) × (-7)}/(5 × 3) = 28/15, is a rational number. 


Commutative property of multiplication of rational numbers:


Two rational numbers can be multiplied in any order. 

Thus, for any rational numbers a/b and c/d, we have: 

(a/b × c/d) = (c/d × a/b) 

For example:

(i) Let us consider the rational numbers 3/4 and 5/7 Then, 

(3/4 × 5/7) = (3 × 5)/(4 × 7) = 15/28 and (5/7 × 3/4) = (5 × 3)/(7 × 4)

= 15/28

Therefore, (3/4 × 5/7) = (5/7 × 3/4) 

(ii) Let us consider the rational numbers -2/5 and 6/7.Then, 

{(-2)/5 × 6/7} = {(-2) × 6}/(5 × 7) = -12/35 and (6/7 × -2/5 ) 

= {6 × (-2)}/(7 × 5) = -12/35

Therefore, (-2/5 × 6/7 ) = (6/7 × (-2)/5)

(iii) Let us consider the rational numbers -2/3 and -5/7 Then, 

(-2)/3 × (-5)/7 = {(-2) × (-5) }/(3 × 7) = 10/21 and (-5/7) × (-2/3) 

= {(-5) × (-2)}/(7 × 3) = 10/21 

Therefore, (-2/3) × (-5/7) = (-5/7) × (-2)/3



Associative property of multiplication of rational numbers:


While multiplying three or more rational numbers, they can be grouped in any order. 

Thus, for any rationals a/b, c/d, and e/f we have: 

(a/b × c/d) × e/f = a/b × (c/d × e/f) 

For example: 

Consider the rationals -5/2, -7/4 and 1/3 we have 

(-5/2 × (-7)/4 ) × 1/3 = {(-5) × (-7)}/(2 × 4) ×1/3} = (35/8 × 1/3)

= (35 × 1)/(8 × 3) = 35/24

and (-5)/2 × (-7/4 × 1/3) = -5/2 × {(-7) × 1}/(4 × 3) = (-5/2 × -7/12)

= {(-5) × (-7)}/(2 × 12) = 35/24

Therefore, (-5/2 × -7/4 ) × 1/3 = (-5/2) × (-7/4 × 1/3) 


Existence of multiplicative identity property:


For any rational number a/b, we have (a/b × 1) = (1 × a/b) = a/b

1 is called the multiplicative identity for rationals. 

For example:

(i) Consider the rational number 3/4. Then, we have 

(3/4 × 1) = (3/4 × 1/1) = (3 × 1)/(4 × 1) = 3/4 and ( 1 × 3/4 )

= (1/1 × 3/4 ) = (1 × 3)/(1 × 4) = 3/4 

Therefore, (3/4 × 1) = (1 × 3/4) = 3/4. 

(ii) Consider the rational -9/13. Then, we have

(-9/13 × 1) = (-9/13 × 1/1) = {(-9) × 1}/(13 × 1) = -9/13 

and (1 × (-9)/13) = (1/1 × (-9)/13) = {1 × (-9)}/(1 × 13) = -9/13

Therefore, {(-9)/13 × 1} = {1 ×(-9)/13} = (-9)/13


Existence of multiplicative inverse property:

Every nonzero rational number a/b has its multiplicative inverse b/a. 

Thus, (a/b × b/a) = (b/a × a/b) = 1

b/a is called the reciprocal of a/b. 

Clearly, zero has no reciprocal. 

Reciprocal of 1 is 1 and the reciprocal of (-1) is (-1) 

For example: 

(i) Reciprocal of 5/7 is 7/5, since (5/7 × 7/5) = (7/5 × 5/7) = 1 

(ii) Reciprocal of -8/9 is -9/8, since (-8/9 × -9/8) = (-9/8 × -8/9 ) =1

(iii) Reciprocal of -3 is -1/3, since

(-3 × (-1)/3) = (-3/1 × (-1)/3) = {(-3) × (-1)}/(1 × 3) = 3/3 = 1 

and (-1/3 × (-3)) = (-1/3 × (-3)/1) = {(-1) × (-3)}/(3 × 1) = 1 

Note: 


Denote the reciprocal of a/b by (a/b)-1

Clearly (a/b)-1 = b/a 


Distributive property of multiplication over addition:

For any three rational numbers a/b, c/d and e/f, we have: 

a/b × (c/d + e/f) = (a/b ×c/d ) + (a/b × e/f) 

For example: 

Consider the rational numbers -3/4, 2/3 and -5/6 we have 

(-3)/4 × {2/3 + (-5)/6} = (-3/4) × {4 + -5/ 6} = (-3/4) × (-1)/6 

= {(-3) × (-1)}/(4 × 6) = 3/24 = 1/8 

again, (-3/4) × 2/3 = {(-3) × 2}/(4 × 3) = -6/12 = -1/2

and

(-3/4) ×(-5/6) = {(-3) × (-5)}/(4 × 6) = 15/24 = 5/8 

Therefore, (-3/4) × 2/3 } + {(-3/4) × (-5/6)} = (-1/2 + 5/8 )

= {(-4) + 5}/8 = 1/8 

Hence, (-3/4) × (2/3 + (-5)/6) = {(-3/4) × 2/3} + {(-3/4) × (-5)/6}.


Multiplicative property of 0: 

Every rational number multiplied with 0 gives 0. 

Thus, for any rational number a/b, we have (a/b × 0) = (0 × a/b) = 0. 

For example: 

(i) (5/18 × 0) = (5/18 × 0/1) = (5 × 0)/(18 × 1) = 0/18 . 

Similarly, (0 × 5/8) = 0 

(ii) {(-12)/17 × 0} = {(-12)/17 × 0/1} = [{(-12) × 0}/{17 × 1}] = 0/17 

= 0. 

Similarly, (0 × (-12)/17) = 0

Rational Numbers

Introduction of Rational Numbers

What is Rational Numbers?

Is Every Rational Number a Natural Number?

Is Zero a Rational Number?

Is Every Rational Number an Integer?

Is Every Rational Number a Fraction?

Positive Rational Number

Negative Rational Number

Equivalent Rational Numbers

Equivalent form of Rational Numbers

Rational Number in Different Forms

Properties of Rational Numbers

Lowest form of a Rational Number

Standard form of a Rational Number

Equality of Rational Numbers using Standard Form

Equality of Rational Numbers with Common Denominator

Equality of Rational Numbers using Cross Multiplication

Comparison of Rational Numbers

Rational Numbers in Ascending Order

Rational Numbers in Descending Order

Representation of Rational Numbers on the Number Line

Rational Numbers on the Number Line

Addition of Rational Number with Same Denominator

Addition of Rational Number with Different Denominator

Addition of Rational Numbers

Properties of Addition of Rational Numbers

Subtraction of Rational Number with Same Denominator

Subtraction of Rational Number with Different Denominator

Subtraction of Rational Numbers

Properties of Subtraction of Rational Numbers

Rational Expressions Involving Addition and Subtraction

Simplify Rational Expressions Involving the Sum or Difference

Multiplication of Rational Numbers

Product of Rational Numbers

Properties of Multiplication of Rational Numbers

Rational Expressions Involving Addition, Subtraction and Multiplication

Reciprocal of a Rational  Number

Division of Rational Numbers

Rational Expressions Involving Division

Properties of Division of Rational Numbers

Rational Numbers between Two Rational Numbers

To Find Rational Numbers





8th Grade Math Practice 

From Properties of Multiplication of Rational Numbers to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Expanded form of Decimal Fractions |How to Write a Decimal in Expanded

    May 07, 25 01:48 AM

    Expanded form of Decimal
    Decimal numbers can be expressed in expanded form using the place-value chart. In expanded form of decimal fractions we will learn how to read and write the decimal numbers. Note: When a decimal is mi…

    Read More

  2. Dividing Decimals Word Problems Worksheet | Answers |Decimals Division

    May 07, 25 01:33 AM

    Dividing Decimals Word Problems Worksheet
    In dividing decimals word problems worksheet we will get different types of problems on decimals division word problems, dividing a decimal by a whole number, dividing a decimals and dividing a decima…

    Read More

  3. How to Divide Decimals? | Dividing Decimals by Decimals | Examples

    May 06, 25 01:23 AM

    Dividing a Decimal by a Whole Number
    Dividing Decimals by Decimals I. Dividing a Decimal by a Whole Number: II. Dividing a Decimal by another Decimal: If the dividend and divisor are both decimal numbers, we multiply both the numbers by…

    Read More

  4. Multiplying Decimal by a Whole Number | Step-by-step Explanation|Video

    May 06, 25 12:01 AM

    Multiplying decimal by a whole number is just same like multiply as usual. How to multiply a decimal by a whole number? To multiply a decimal by a whole number follow the below steps

    Read More

  5. Word Problems on Decimals | Decimal Word Problems | Decimal Home Work

    May 05, 25 01:27 AM

    Word problems on decimals are solved here step by step. The product of two numbers is 42.63. If one number is 2.1, find the other. Solution: Product of two numbers = 42.63 One number = 2.1

    Read More

Rational Numbers - Worksheets

Worksheet on Rational Numbers

Worksheet on Equivalent Rational Numbers

Worksheet on Lowest form of a Rational Number

Worksheet on Standard form of a Rational Number

Worksheet on Equality of Rational Numbers

Worksheet on Comparison of Rational Numbers

Worksheet on Representation of Rational Number on a Number Line

Worksheet on Adding Rational Numbers

Worksheet on Properties of Addition of Rational Numbers

Worksheet on Subtracting Rational Numbers

Worksheet on Addition and Subtraction of Rational Number

Worksheet on Rational Expressions Involving Sum and Difference

Worksheet on Multiplication of Rational Number

Worksheet on Properties of Multiplication of Rational Numbers

Worksheet on Division of Rational Numbers

Worksheet on Properties of Division of Rational Numbers

Worksheet on Finding Rational Numbers between Two Rational Numbers

Worksheet on Word Problems on Rational Numbers

Worksheet on Operations on Rational Expressions

Objective Questions on Rational Numbers