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Assignments For Class 8 Mathematics Rational Numbers

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Class 8 Mathematics Rational Numbers Worksheets

We have provided below free printable  Class 8 Mathematics Rational Numbers Worksheets  for Download in PDF. The worksheets have been designed based on the latest  NCERT Book for Class 8 Mathematics Rational Numbers . These Worksheets for Grade 8 Mathematics Rational Numbers  cover all important topics which can come in your standard 8 tests and examinations.  Free worksheets for CBSE Class 8 Mathematics Rational Numbers , school and class assignments, and practice test papers have been designed by our highly experienced class 8 faculty. You can free download CBSE NCERT printable worksheets for Mathematics Rational Numbers Class 8 with solutions and answers. All worksheets and test sheets have been prepared by expert teachers as per the latest Syllabus in Mathematics Rational Numbers Class 8. Students can click on the links below and download all Pdf  worksheets for Algebraic Expressions and Identities class 8  for free. All latest Kendriya Vidyalaya  Class 8 Mathematics Rational Numbers Worksheets  with Answers and test papers are given below.

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Chapter 1 Class 8 Rational Numbers

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Get solutions of all NCERT Exercises and examples of Chapter 1 Class 8 Rational numbers free at teachoo. Answers of all NCERT questions are given, and Concepts are also explained.

In this chapter, we will study

• What is a Rational number
• Properties of rational number, like Closure , Commutativity, Associativity
• as well Distributivity
• We also see what an additive identity and an additive inverse is (Additive identity is 0, and additive inverse is negative of the number)
• And what a multiplicative identity and inverse is (Multiplicative identity is 1, and multiplicative inverse is reciprocal)
• Then, we represent Rational numbers on the number line
• And find Rational numbers between two rational numbers

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Home » 8th Class » NCERT Book Class 8 Maths Chapter 1 Rational Numbers (PDF)

NCERT Book Class 8 Maths Chapter 1 Rational Numbers (PDF)

NCERT Book Class 8 Maths Chapter 1 Rational Numbers is here. You can read and download Class 8 Maths Chapter 1 PDF from this page of aglasem.com. Rational Numbers is one of the many lessons in NCERT Book Class 8 Maths in the new , updated version of 2023-24 . So if you are in 8th standard , and studying Maths textbook (named Mathematics ), then you can read Ch 1 here and afterwards use NCERT Solutions to solve questions answers of Rational Numbers.

NCERT Book Class 8 Maths Chapter 1 Rational Numbers

The complete Chapter 1 , which is Rational Numbers , from NCERT Books for Class 8 Maths is as follows.

NCERT Book Class 8 Maths Chapter 1 Rational Numbers PDF Download Link – Click Here To Download The Complete Chapter PDF

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NCERT Book Class 8 Maths Chapter 1 Rational Numbers PDF

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NCERT Book for Class 8 Maths

Besides the chapter on Rational Numbers, you can read or download the NCERT Class 8 Maths PDF full book from aglasem. Here is the complete book:

• Chapter 1 Rational Numbers
• Chapter 2 Linear Equations in One Variable
• Chapter 3 Understanding Quadrilaterals
• Chapter 4 Data Handling
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• Chapter 6 Cubes and Cube Roots
• Chapter 7 Comparing Quantities
• Chapter 8 Algebraic Expressions and Identities
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• Chapter 13 Introduction to Graphs
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Class 8 Maths Chapter 1 Rational Numbers NCERT Textbook – An Overview

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Class viii math, class-8 rational numbers.

Introduction to Rational Number

Facts of Rational Numbers

Addition of rational numbers, addition properties of rational numbers, subtraction of rational number, subtraction properties of rational number, multiplication of rational numbers, multiplication properties of rational number, division of rational number, division properties of rational number, rational number between any two given rational numbers.

Rational Numbers Test

Rational Number Worksheet

Answer Sheet

Introduction to rational numbers.

• Define rational number
• Number line representation of rational numbers
• Comparison of rational numbers
• Operation like addition, subtraction, multiplication, and division

Numbers in form of a ⁄ b where 'a' and 'b' are integer. Here, 'b' is not equal to zero, known as rational numbers. some examples are 5 ⁄ 9 , -6 ⁄ 11 , 9, -3, 0, 3 ⁄ -11 etc.

Every integer is a rational number, but every rational number is not an integer. It is because, every integer 'i' can be written as i ⁄ 1 which is in fraction form. For example, -3 ⁄ 8 is a rational number but not an integer.

Every rational number is not a fraction, but every fraction is a rational number. For example, -5 ⁄ 9 is a rational number but not a fraction.

The rational number -a ⁄ b is same as − a ⁄ b .

Every rational number should be written with positive integer as denominator.

Two rational numbers a ⁄ b = c ⁄ d , if a × d = b × c .

Rational number is positive if both numerator and denominator are positive integers, or both are negative integers. If one is positive integer and other is negative integer then rational number is said to be negative.

All finite decimal numbers are rational numbers. Let's see some examples 0.5 = 1 ⁄ 2 , 0.26 = 13 ⁄ 50 and 0.625 = 5 ⁄ 8 .

When we add two rational numbers, first convert each rational number with positive denominator. Rational Numbers with same denominator If a ⁄ b and c ⁄ d are two rational numbers, then ( a ⁄ b ) + ( c ⁄ b ) = (a+c) ⁄ b .

Example 1. Add 6 ⁄ 9 and 8 ⁄ -9 . Solution. 6 ⁄ 9 + 8 ⁄ -9 Here, first convert 8 ⁄ -9 into positive denominator. 8 ⁄ -9 = {8×(-1)} ⁄ {-9×(-1)}       = -8 ⁄ 9 Then, 6 ⁄ 9 + (- 8 ⁄ 9 ) = 6 ⁄ 9 − 8 ⁄ 9 = (6-8) ⁄ 9 = -2 ⁄ 9 .

Rational Numbers With Different Denominator Here we first find LCM of denominators of two rational numbers. Then we covert the denominator of each rational number to have same value as LCM. At last, add the rational numbers following the process of same denominator.

Example 1. Add -5 ⁄ 7 + 7 ⁄ 3 . Solution. LCM of 7 and 3 = 21 -5 ⁄ 7 = (-5×3) ⁄ (7×3) = -15 ⁄ 21 7 ⁄ 3 = (7×7) ⁄ (3×7) = 49 ⁄ 21 Then, -15 ⁄ 21 + 49 ⁄ 21 = (-15+49) ⁄ 21 = 34 ⁄ 21 .

• Closure property

Commutative

Associative.

• Special properties

Closure Property

Example 1. Add 1 ⁄ 2 and 2 ⁄ 5 . Solution. 1 ⁄ 2 + 2 ⁄ 5 = (5+4) ⁄ 10 = 9 ⁄ 10 So, 9 ⁄ 10 is a rational number.

Example 2. Add -5 ⁄ 9 and -2 ⁄ 3 Solution. -5 ⁄ 9 + -2 ⁄ 3 = {-5+(-6)} ⁄ 9 = -11 ⁄ 9

Example 3. Add -6 ⁄ 5 and 1 ⁄ 4 . Solution. -6 ⁄ 5 + 1 ⁄ 4 = (-6×4+1×5) ⁄ 20 = (-24+5) ⁄ 20 = -19 ⁄ 20

Example 1. -5 ⁄ 7 + 3 ⁄ 5 = 3 ⁄ 5 + -5 ⁄ 7 Solution. We have to prove LHS = RHS LHS = -5 ⁄ 7 + 3 ⁄ 5 = (-25+21) ⁄ 35 = -4 ⁄ 35 RHS = 3 ⁄ 5 + -5 ⁄ 7 = {21+(-25)} ⁄ 35 = -4 ⁄ 35 LHS = RHS. So, it is following commutative properties.

Example 1. ( 1 ⁄ 2 + 2 ⁄ 3 ) + -3 ⁄ 5 = 1 ⁄ 2 + ( 2 ⁄ 3 + -3 ⁄ 5 ). Solution. We have to prove LHS = RHS LHS = ( 1 ⁄ 2 + 2 ⁄ 3 ) + -3 ⁄ 5 = (3+4) ⁄ 6 + ( -3 ⁄ 5 ) = 7 ⁄ 6 - 3 ⁄ 5 = (35−18) ⁄ 30 = 17 ⁄ 30 RHS = 1 ⁄ 2 + ( 2 ⁄ 3 + -3 ⁄ 5 ) = 1 ⁄ 2 + {10+(-9)} ⁄ 15 = 1 ⁄ 2 + 1 ⁄ 15 = (15+2) ⁄ 30 = 17 ⁄ 30 LHS = RHS, hence it is proved that it follows associative property.

Special Properties of Rational Numbers

Example 1. Additive inverse of -3 ⁄ 5 = −( -3 ⁄ 5 ) = 3 ⁄ 5 .

Example 2. Additive inverse of -4 ⁄ -9 = −( -4 ⁄ -9 ) = -4 ⁄ 9 .

Subtraction of rational numbers can be done by adding additive inverse of subtrahend with minuend. That is, a ⁄ b − c ⁄ d = a ⁄ b + ( -c ⁄ d ) .here -c ⁄ d is additive inverse of c ⁄ d .

Example 1. Subtract 5 ⁄ 9 from -3 ⁄ 7 . Solution. -3 ⁄ 7 − 5 ⁄ 9         = -3 ⁄ 7 + additive inverse of 5 ⁄ 9         = -3 ⁄ 7 + ( -5 ⁄ 9 )         = (-3×9−5×7) ⁄ 63         = {-27+(-35} ⁄ 63         = -62 ⁄ 63

Example 2. subtract -3 ⁄ 25 from 3 1 ⁄ 5 . Solution. 3 1 ⁄ 5 − ( -3 ⁄ 25 )         = 16 ⁄ 5 + additive inverse of ( -3 ⁄ 25 )         = 16 ⁄ 5 + 3 ⁄ 25         = (80+3) ⁄ 25         = 83 ⁄ 25         = 3 8 ⁄ 25

Not Commutative

Not associative.

Example 1. 6 ⁄ 9 − 3 ⁄ 4 = (24-27) ⁄ 36 = -3 ⁄ 36 = -1 ⁄ 12 -1 ⁄ 12 is a rational number. Example 2. -7 ⁄ 9 − 4 ⁄ 3 = (-7-12) ⁄ 9 = -19 ⁄ 9 = −2 1 ⁄ 9 −2 1 ⁄ 9 is a rational number.

Example 1. Prove 1 ⁄ 3 − 5 ⁄ 6 ≠ 5 ⁄ 6 − 1 ⁄ 3 . Solution. LHS = 1 ⁄ 3 − 5 ⁄ 6 = (2-5) ⁄ 6 = -1 ⁄ 2 RHS = 5 ⁄ 6 − 1 ⁄ 3 = (5-2) ⁄ 6 = 1 ⁄ 2 LHS ≠ RHS i.e. -1 ⁄ 2 ≠ 1 ⁄ 2 Hence it is proved that 1 ⁄ 3 − 5 ⁄ 6 ≠ 5 ⁄ 6 − 1 ⁄ 3 .

Example 1. prove ( -3 ⁄ 5 − 2 ⁄ 7 ) − ( -1 ⁄ 5 ) ≠ -3 ⁄ 5 − ( 2 ⁄ 7 − -1 ⁄ 5 ). Solution. LHS = ( -3 ⁄ 5 − 2 ⁄ 7 ) − ( -1 ⁄ 5 )         = (-21-10) ⁄ 35 + 1 ⁄ 5         = (-31) ⁄ 35 + 1 ⁄ 5         = (-31+7) ⁄ 35 = -24 ⁄ 35 RHS = -3 ⁄ 5 − ( 2 ⁄ 7 − -1 ⁄ 5 )         = -3 ⁄ 5 − ( 2 ⁄ 7 + 1 ⁄ 5 )         = -3 ⁄ 5 − (10+7) ⁄ 35         = -3 ⁄ 5 − 17 ⁄ 35         = (-21-17) ⁄ 35 = -38 ⁄ 35 -24 ⁄ 35 ≠ -38 ⁄ 35 i.e. LHS ≠ RHS Hence, it is proved that subtraction does not follow associative property.

To get the product of two rational number, numerators are multiplied, and denominators are multiplied. Let's see some examples.

Example 1. Multiply -3 ⁄ 7 and 8 ⁄ 9 Solution. -3 ⁄ 7 × 8 ⁄ 9         = (-3×8) ⁄ 7×9         = -8 ⁄ 21

Example 2. Multiply 3 1 ⁄ 7 and -4 3 ⁄ 4 Solution. 3 1 ⁄ 7 × (-4 3 ⁄ 4 )         = 22 ⁄ 7 × ( -19 ⁄ 4 )         = {22×(-19)} ⁄ 7×4         = -418 ⁄ 28         = -14 26 ⁄ 28

Commutative Property

Associative property.

• Distributive Properties

Multiplicative Identity

Multiplicative inverse.

Example 1. 2 ⁄ 5 × 7 ⁄ 9 = 2×7 ⁄ 5×9 = 14 ⁄ 45 14 ⁄ 45 is a rational number. Example 2. 3 ⁄ 4 × 1 ⁄ 6 = 3×1 ⁄ 4×6 = 3 ⁄ 24 = 1 ⁄ 8 1 ⁄ 8 is a rational number. Example 3. 4 ⁄ 7 × 5 ⁄ 8 = 4×5 ⁄ 7×8 = 5 ⁄ 14 5 ⁄ 14 is a rational number.

Example 1. Prove 2 ⁄ 3 × ( -4 ⁄ 5 ) = ( -4 ⁄ 5 ) × 2 ⁄ 3 . Solution. LHS = 2 ⁄ 3 × ( -4 ⁄ 5 ) = {2×(-4)} ⁄ (3×5) = -8 ⁄ 15 RHS = -4 ⁄ 5 × 2 ⁄ 3 = (-4×2) ⁄ 5×3 = -8 ⁄ 15 LHS = RHS i.e. 2 ⁄ 3 × ( -4 ⁄ 5 ) = ( -4 ⁄ 5 ) × 2 ⁄ 3 Hence, it is proved that multiplication of rational number is commutative.

Example 1. Prove ( 5 ⁄ 7 × -3 ⁄ 2 ) × -1 ⁄ 7 = 5 ⁄ 7 × ( -3 ⁄ 2 × -1 ⁄ 7 ) Solution. LHS = ( 5 ⁄ 7 × -3 ⁄ 2 ) × -1 ⁄ 7 = -15 ⁄ 14 × -1 ⁄ 7 = 15 ⁄ 98 RHS = 5 ⁄ 7 × ( -3 ⁄ 2 × -1 ⁄ 7 ) = 5 ⁄ 7 × 3 ⁄ 14 = 15 ⁄ 98 LHS = RHS i.e. ( 5 ⁄ 7 × -3 ⁄ 2 ) × -1 ⁄ 7 = 5 ⁄ 7 × ( -3 ⁄ 2 × -1 ⁄ 7 ) Hence, multiplication of fraction is associative.

Distributive Property

Example 1. Find multiplicative invers of 7 ⁄ 9 × 10 ⁄ 7 . Solution. First, we multiply 7 ⁄ 9 and 10 ⁄ 7 , then convert into its reciprocal          7 ⁄ 9 × 10 ⁄ 7         = (7×10) ⁄ (9×7)         = 70 ⁄ 63 = 10 ⁄ 9 Multiplicative invers of 10 ⁄ 9 = 9 ⁄ 10

When two rational numbers are divided, first we find out reciprocal of the divisor, then it is multiplied with the dividend. In other words, if a ⁄ b and c ⁄ d are rational numbers ,then a ⁄ b ÷ c ⁄ d = a ⁄ b × d ⁄ c (reciprocal of c ⁄ d ) = a×d ⁄ b×c . Let's see some examples.

Example 1. 2 1 ⁄ 5 ÷ ( -1 ⁄ 3 ). Solution. 2 1 ⁄ 5 ÷ ( -1 ⁄ 3 )         = 11 ⁄ 5 ÷ ( -1 ⁄ 3 ) Reciprocal of -1 ⁄ 3 is equal to -3.         = 11 ⁄ 5 × (-3)         = -33 ⁄ 5

Example 2. Divide the sum of 1 ⁄ 7 and 1 ⁄ 2 by the product of 9 ⁄ 2 and 3 ⁄ 7 Solution. ( 1 ⁄ 7 + 1 ⁄ 2 ) ÷ ( 9 ⁄ 2 × 3 ⁄ 7 )          = (2+7) ⁄ 14 ÷ (9×3) ⁄ (2×7)          = 9 ⁄ 14 ÷ 27 ⁄ 14 Reciprocal of 27 ⁄ 14 is 14 ⁄ 27          = 9 ⁄ 14 × 14 ⁄ 27          = (9×14) ⁄ (14×27)          = 9 ⁄ 27          = 1 ⁄ 3

Division is Non-commutative

Division is non-associative.

In order to know rational number between two rational numbers, we have to calculate the mean of two given rational numbers. If a ⁄ b and c ⁄ d are two rational numbers. Then mean of a ⁄ b and c ⁄ d is written as ( a ⁄ b + c ⁄ d ) ⁄ 2 .

Example 1. Find a rational number between -4 ⁄ 5 and 5 ⁄ 6 . Solution. First, we calculate mean of ( -4 ⁄ 5 + 5 ⁄ 6 ) ÷ 2         = (-24+25) ⁄ 30 ÷ 2         = 1 ⁄ 30 × 1 ⁄ 2 = 1 ⁄ 60 So, 1 ⁄ 60 is a rational number between -4 ⁄ 5 and 5 ⁄ 6 .

Class-8 Rational Numbers Test

Rational Numbers - 1

Rational Numbers - 2

Rational Numbers - 3

Class-8 Rational Numbers Worksheet

Rational Numbers Worksheet - 1

Rational Numbers Worksheet - 2

Rational Numbers Worksheet - 3

Rational Numbers Worksheet - 4

Rational-Numbers-Answer Download the pdf

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NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers

• NCERT Solutions
• Chapter 1 Rational Numbers

Practice and Master the Concepts of Real Numbers with NCERT Solutions for Class 10 Maths Chapter 1

A detailed step-by-step explanation of NCERT Solutions for Class 8 Maths Chapter 1 are given in this article along with all the required study material. The Class 8 Maths Chapter 1 of NCERT is related to rational numbers and their applications. By learning and understanding the concepts of rational numbers deeply, you are not only able to score good marks but also develop a ground to learn new concepts introduced in class 8. You can also download the PDF of the solutions of chapter 1 Class 8 Maths NCERT from the link given below.

Important Topics Covered in Class 8 Maths Chapter 1 Rational Numbers

The following list of important topics under NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers has been provided for students to get a hang of the major concepts in this chapter before going through the solutions to the important questions.

Introduction

1.2 Properties of Rational Numbers

Commutativity

Associativity

The role of zero

The role of 1

Negative of a number

Distributivity of multiplication over addition for rational numbers.

Representation of Rational Numbers on the Number Line

Rational Numbers between Two Rational Numbers

Access NCERT Solutions For Class 8 Maths Chapter 1 – Rational Numbers

Exercise 1.1

1. Using appropriate properties find:

i. \[\text{-}\dfrac{\text{2}}{\text{3}}\text{ }\!\!\times\!\!\text{ }\dfrac{\text{3}}{\text{5}}\text{+}\dfrac{\text{5}}{\text{2}}\text{-}\dfrac{\text{3}}{\text{5}}\text{ }\!\!\times\!\!\text{ }\dfrac{\text{1}}{\text{6}}\]

Ans: Given 

\[\text{-}\dfrac{\text{2}}{\text{3}}\text{ }\!\!\times\!\!\text{ }\dfrac{\text{3}}{\text{5}}\text{+}\dfrac{\text{5}}{\text{2}}\text{-}\dfrac{\text{3}}{\text{5}}\text{ }\!\!\times\!\!\text{ }\dfrac{\text{1}}{\text{6}}\]  

By using commutative property of rational numbers i.e $\text{a+b=b+a}$

\[\text{=-}\dfrac{\text{2}}{\text{3}}\text{ }\!\!\times\!\!\text{ }\dfrac{\text{3}}{\text{5}}\text{-}\dfrac{\text{3}}{\text{5}}\text{ }\!\!\times\!\!\text{ }\dfrac{\text{1}}{\text{6}}\text{+}\dfrac{\text{5}}{\text{2}}\]                     

Now, by using distributive property of rational numbers i.e $\text{a }\!\!\times\!\!\text{ b+b }\!\!\times\!\!\text{ c=b }\!\!\times\!\!\text{ }\left( \text{a+c} \right)$

\[\text{=}\left( \text{-}\dfrac{\text{3}}{\text{5}} \right)\text{ }\!\!\times\!\!\text{ }\left( \dfrac{\text{2}}{\text{3}}\text{+}\dfrac{\text{1}}{\text{6}} \right)\text{+}\dfrac{\text{5}}{\text{2}}\]                  

\[\text{=}\left( \text{-}\dfrac{\text{3}}{\text{5}} \right)\text{ }\!\!\times\!\!\text{ }\left( \dfrac{\text{2 }\!\!\times\!\!\text{ 2+1}}{\text{6}} \right)\text{+}\dfrac{\text{5}}{\text{2}}\]

\[\text{=}\left( \text{-}\dfrac{\text{3}}{\text{5}} \right)\text{ }\!\!\times\!\!\text{ }\left( \dfrac{\text{5}}{\text{6}} \right)\text{+}\dfrac{\text{5}}{\text{2}}\]

\[\text{=}\left( \text{-}\dfrac{\text{3}}{\text{6}} \right)\text{+}\dfrac{\text{5}}{\text{2}}\]

\[\text{=}\left( \dfrac{\text{-3+5 }\!\!\times\!\!\text{ 3}}{\text{6}} \right)\]

\[\text{=}\left( \dfrac{\text{-3+15}}{\text{6}} \right)\]

\[\text{=}\dfrac{\text{12}}{\text{6}}\]

\[\text{=2}\]

ii. \[\dfrac{\text{2}}{\text{5}}\text{ }\!\!\times\!\!\text{ }\left( \text{-}\dfrac{\text{3}}{\text{7}} \right)\text{-}\dfrac{\text{1}}{\text{6}}\text{ }\!\!\times\!\!\text{ }\dfrac{\text{3}}{\text{2}}\text{+}\dfrac{\text{1}}{\text{14}}\text{ }\!\!\times\!\!\text{ }\dfrac{\text{2}}{\text{5}}\]

\[\dfrac{\text{2}}{\text{5}}\text{ }\!\!\times\!\!\text{ }\left( \text{-}\dfrac{\text{3}}{\text{7}} \right)\text{-}\dfrac{\text{1}}{\text{6}}\text{ }\!\!\times\!\!\text{ }\dfrac{\text{3}}{\text{2}}\text{+}\dfrac{\text{1}}{\text{14}}\text{ }\!\!\times\!\!\text{ }\dfrac{\text{2}}{\text{5}}\]

\[\text{=}\dfrac{\text{2}}{\text{5}}\text{ }\!\!\times\!\!\text{ }\left( \text{-}\dfrac{\text{3}}{\text{7}} \right)\text{+}\dfrac{\text{1}}{\text{14}}\text{ }\!\!\times\!\!\text{ }\dfrac{\text{2}}{\text{5}}\text{-}\dfrac{\text{1}}{\text{6}}\text{ }\!\!\times\!\!\text{ }\dfrac{\text{3}}{\text{2}}\]

\[\text{=}\dfrac{\text{2}}{\text{5}}\text{ }\!\!\times\!\!\text{ }\left( \text{-}\dfrac{\text{3}}{\text{7}}\text{+}\dfrac{\text{1}}{\text{14}} \right)\text{-}\dfrac{\text{1}}{\text{4}}\]                          

\[\text{=}\dfrac{\text{2}}{\text{5}}\text{ }\!\!\times\!\!\text{ }\left( \dfrac{\text{-3 }\!\!\times\!\!\text{ 2+1}}{\text{14}} \right)\text{-}\dfrac{\text{1}}{\text{4}}\]

\[\text{=}\dfrac{\text{2}}{\text{5}}\text{ }\!\!\times\!\!\text{ }\left( \dfrac{\text{-5}}{\text{14}} \right)\text{-}\dfrac{\text{1}}{\text{4}}\]

\[\text{=-}\dfrac{\text{1}}{\text{7}}\text{-}\dfrac{\text{1}}{\text{4}}\]

\[\text{=}\dfrac{\text{-4-7}}{\text{28}}\]

  \[\text{=}\dfrac{\text{-11}}{\text{28}}\]

2. Write the additive inverse of each of the following:

i. \[\dfrac{\text{2}}{\text{8}}\]

Ans: Since, additive inverse of $\dfrac{\text{a}}{\text{b}}\text{=}\dfrac{\text{-a}}{\text{b}}$

Therefore, additive inverse of  \[\dfrac{\text{2}}{\text{8}}\]is \[\text{-}\dfrac{\text{2}}{\text{8}}\].

ii. \[\text{-}\dfrac{\text{5}}{\text{9}}\]

Ans: Since, additive inverse of $\text{-}\dfrac{\text{a}}{\text{b}}\text{=}\dfrac{\text{a}}{\text{b}}$

Therefore, additive inverse of  \[\text{-}\dfrac{\text{5}}{\text{9}}\]is \[\dfrac{\text{5}}{\text{9}}\].

iii. \[\dfrac{\text{-6}}{\text{-5}}\]

Ans: Since, additive inverse of $\dfrac{\text{-a}}{\text{-b}}\text{=-}\dfrac{\text{a}}{\text{b}}$

Therefore, additive inverse of \[\dfrac{\text{-6}}{\text{-5}}\]is \[\text{-}\dfrac{\text{6}}{\text{5}}\].

iv. \[\dfrac{\text{2}}{\text{-9}}\]

Ans: Since, additive inverse of $\dfrac{\text{a}}{\text{-b}}\text{=}\dfrac{\text{a}}{\text{b}}$

Therefore, additive inverse of  \[\dfrac{\text{2}}{\text{-9}}\] is  \[\dfrac{\text{2}}{\text{9}}\].

v. \[\dfrac{\text{19}}{\text{-6}}\]

Therefore, additive inverse of  \[\dfrac{\text{19}}{\text{-6}}\] is \[\dfrac{\text{19}}{\text{6}}\].

3. Verify that \[\text{-(-x)=x}\] for.

i. \[\text{x=}\dfrac{\text{11}}{\text{15}}\]

Ans: The additive inverse of  \[\text{x=}\dfrac{\text{11}}{\text{15}}\] is \[\text{-x=-}\dfrac{\text{11}}{\text{15}}\] 

Now,  \[\dfrac{\text{11}}{\text{15}}\text{+}\left( \text{-}\dfrac{\text{11}}{\text{15}} \right)\text{=0}\]

Thus , it represents that the additive inverse of \[\text{-}\dfrac{\text{11}}{\text{15}}\] is \[\dfrac{\text{11}}{\text{15}}\] i.e., \[\text{-}\left( \text{-}\dfrac{\text{11}}{\text{15}} \right)\text{=}\dfrac{\text{11}}{\text{15}}\] .

Hence, \[\text{-(-x)=x}\] holds for \[\text{x=}\dfrac{\text{11}}{\text{15}}\].

ii. \[\text{x=-}\dfrac{\text{13}}{\text{17}}\]

Ans: The additive inverse of \[\operatorname{x}=-\dfrac{13}{17}\] is \[\text{-x=}\dfrac{\text{13}}{\text{17}}\]

Now,  \[\text{-}\dfrac{\text{13}}{\text{17}}\text{+}\dfrac{\text{13}}{\text{17}}\text{=0}\]

Thus , it represents that the additive inverse of \[\dfrac{\text{13}}{\text{17}}\] is \[\text{-}\dfrac{\text{13}}{\text{17}}\] i.e., \[\text{-}\left( \text{-}\dfrac{\text{13}}{\text{17}} \right)\text{=}\dfrac{\text{13}}{\text{17}}\] .

Hence, \[\text{-(-x)=x}\] holds for \[\text{x=-}\dfrac{\text{13}}{\text{17}}\].

4. Find the multiplicative inverse of the following.

i. \[\text{-13}\]

Ans: Since, multiplicative inverse of $\text{-a=-}\dfrac{\text{1}}{\text{a}}$.

Therefore, the multiplicative inverse of  \[\text{-13}\] is \[\text{-}\dfrac{\text{1}}{\text{13}}\].

ii. \[\text{-}\dfrac{\text{13}}{\text{19}}\]

Ans: Since, multiplicative inverse of $\text{-}\dfrac{\text{a}}{\text{b}}\text{=-}\dfrac{\text{b}}{\text{a}}$.

Therefore, multiplicative inverse of  \[\text{-}\dfrac{\text{13}}{\text{19}}\] is \[\text{-}\dfrac{\text{19}}{\text{13}}\].

iii. \[\dfrac{\text{1}}{\text{5}}\]

Ans: Since, multiplicative inverse of $\dfrac{\text{a}}{\text{b}}\text{=}\dfrac{\text{b}}{\text{a}}$.

Therefore, the multiplicative inverse of  \[\dfrac{\text{1}}{\text{5}}\] is \[\text{5}\].

iv. \[\text{-}\dfrac{\text{5}}{\text{8}}\text{ }\!\!\times\!\!\text{ -}\dfrac{\text{3}}{\text{7}}\]

Ans: It can be written as \[\text{-}\dfrac{\text{5}}{\text{8}}\text{ }\!\!\times\!\!\text{ -}\dfrac{\text{3}}{\text{7}}\text{=}\dfrac{\text{15}}{\text{56}}\].

Since, multiplicative inverse of $\dfrac{\text{a}}{\text{b}}\text{=}\dfrac{\text{b}}{\text{a}}$.

Therefore, multiplicative inverse of  \[\text{-}\dfrac{\text{5}}{\text{8}}\text{ }\!\!\times\!\!\text{ -}\dfrac{\text{3}}{\text{7}}\] is \[\dfrac{\text{56}}{\text{15}}\].

v. \[\text{-1 }\!\!\times\!\!\text{ -}\dfrac{\text{2}}{\text{5}}\]

Ans: It can be written as\[\text{-1 }\!\!\times\!\!\text{ -}\dfrac{\text{2}}{\text{5}}\text{=}\dfrac{\text{2}}{\text{5}}\] .

Therefore, multiplicative inverse of  \[\text{-1 }\!\!\times\!\!\text{ -}\dfrac{\text{2}}{\text{5}}\] is \[\dfrac{\text{5}}{\text{2}}\].

vi. \[\text{-1}\]

Therefore, the multiplicative inverse of  \[\text{-1}\] is \[\text{-1}\].

5. Name the property under multiplication used in each of the following:

i. \[\dfrac{\text{-4}}{\text{5}}\text{ }\!\!\times\!\!\text{ 1=1 }\!\!\times\!\!\text{ }\dfrac{\text{-4}}{\text{5}}\text{=}\dfrac{\text{-4}}{\text{5}}\]

Ans: Since, after multiplying by \[\text{1}\] , we are getting the same number.

Therefore, \[\text{1}\] is the multiplicative identity.

ii. \[\text{-}\dfrac{\text{13}}{\text{17}}\text{ }\!\!\times\!\!\text{ }\dfrac{\text{-2}}{\text{7}}\text{=}\dfrac{\text{-2}}{\text{7}}\text{ }\!\!\times\!\!\text{ }\dfrac{\text{-13}}{\text{17}}\]

Ans: Since, $\text{a }\!\!\times\!\!\text{ b=b }\!\!\times\!\!\text{ a}$.

Therefore, its Commutative property.

iii. \[\text{-}\dfrac{\text{19}}{\text{29}}\text{ }\!\!\times\!\!\text{ -}\dfrac{\text{29}}{\text{19}}\text{=1}\]

Ans: Since, \[\text{-a }\!\!\times\!\!{\dfrac{1}{-a}}\text{=1}\].

Therefore, the property is Multiplicative inverse.

6. Multiply \[\dfrac{\text{6}}{\text{13}}\] by the reciprocal of \[\text{-}\dfrac{\text{7}}{\text{16}}\]

Ans: Reciprocal of \[\text{-}\dfrac{\text{7}}{\text{16}}\]  is \[\text{-}\dfrac{\text{16}}{\text{7}}\] .

Thus, \[\dfrac{\text{6}}{\text{13}}\text{ }\!\!\times\!\!\text{ -}\dfrac{\text{16}}{\text{7}}\]

 \[\text{=-}\dfrac{\text{96}}{\text{91}}\] .

7. Tell what property allows you to compute \[\dfrac{\text{1}}{\text{3}}\text{ }\!\!\times\!\!\text{ }\left( \text{6 }\!\!\times\!\!\text{ }\dfrac{\text{4}}{\text{3}} \right)\] as \[\left( \dfrac{\text{1}}{\text{3}}\text{ }\!\!\times\!\!\text{ 6} \right)\text{ }\!\!\times\!\!\text{ }\dfrac{\text{4}}{\text{3}}\] .

Ans: Since, $\text{a }\!\!\times\!\!\text{ }\left( \text{b }\!\!\times\!\!\text{ c} \right)\text{=}\left( \text{a }\!\!\times\!\!\text{ b} \right)\text{ }\!\!\times\!\!\text{ c}$ .

Therefore, its associative property.

8. Is \[\dfrac{\text{8}}{\text{9}}\] the multiplicative inverse of \[\text{-1}\dfrac{\text{1}}{\text{8}}\]? Why or why not?

Ans: We know, If it is the multiplicative inverse, then the product should be \[\text{1}\].

\[\dfrac{\text{8}}{\text{9}}\text{ }\!\!\times\!\!\text{ }\left( \text{-1}\dfrac{\text{1}}{\text{8}} \right)\text{=}\dfrac{\text{8}}{\text{9}}\text{ }\!\!\times\!\!\text{ }\left( \text{-}\dfrac{\text{9}}{\text{8}} \right)\]

\[\text{=-1}\] 

Since, the product is not \[\text{1}\] .

Therefore, \[\dfrac{\text{8}}{\text{9}}\] is not the multiplicative inverse of \[\text{-1}\dfrac{\text{1}}{\text{8}}\] .

9. Is \[\text{0}\text{.3}\] the multiplicative inverse of \[\text{3}\dfrac{\text{1}}{\text{3}}\]? Why or why not?

\[\text{0}\text{.3 }\!\!\times\!\!\text{ 3}\dfrac{\text{1}}{\text{3}}\text{=0}\text{.3 }\!\!\times\!\!\text{ }\dfrac{\text{10}}{\text{3}}\]

\[\text{=}\dfrac{\text{3}}{\text{10}}\text{ }\!\!\times\!\!\text{ }\dfrac{\text{10}}{\text{3}}\]

\[\text{=1}\]

Since, the product is \[\text{1}\] .

Therefore, \[\text{0}\text{.3}\] is the multiplicative inverse of \[\text{3}\dfrac{\text{1}}{\text{3}}\].

i. The rational number that does not have a reciprocal.

Ans: \[\text{0}\] is a rational number but its reciprocal is not defined.

ii. The rational numbers that are equal to their reciprocals.

Ans: \[\text{1}\] and \[\text{-1}\] are the rational numbers that are equal to their reciprocals.

iii. The rational number that is equal to its negative.

Ans: \[\text{0}\] is the rational number that is equal to its negative.

11. Fill in the blanks.

i. Zero has __________ reciprocal.

ii. The numbers __________ and __________ are their own reciprocals.

Ans: \[\text{1,-1}\]

iii. The reciprocal of \[\text{-5}\] is __________.

Ans: \[\text{-}\dfrac{\text{1}}{\text{5}}\]

iv. Reciprocal of \[\dfrac{\text{1}}{\text{x}}\], where \[\text{x}\ne \text{0}\] is __________.

Ans: \[\text{x}\]

v. The product of two rational numbers is always a __________.

Ans: Rational number

vi. The reciprocal of a positive rational number is __________.

Ans: Positive rational number

Refer to page 9 - 14 for exercise 1.1 in the PDF

Exercise 1.2

1. Represent these numbers on the number line.

i. $\dfrac{7}{4}$

Ans: Since, $\dfrac{7}{4}$ is approximately equal to $1.75$.

Therefore, $\dfrac{7}{4}$ can be represented on the number line as follows:

ii. $-\dfrac{5}{6}$

Ans: Since, $-\dfrac{5}{6}$ is approximately equal to $-0.833$.

Thus, $-\dfrac{5}{6}$ can be represented on the number line as follows:

2. Represent \[\text{-}\dfrac{\text{2}}{\text{11}}\text{,-}\dfrac{\text{5}}{\text{11}}\text{,-}\dfrac{\text{9}}{\text{11}}\] on the number line.

Ans: We can divide interval between $\text{0}$ and $\text{-1}$ in $\text{11}$ parts to get \[\text{-}\dfrac{\text{2}}{\text{11}}\text{,-}\dfrac{\text{5}}{\text{11}}\text{,-}\dfrac{\text{9}}{\text{11}}\] on number line.

Thus, \[\text{-}\dfrac{\text{2}}{\text{11}}\text{,-}\dfrac{\text{5}}{\text{11}}\text{,-}\dfrac{\text{9}}{\text{11}}\] can be represented on the number line as follows:

3. Write five rational numbers which are smaller than \[\text{2}\].

Ans: Since, we have to write five rational numbers which are less than \[\text{2}\] .

Therefore, we can multiply and divide \[\text{2}\] by \[7\].

Now, \[\text{2}\] becomes \[\dfrac{\text{14}}{\text{7}}\].

Thus, five rational numbers which are smaller than \[\text{2}\] are given as - 

\[\dfrac{\text{13}}{\text{7}}\text{,}\dfrac{\text{12}}{\text{7}}\text{,}\dfrac{\text{11}}{\text{7}}\text{,}\dfrac{\text{10}}{\text{7}}\text{,}\dfrac{\text{9}}{\text{7}}\].

4. Find ten rational numbers between \[\dfrac{\text{-2}}{\text{5}}\] and \[\dfrac{\text{1}}{\text{2}}\].

Ans: We can make denominator of  \[\dfrac{\text{-2}}{\text{5}}\] and \[\dfrac{\text{1}}{\text{2}}\] same.

Therefore, multiplying and dividing \[\dfrac{\text{-2}}{\text{5}}\] by $\text{4}$ and \[\dfrac{\text{1}}{\text{2}}\] by $\text{10}$.

Thus, now \[\dfrac{\text{-2}}{\text{5}}\] becomes \[\dfrac{\text{-8}}{\text{20}}\]  and  \[\dfrac{\text{1}}{\text{2}}\] becomes \[\dfrac{\text{10}}{\text{20}}\] .

Hence, ten rational numbers between \[\dfrac{\text{-2}}{\text{5}}\] and \[\dfrac{\text{1}}{\text{2}}\] are-

\[\text{-}\dfrac{\text{7}}{\text{20}}\text{,-}\dfrac{\text{6}}{\text{20}}\text{,-}\dfrac{\text{5}}{\text{20}}\text{,-}\dfrac{\text{4}}{\text{20}}\text{,-}\dfrac{\text{3}}{\text{20}}\text{,-}\dfrac{\text{2}}{\text{20}}\text{,-}\dfrac{\text{1}}{\text{20}}\text{,0,}\dfrac{\text{1}}{\text{20}}\text{,}\dfrac{\text{2}}{\text{20}}\].

5. Find five rational numbers between-

i. \[\dfrac{\text{2}}{\text{3}}\] and \[\dfrac{\text{4}}{\text{5}}\]

Ans: We can make denominator of  \[\dfrac{\text{2}}{\text{3}}\] and \[\dfrac{\text{4}}{\text{5}}\] same.

Therefore, multiplying and dividing \[\dfrac{\text{2}}{\text{3}}\] by $\text{15}$ and \[\dfrac{\text{4}}{\text{5}}\] by $9$.

Thus, now \[\dfrac{\text{2}}{\text{3}}\] becomes \[\dfrac{\text{30}}{\text{45}}\]  and  \[\dfrac{\text{4}}{\text{5}}\] becomes \[\dfrac{\text{36}}{\text{45}}\] .

Hence, ten rational numbers between \[\dfrac{\text{2}}{\text{3}}\] and \[\dfrac{\text{4}}{\text{5}}\] are-

\[\dfrac{\text{31}}{\text{45}}\text{,}\dfrac{\text{32}}{\text{45}}\text{,}\dfrac{\text{33}}{\text{45}}\text{,}\dfrac{\text{34}}{\text{45}}\text{,}\dfrac{\text{35}}{\text{45}}\].

ii. \[\dfrac{\text{-3}}{\text{2}}\] and \[\dfrac{\text{5}}{\text{3}}\].

Ans: We can make denominator of  \[\text{-}\dfrac{\text{3}}{\text{2}}\] and \[\dfrac{\text{5}}{\text{3}}\] same.

Therefore, multiplying and dividing \[\text{-}\dfrac{\text{3}}{\text{2}}\] by $3$ and \[\dfrac{\text{5}}{\text{3}}\] by $2$.

Thus, now \[\text{-}\dfrac{\text{3}}{\text{2}}\] becomes \[\text{-}\dfrac{\text{9}}{\text{6}}\]  and  \[\dfrac{\text{5}}{\text{3}}\] becomes \[\dfrac{\text{10}}{\text{6}}\] .

Hence, ten rational numbers between \[\text{-}\dfrac{\text{3}}{\text{2}}\] and \[\dfrac{\text{5}}{\text{3}}\] are-

\[\text{-}\dfrac{\text{8}}{\text{6}}\text{,-}\dfrac{\text{7}}{\text{6}}\text{,-1,-}\dfrac{\text{5}}{\text{6}}\text{,-}\dfrac{\text{4}}{\text{6}}\].

iii. \[\dfrac{\text{1}}{\text{4}}\] and \[\dfrac{\text{1}}{\text{2}}\]

Ans: We can make denominator of  \[\dfrac{\text{1}}{\text{4}}\] and \[\dfrac{\text{1}}{\text{2}}\] same.

Therefore, multiplying and dividing \[\dfrac{\text{1}}{\text{4}}\] by $8$ and \[\dfrac{\text{1}}{\text{2}}\] by $16$.

Thus, now \[\dfrac{\text{1}}{\text{4}}\] becomes \[\dfrac{\text{8}}{\text{32}}\]  and  \[\dfrac{\text{1}}{\text{2}}\] becomes \[\dfrac{\text{16}}{\text{32}}\] .

Hence, ten rational numbers between \[\dfrac{\text{1}}{\text{4}}\] and \[\dfrac{\text{1}}{\text{2}}\] are-

\[\dfrac{\text{9}}{\text{32}}\text{,}\dfrac{\text{10}}{\text{32}}\text{,}\dfrac{\text{11}}{\text{32}}\text{,}\dfrac{\text{12}}{\text{32}}\text{,}\dfrac{\text{13}}{\text{32}}\].

6. Write five rational numbers greater than \[\text{-2}\].

Ans: Since, we have to write five rational numbers which are greater than \[\text{-2}\].

Therefore, we can multiply and divide \[\text{-2}\] by \[7\].

Now, \[\text{-2}\] becomes \[\text{-}\dfrac{\text{14}}{\text{7}}\].

Thus, five rational numbers greater than \[\text{-2}\] are given as-

\[\text{-}\dfrac{\text{13}}{\text{7}}\text{,-}\dfrac{\text{12}}{\text{7}}\text{,-}\dfrac{\text{11}}{\text{7}}\text{,-}\dfrac{\text{10}}{\text{7}}\text{,-}\dfrac{\text{9}}{\text{7}}\].

7. Find ten rational numbers between \[\dfrac{\text{3}}{\text{5}}\] and \[\dfrac{\text{3}}{\text{4}}\].

Ans: We can make denominator of  \[\dfrac{\text{3}}{\text{5}}\] and \[\dfrac{\text{3}}{\text{4}}\] same.

Therefore, multiplying and dividing \[\dfrac{\text{3}}{\text{5}}\] by $16$ and \[\dfrac{\text{3}}{\text{4}}\] by $20$.

Thus, now \[\dfrac{\text{3}}{\text{5}}\] becomes \[\dfrac{\text{48}}{\text{80}}\]  and  \[\dfrac{\text{3}}{\text{4}}\] becomes \[\dfrac{\text{60}}{\text{80}}\] .

Hence, ten rational numbers between \[\dfrac{\text{3}}{\text{5}}\] and \[\dfrac{\text{3}}{\text{4}}\] are-

\[\dfrac{\text{49}}{\text{80}}\text{,}\dfrac{\text{50}}{\text{80}}\text{,}\dfrac{\text{51}}{\text{80}}\text{,}\dfrac{\text{52}}{\text{80}}\text{,}\dfrac{\text{53}}{\text{80}}\text{,}\dfrac{\text{54}}{\text{80}}\text{,}\dfrac{\text{55}}{\text{80}}\text{,}\dfrac{\text{56}}{\text{80}}\text{,}\dfrac{\text{57}}{\text{80}}\text{,}\dfrac{\text{58}}{\text{80}}\].

NCERT Solutions For Class 8 Maths Chapter 1 Rational Numbers - PDF Download

Points to remember to solve chapter 1 of class 8 ncert.

Rational Number: Any number that can be expressed in the form of p/q, where p and q are integers and q ≠ 0, is known as a rational number. The collection or group of rational numbers is denoted by Q.

Properties of a Rational Number

Example: Let p and q be any two rational numbers. Then their sum, difference and product will also be a rational number. This is known as the Closure property.

Commutativity: Rational numbers will be commutative under addition and multiplication. 

Let p and q be any two rational numbers, then

Commutative law under addition is p + q = q + p.

Commutative law under multiplication is p x q = q x p.

(Note: Rational numbers, integers and whole numbers are commutative under addition and multiplication. Also, they are non-commutative under subtraction and division.)

Associativity: Rational numbers will be associative under addition and multiplication. 

Let p, q and r be the rational numbers, then

Associative property under addition is: p + (q + r) = (p + q) + r

Associative property under multiplication is: p(qr) = (pq)r

Role of Zero and One: 0 will be the additive identity for rational numbers. 1 will be the multiplicative identity for the rational numbers.

Multiplicative Inverse: When the product of two rational numbers is 1, then they are called as the multiplicative inverse of each other.

We Cover All The Exercises in Chapter Given Below:

Exercise 1.1 - 11 Questions with Solutions.

Exercise 1.2 - 7 Questions with Solutions.

NCERT Solutions for Class 8 Maths - Chapterwise Solutions

Chapter 2 - Linear Equations in One Variable

Chapter 3 - Understanding Quadrilaterals

Chapter 4 - Practical Geometry

Chapter 5 - Data Handling

Chapter 6 - Squares and Square Roots

Chapter 7 - Cubes and Cube Roots

Chapter 8 - Comparing Quantities

Chapter 9 - Algebraic Expressions and Identities

Chapter 10 - Visualising Solid Shapes

Chapter 11 - Mensuration

Chapter 12 - Exponents and Powers

Chapter 13 - Direct and Inverse Proportions

Chapter 14 - Factorisation

Chapter 15 - Introduction to Graphs

Chapter 16 - Playing with Numbers

Along with this, students can also download additional study materials provided by Vedantu, for Chapter 1 of CBSE Class 8 Maths Solutions –

Chapter 1: Important Questions

Chapter 1: Important Formulas

Chapter 1: Revision Notes

Chapter 1: NCERT Exemplar Solutions

Chapter 1: RD Sharma Solutions

Benefits of NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers

Vedantu’s NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers are curated and prepared by our best teachers to cater to students’ need for concise and accurate maths solutions to all the questions given in the NCERT textbook for this chapter. Students going through these solutions will have a higher chance of scoring well in their Class 8 Maths exams.

Quick Revision

The following are the key points to remember while studying the chapter on Rational Numbers.

Rational numbers are enclosed within addition, subtraction, and multiplication operations.

The operations related to addition and multiplication are:

Commutative property of rational numbers

Associative property of rational numbers

The rational number 0 is the additive identity for all rational numbers.

The rational number 1 is the multiplicative identity for all rational numbers.

For all rational numbers a, b, and c, a (b + c) = ab + ac and a (b – c) = ab – ac.

A number line can be used to represent rational numbers.

There can be countless rational numbers between any two given rational numbers. Mean aids in determining rational numbers between two rational numbers.

List of Formulas

The following is a list of important formulas that students need to keep in mind while studying the chapter on Rational Numbers. 

$ Q = {\dfrac{p}{q} : p, q \epsilon Z;  q \neq 0} $

$ \dfrac{x}{y} \pm \dfrac{m}{n} = \dfrac{xn \pm ym}{yn} $

$ \dfrac{x}{y} \times \dfrac{m}{n} = \dfrac{xm}{yn} $

$ \dfrac{x}{y} \div \dfrac{m}{n} = \dfrac{xn}{ym} $

Did You Know?

Pythagoras was confident that using whole numbers of a small enough unit would measure everything. That means any two measurements or lengths will have a rational ratio. Through Pythagoras Theorem, they discovered that a rational number could not represent a unit square’s diagonal measurement. They had to come up with new solutions that can avoid showing their false assumptions and complied with facts. This historical event is a first of its kind which gives strength to the fact that mathematicians always look for evidence. You will never be able to understand this without learning Rational Numbers!

Conclusion:

NCERT Solutions for Class 8 Maths Rational Numbers offer comprehensive guidance for students to master this important mathematical concept. With clear explanations, practice exercises, and problem-solving techniques, these solutions facilitate a deeper understanding of rational numbers. By utilizing these solutions, students can build a solid foundation in mathematics and excel in their academic endeavors.

FAQs on NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers

1. What is the importance of learning the Class 8 Maths Chapter 1 Rational Number?

Numbers are the building block of mathematics. In lower classes, the students main focus is in teaching him about the different types of numbers that include - natural numbers, whole numbers, integers etc. Chapter 1 of Class 8 is designed to teach students another set of numbers, namely - the rational numbers. “A number which can be written in the form p/q, where p and q are integers and q ≠ 0 is called a rational number”. This chapter explains in delta about all the concepts that a student of Class 8 needs to learn about the rational numbers. Along with these, Chapter 1 of Class 8 also explains to the students the method of representing a rational number on a number line as well as the method of finding a rational number between any 2 rational numbers.

2. How can you identify a rational number?

A rational number is a number that can be written in the form of a ratio. This implies that it can be written as a fraction. A fraction in which both the numerator (the number on top), as well as the denominator (the number on the bottom), are whole numbers. For better understanding here are a few examples: 

The number 14  is a rational number. This is because it can be written as the fraction - 14/1.

Likewise, 13/24 is a rational number as it is already written as a fraction.

Even a large fraction like 3478987/784362 is rational, only because it can be written as a fraction.

Even decimals such as 23.4 is a rational number as it can be represented as a fraction - 234/10

3. What are the best study materials for scoring well in maths?

Irrespective of how well they are prepared, Maths is a nightmare for most of the subjects. This mainly because maths is an application-oriented subject which cannot be mastered overnight. It needs practice and hard work to excel in Maths. Along with it, students need the right mindset in order to be able to tackle the subject successfully in the exam. Following are the few exam study materials which when incorporated into the study process makes it easy for students to  score well in the exams : 

Previous years question papers with solutions.

Mock papers with solutions 

NCERT Solutions for Class 8 Maths by Vedantu

Sample papers for Class 8 Maths

4. What are the subtopics in Mathematics Chapter 1 Rational Numbers Class 8?

The topics in Chapter 1- Rational Numbers are as follows.

Topic 1: Introduction of Rational numbers.

Topic 2: Properties of rational numbers.

Topic 3: Representation of Rational Numbers on a number line.

Topic 4: Rational Numbers between the two Rational Numbers.

You can download Vedantu’s app to access the study material related to this chapter. All the resources are free of cost. 

5. How can NCERT Solutions help in the preparation of the chapter 1 Maths of Class 8?

NCERT Solutions Class 8 Mathematics Chapter 1 Rational Numbers are the best for the preparation. Each step is explained in a detailed manner. The chapter is basic and very important. Students should have a thorough understanding of the concepts and topics in the chapter. If the questions given in the NCERT are practised, then one can excel in the chapter. 

6. What is the importance of rational numbers?

Rational numbers are the basic and important part of the curriculum of Mathematics, which are to be learned properly. The further chapters are related to Rational numbers too. If a student excels in this chapter, it becomes easier for him to understand the other chapters which involve the topics of rational numbers. A rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p, and a non-zero denominator q. 

7. What are the properties of rational numbers?

The properties of rational numbers are given below.

Closure property.

Commutative property.

Associative Property.

Distributive Property.

Identity Property.

Inverse Property.

The above properties are the six important properties of rational numbers. Use the official website of Vedantu to access the study material related to Chapter 1, Rational Numbers.

8. What is the additive inverse property of rational numbers?

The opposite, or additive inverse, of a number, is the same distance from zero on a number line as the original number but on the other side of zero. Zero is its own additive inverse. In other words, the additive inverse of a rational number is the same number with the opposite sign. There are many problems connected to this chapter. Practising those problems will help the students understand the concept of rational numbers and their properties.

NCERT Solutions for Class 8 Maths

Ncert solutions for class 8.

NCERT Solutions for Class 6, 7, 8, 9, 10, 11 and 12

Rational Numbers Class 8 Extra Questions Maths Chapter 1

October 28, 2020 by Veerendra

If you are looking for Extra questions for class 8 maths Rational Numbers , You have reached the correct page. You can also use these extra questions like Rational numbers class 8 worksheets with answers.

Extra Questions for Class 8 Maths Chapter 1 Rational Numbers

Rational Numbers Class 8 Extra Questions Very Short Answer Type

Question 1. Pick up the rational numbers from the following numbers. \(\frac { 6 }{ 7 }\), \(\frac { -1 }{ 2 }\), 0, \(\frac { 1 }{ 0 }\), \(\frac { 100 }{ 0 }\) Solution: Since rational numbers are in the form of \(\frac { a }{ b }\) where b ≠ 0. Only \(\frac { 6 }{ 7 }\), \(\frac { -1 }{ 2 }\) and 0 are the rational numbers.

Question 2. Find the reciprocal of the following rational numbers: (a) \(\frac { -3 }{ 4 }\) (b) 0 (c) \(\frac { 6 }{ 11 }\) (d) \(\frac { 5 }{ -9 }\) Solution: (a) Reciprocal of \(\frac { -3 }{ 4 }\) is \(\frac { -4 }{ 3 }\) (b) Reciprocal of 0, i.e. \(\frac { 1 }{ 0 }\) is not defined. (c) Reciprocal of \(\frac { 6 }{ 11 }\) is \(\frac { 11 }{ 6 }\) (d) Reciprocal of \(\frac { 5 }{ -9 }\) = \(\frac { -9 }{ 5 }\)

Question 3. Write two such rational numbers whose multiplicative inverse is same as they are. Solution: Reciprocal of 1 = \(\frac { 1 }{ 1 }\) = 1 Reciprocal of -1 = \(\frac { 1 }{ -1 }\) = -1 Hence, the required rational numbers are -1 and 1.

Question 4. What properties, the following expressions show? (i) \(\frac { 2 }{ 3 } +\frac { 4 }{ 5 } =\frac { 4 }{ 5 } +\frac { 2 }{ 3 }\) (ii) \(\frac { 1 }{ 3 } \times \frac { 2 }{ 3 } =\frac { 2 }{ 3 } \times \frac { 1 }{ 3 }\) Solution: (i) \(\frac { 2 }{ 3 } +\frac { 4 }{ 5 } =\frac { 4 }{ 5 } +\frac { 2 }{ 3 }\) shows the commutative property of addition of rational numbers. (ii) \(\frac { 1 }{ 3 } \times \frac { 2 }{ 3 } =\frac { 2 }{ 3 } \times \frac { 1 }{ 3 }\) shows the commutative property of multiplication of rational numbers.

Question 5. What is the multiplicative identity of rational numbers? Solution: 1 is the multiplicating identity of rational numbers.

Question 6. What is the additive identity of rational numbers? Solution: 0 is the additive identity of rational numbers.

Question 12. Identify the rational number which is different from the other three : \(\frac { 2 }{ 3 }\), \(\frac { -4 }{ 5 }\), \(\frac { 1 }{ 2 }\), \(\frac { 1 }{ 3 }\). Explain your reasoning. Solution: \(\frac { -4 }{ 5 }\) is the rational number which is different from the other three, as it lies on the left side of zero while others lie on the right side of zero on the number line.

Rational Numbers Class 8 Extra Questions Short Answer Type

Rational Numbers Class 8 Extra Questions Higher Order Thinking Skills (HOTS)

Question 23. One-third of a group of people are men. If the number of women is 200 more than the men, find the total number of people. Solution: Number of men in the group = \(\frac { 1 }{ 3 }\) of the group Number of women = 1 – \(\frac { 1 }{ 3 }\) = \(\frac { 2 }{ 3 }\) Difference between the number of men and women = \(\frac { 2 }{ 3 }\) – \(\frac { 1 }{ 3 }\) = \(\frac { 1 }{ 3 }\) If difference is \(\frac { 1 }{ 3 }\), then total number of people = 1 If difference is 200, then total number of people = 200 ÷ \(\frac { 1 }{ 3 }\) = 200 × 3 = 600 Hence, the total number of people = 600

Extra Questions for Class 8 Maths

Ncert solutions for class 8 maths, free resources.

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Rational Numbers Worksheet Class 8 PDF with Answers

These Rational Numbers worksheet PDF can be helpful for both teachers and students. Teachers can track their student’s performance in the chapter Rational Numbers. Students can easily identify their strong points and weak points by solving questions from the worksheet. Accordingly, students can work on both weak points and strong points. 

All students studying in CBSE class 8th, need to practise a lot of questions for the chapter Rational Numbers. Students can easily practise questions from the Rational Numbers problems worksheet PDF. By practising a lot of questions, students can improve their confidence level. With the help of confidence level, students can easily cover all the concepts included in the chapter Rational Numbers. 

Rational Numbers Worksheets with Solutions

Solutions is the written reply for all questions included in the worksheet. With the help of Rational Numbers worksheets with solutions, students can solve all doubts regarding questions. Students can have deep learning in the chapter Rational Numbers by solving all their doubts. By solving doubts, students can also score well in the chapter Rational Numbers. 

Rational Numbers Worksheet PDF

Worksheet is a sheet which includes many questions to solve for class 8th students. The Rational Numbers worksheet PDF provides an opportunity for students to enhance their learning skills. Through these skills, students can easily score well in the chapter Rational Numbers. Students can solve the portable document format (PDF) of the worksheet from their own comfort zone. 

How to Download the Rational Numbers Worksheet PDF? 

To solve questions from the Rational Numbers worksheet PDF, students can easily go through the given steps. Those steps are- 

• Open Selfstudys website.

• Bring the arrow towards CBSE which can be seen in the navigation bar. 

• Drop down menu will appear, select KVS NCERT CBSE Worksheet. 

• A new page will appear, select class 8th from the given list of classes. 
• Select Mathematics from the given list of subjects. Now click the chapter’s name that is Rational Numbers. 

Features of the Rational Numbers Worksheet PDF

Before starting to solve questions from the Rational Numbers problems worksheet PDF, students need to know everything about the worksheet. Those features are- 

• Variety of questions are included:  The Rational Numbers Maths Worksheet for Class 8 includes varieties of questions. Those varieties of questions are- one mark questions, two mark questions, three mark questions, etc. 
• Solutions are provided:  Doubts regarding each question can be easily solved through the solutions given. Through solving questions, a student's comprehensive skill can be increased. 
• All concepts are covered:  By solving questions from the Rational Numbers problems worksheet, students can easily cover all the concepts included in the chapter.  
• Created by Expert:  These worksheets are personally created by the subject experts. These Rational Numbers worksheet pdf are created with proper research. 
• Provides plenty of questions:  The Rational Numbers worksheet provides plenty of questions to practise. Through good practice, students can get engaged in the learning process. 

Benefits of the Rational Numbers Worksheet PDF

With the help of Rational Numbers problems worksheet PDF, students can easily track their performance. This is the most crucial benefit, other than this there are more benefits. Those benefits are- 

• Builds a strong foundation:  Regular solving questions from the worksheet can help students to build a strong foundation. Through the strong foundation, students can score well in the chapter Rational Numbers. 
• Improves speed and accuracy:  While solving questions from the chapter Rational Numbers, students need to maintain the speed and accuracy. Speed and accuracy can be easily maintained and improved by solving questions from the Rational Numbers worksheet PDF. 
• Acts as a guide:  Rational Numbers worksheets with solutions acts as guide for both the teachers and students. Through the worksheet, teachers can guide their students according to the answers given by them. Students can also analyse themselves with the help of answers and can improve accordingly. 
• Enhances the learning process:  Regular solving of questions from the worksheet can help students enhance their learning process. According to the learning skills, students can easily understand all topics and concepts included in the chapter Rational Numbers.  
• Improvisation of grades:  Regular solving of questions from the worksheet can help students to improve their marks and grades. With the help of good marks and good grades, students can select their desired field further. 

Tips to Score Good Marks in Rational Numbers Worksheet

Students are requested to follow some tips to score good marks in the Rational Numbers worksheet. Those tips are-

• Complete all the concepts:  First and the most crucial step is to understand all the concepts included in the chapter Rational Numbers.  
• Practise questions:  Next step is to practise questions from the Rational Numbers problems worksheet. Through this students can identify all types of questions: easy, moderate, difficult, etc.  
• Note down the mistakes:  After practising questions, students need to note down the wrong sums that have been done earlier. 
• Rectify the mistakes:  After noting down the mistakes, students need to rectify all the mistakes made. 
• Maintain a positive attitude:  Students are requested to maintain a positive attitude while solving worksheets. By maintaining a positive attitude, students can improve speed and accuracy while solving the worksheets. 
• Remain focused:  Students need to remain focused while solving questions from the Rational Numbers problems worksheet pdf. As it helps students to solve the questions as fast as possible. 

When should a student start solving the Rational Numbers Worksheet PDF?

Students studying in class 8 should start solving worksheets after covering each and every concept included in the chapter. Regular solving questions from the Rational Numbers worksheet PDF, can help students to have a better understanding of the chapter. Better understanding of the chapter Rational Numbers can help students to score well in the class 8th board exam. 

Regular solving questions from the Rational Numbers Worksheet PDF can help students to build a strong foundation for the chapter Rational Numbers. Strong foundation of the chapter Rational Numbers can help students to understand further chapters. 

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