# NCERT Solutions Class 8 Maths Chapter 1

## NCERT Solutions for Class 8 Mathematics Chapter 1 – Rational Numbers

Mathematics is a subject of logical and analytical thinking. As a result, students must have the proper knowledge and understanding of Mathematics. Extramarks NCERT Solutions for Class 8 Mathematics Chapter 1 has a detailed explanation for the entire chapter.

The chapter covers topics like introduction to Whole numbers, Natural numbers, Integers, and Rational numbers. Students also learn the different properties of Rational numbers, ways to find rational numbers between two numbers represented on a number line, and negatives and reciprocals of a rational number.

Our NCERT Solutions for Class 8 Mathematics Chapter 1 is specially designed, keeping in mind all the core concepts of this chapter. It has highlighted points from the NCERT textbook, making it easier for the students to recall the concepts whenever required. It is based on the latest CBSE syllabus 2021-2022,  so that students don’t lack the fundamental aspects of  Mathematics while solving problems.

Extramarks’ website is a reliable source for NCERT Solutions, NCERT study materials, revision notes and mock tests. Students are requested to regularly visit Extramarks’ website in order to excel in their tests and examinations.

## Key Topics Covered In NCERT Solutions for Class 8 Mathematics Chapter 1

NCERT Class 8 Mathematics Chapter 1 has all the basic concepts which will be required to cover other chapters of Class 8 Mathematics. Hence, students are requested to study this chapter with proper understanding. The core concepts like representing a number line and properties of the rational number will be useful for them in Class 9 and Class 10 Mathematics. By thoroughly covering this chapter you will strengthen the foundation of your Class 8 Mathematics which will be required in higher classes as well.

You can find complete details about this chapter in our NCERT Solutions for Class 8 Mathematics Chapter 1.  and you will be able to understand every concept and answer any question easily. This encourages the students to master the topic and increases their confidence in achieving a higher grade.

This chapter covers the following topics:

 Topic 1.1 Introduction 1.2 Properties of Rational Numbers 1.3 Representation of Rational Numbers on the Number Line 1.4 Rational Numbers between Two Rational Numbers 1.5 Summary

This chapter provides a great understanding of Solid Shapes and establishes a strong foundation in various Mathematical applications.

### 1.1 Introduction

The chapter begins with the introduction of different sets of numbers like Whole numbers, Natural numbers, Integers and Rational numbers, which forms the base for Mathematics in higher classes. .

To know these categories of numbers properly, one should refer to NCERT Solutions for Class 8 Mathematics Chapter 1  so, try to clear your r concepts and don’t memorize.

### 1.2 Properties of Rational Numbers

Properties of operations on different types of numbers always play a great role in Mathematics. As a result, students should have a thorough knowledge of all the properties and its application to make their calculations strong.

This section covers some properties of rational numbers like Closure Property, Commutative Property, Associative Property, and Distributive Property of multiplication over addition for rational numbers. These properties will be of great help in other algebra-based chapters too. It also covers the role of both zero and one, as well as the negatives and reciprocals of a number.

### 1.3 Representation of Rational Numbers on Number Line

Students have been learning ways to represent a number line from their lower grades. It has always been a challenging task for students. Hence, to get better in this section, students must regularly practice and refer to NCERT Solutions for Class 8 Mathematics Chapter 1.

This section particularly explains the representation of rational numbers on the number line.. The various steps, if carefully followed by the students, can never stop them from getting the best possible score in this section.

### 1.4 Rational Numbers Between Two Rational Numbers

There are hundreds of rational numbers between two rational numbers. So, it becomes quite challenging to find rational numbers between two rational numbers.

For example: Between 1 and 2, you will find rational numbers like ⅔,¾,⅘ and so on.

But if students know proper tips and tricks to find rational numbers between two rational numbers, they will be able to solve such types of problems easily. The different illustrations and problems given in our NCERT Solutions for Class 8 Mathematics Chapter 1 will also help students in learning and mastering the concept rather than simply memorizing it..

### 1.5 Summary

To sum up, this chapter focuses on the algebraic part of Mathematics. The core concepts like negative and reciprocals representing a rational number on a number line are important in this chapter.  Students should master these concepts before they move on to the next chapter.

## NCERT Solutions for Class 8 Mathematics Chapter 1 Exercise &  Solutions

NCERT Solutions for Class 8 Mathematics Chapter 1 Rational numbers are available on the Extramarks’ website. You can find complete theory and step-by-step solutions to all numerical problems from the NCERT textbook. Students should leverage the solution guide to get through the chapter for the Class 8 CBSE examinations.

The solution guide on the Extramarks’ website for the Class 8 Mathematics Chapter 1 covers topics on properties of rational numbers, representation of rational numbers and their various applications. Subject experts advise students to revise this chapter multiple times as it’s a core topic for the Class 8 Mathematics syllabus. The topics covered in this chapter will go a long way in preparing a strong foundation for NCERT Class 9 and NCERT Class 10 Mathematics syllabi.

Click on the  links given below to view exercise-specific questions and solutions for NCERT Solutions for Class 8 Mathematics Chapter 1:

By choosing Extramarks, students can enjoy a smooth and complete learning experience. NCERT Solutions for Class 8 Mathematics Chapter 1 is written in easy-to-understand language designed by subject matter experts after extensive research and following the CBSE guidelines.   Students may even refer to  CBSE past year question papers and stay ahead of the competition.

Along with Class 8 Mathematics Solutions, you can explore NCERT Solutions on our Extramarks website for all primary and secondary classes.

## NCERT Exemplar Class 8  Mathematics

NCERT Exemplar Class 8 Mathematics is a complete source of information for CBSE students preparing for their grade 8 standard exams. It contains various examples and solutions, and question answers to leverage their performance. While revising from NCERT Exemplar, students gather complete knowledge of the Mathematical concepts and gain deeper insights on a variety of interlinked topics between various chapters.

Exemplar books have not only proven beneficial for CBSE students but also for other boards as well. . It covers complex theories in such an easy-to-understand manner so that students can confidently face the upcoming examinations. Students appearing for Class 8 are advised to refer to the NCERT Exemplar Class 8 Mathematics and incorporate it into their core study material. Students will understand how to solve complex problems which will help them to prepare for boards and other entrance examinations.

After referring to the NCERT Solutions and NCERT Exemplar, the students can think logically to solve a problem. As a result, students can easily switch to more advanced and higher-level conceptual questions.

### Key Features of NCERT Solutions for Class 8 Mathematics Chapter 1

It is necessary that you have authentic and reliable resources to boost your performance and beat your previous records.  Therefore, NCERT Solutions for Class 8 Mathematics Chapter 1 offers a complete solution for all problems. The key features  are:

• Extramarks NCERT Solutions for Class 8 Mathematics Chapter 1 has all the highlighted conceptual topics to enhance your learning experience and master the topic.
• The core concepts covered in Extramarks NCERT Solutions for Class 8 Mathematics Chapter 1 will help students to solve their examination papers in a more thoughtful and logical manner
• With the rational number chapter solutions, students will be able to apply the concepts  required to solve  any algebraic related problems

Q.1 Using appropriate properties find:

$\begin{array}{l}\left(\mathrm{i}\right)\frac{-2}{3}×\frac{3}{5}+\frac{5}{2}-\frac{3}{5}×\frac{1}{6}\end{array}$ $\begin{array}{l}\left(\mathrm{ii}\right)\frac{2}{5}×\left(-\frac{3}{7}\right)-\frac{1}{6}×\frac{3}{2}+\frac{1}{14}×\frac{2}{5}\end{array}$

Ans-

$\begin{array}{l}\text{(i)}\frac{-2}{3}×\frac{3}{5}+\frac{5}{2}-\frac{3}{5}×\frac{1}{6}\\ =\frac{-2}{3}×\frac{3}{5}-\frac{3}{5}×\frac{1}{6}+\frac{5}{2}\text{}\left[\text{By using Commutative property}\right]\\ =\left(-\frac{3}{5}\right)×\left(\frac{2}{3}+\frac{1}{6}\right)+\frac{5}{2}\text{}\left[\text{By using Distributive property}\right]\\ =\left(-\frac{3}{5}\right)×\left(\frac{4+1}{6}\right)+\frac{5}{2}\end{array}$ $\begin{array}{l}=\left(-\frac{3}{5}\right)×\left(\frac{5}{6}\right)+\frac{5}{2}\\ =\left(-\frac{3}{6}\right)+\frac{5}{2}\\ =\left(\frac{-3+15}{6}\right)=\frac{12}{6}=2\end{array}$ $\begin{array}{l}\text{(ii)}\frac{2}{5}×\left(-\frac{3}{7}\right)-\frac{1}{6}×\frac{3}{2}+\frac{1}{14}×\frac{2}{5}\\ =\frac{2}{5}×\left(-\frac{3}{7}\right)+\frac{1}{14}×\frac{2}{5}-\frac{1}{6}×\frac{3}{2}\text{}\left[\text{By using Commutative property}\right]\\ =\frac{2}{5}×\left(-\frac{3}{7}+\frac{1}{14}\right)-\frac{1}{4}\\ =\frac{2}{5}×\left(\frac{-6+1}{14}\right)-\frac{1}{4}\\ =\frac{2}{5}×\left(\frac{-5}{14}\right)-\frac{1}{4}\\ =\left(\frac{-2}{14}\right)-\frac{1}{4}\\ =\frac{-1}{7}-\frac{1}{4}\\ =\frac{-4-7}{28}=\frac{-11}{28}\end{array}$

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

$\left(\mathrm{i}\right)\frac{2}{8}$ $\left(\mathrm{ii}\right)\frac{-5}{9}$ $\left(\mathrm{iii}\right)\frac{-6}{-5}$ $\left(\mathrm{iv}\right)\frac{2}{-9}$ $\left(\mathrm{v}\right)\frac{19}{-6}$

Ans-

$\begin{array}{l}\left(\text{i}\right)\text{Additive inverse of}\text{}\frac{2}{8}\text{}\text{=}-\frac{2}{8}\text{}\\ \left(\text{ii}\right)\text{Additive inverse of}\frac{-5}{9}\text{=}\frac{5}{9}\\ \left(\text{iii}\right)\text{Additive inverse of}\frac{-6}{-5}\text{}\text{=}\frac{-6}{5}\\ \left(\text{iv}\right)\text{Additive inverse of}\frac{2}{-9}\text{=}\frac{2}{9}\\ \left(\text{v}\right)\text{Additive inverse of}\frac{19}{-6}\text{=}\frac{19}{6}\end{array}$

Q.3 Verify that – (– x) = x for.

$\left(\mathrm{i}\right)\mathrm{x}=\frac{11}{15}\left(\mathrm{ii}\right)\mathrm{x}=-\frac{13}{17}$

Ans-

$\begin{array}{l}\left(\text{i}\right)\text{x=}\frac{11}{15}\\ \text{L.H.S=}-\left(-\mathrm{x}\right)\\ =-\left(-\frac{11}{15}\right)\\ =\frac{11}{15}\left(\text{as two minus becomes plus}\right)\\ =\mathrm{x}\\ =\text{R.H.S}\\ \text{}\therefore \text{L.H.S=R.H.S}\\ \text{Hence proved.}\\ \\ \left(\text{ii}\right)\text{x=}-\frac{13}{17}\\ \text{L.H.S=}-\left(-\mathrm{x}\right)\\ =-\left(-\frac{13}{17}\right)\\ =\frac{13}{17}\text{}\left(\text{as two minus becomes plus}\right)\\ =\mathrm{x}\\ =\text{R.H.S}\\ \text{}\therefore \text{L.H.S=R.H.S}\\ \text{Hence proved.}\end{array}$

Q.4 Find the multiplicative inverse of the following.

$\begin{array}{l}\left(\mathrm{i}\right)-13\left(\mathrm{ii}\right)\frac{-13}{19}\left(\mathrm{iii}\right)\frac{1}{5}\left(\mathrm{iv}\right)\frac{-5}{8}×\frac{-3}{7}\\ \left(\mathrm{v}\right)-1×\frac{-2}{5}\left(\mathrm{vi}\right)-1\end{array}$

Ans-

$\begin{array}{l}\left(\mathrm{i}\right)-13\\ \mathrm{The}\text{}\mathrm{multiplicative}\text{}\mathrm{inverse}\text{}\mathrm{of}-13=-\frac{1}{13}\\ \left(\mathrm{ii}\right)\frac{-13}{19}\\ \mathrm{The}\text{}\mathrm{multiplicative}\text{}\mathrm{inverse}\text{}\mathrm{of}\frac{-13}{19}=\frac{-19}{13}\\ \left(\mathrm{iii}\right)\frac{1}{5}\\ \mathrm{The}\text{}\mathrm{multiplicative}\text{}\mathrm{inverse}\text{}\mathrm{of}\frac{1}{5}=5\\ \left(\mathrm{iv}\right)\frac{-5}{8}×\frac{-3}{7}=\frac{15}{56}\\ \mathrm{The}\text{}\mathrm{multiplicative}\text{}\mathrm{inverse}\text{}\mathrm{of}\frac{15}{56}=\frac{56}{15}\\ \left(\mathrm{v}\right)-1×\frac{-2}{5}=\frac{2}{5}\\ \mathrm{The}\text{}\mathrm{multiplicative}\text{}\mathrm{inverse}\text{}\mathrm{of}\frac{2}{5}=\frac{5}{2}\\ \left(\mathrm{vi}\right)-1\\ \mathrm{The}\text{}\mathrm{multiplicative}\text{}\mathrm{inverse}\text{}\mathrm{of}-1=-1\end{array}$

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

$\begin{array}{l}\left(\mathrm{i}\right)\frac{-4}{5}×1=1×\frac{-4}{5}=\frac{-4}{5}\left(\mathrm{ii}\right)-\frac{13}{17}×\frac{-2}{7}=\frac{-2}{7}×\frac{-13}{17}\\ \left(\mathrm{iii}\right)\frac{-19}{29}×\frac{29}{-19}=1\end{array}$

Ans-

$\begin{array}{l}\left(\mathrm{i}\right)\frac{-4}{5}×1=1×\frac{-4}{5}=\frac{-4}{5}\\ \text{Here, 1 is the multiplicative identity.}\\ \left(\mathrm{ii}\right)-\frac{13}{17}×\frac{-2}{7}=\frac{-2}{7}×\frac{-13}{17}\\ \text{Here, commutative property is used.}\\ \left(\mathrm{iii}\right)\frac{-19}{29}×\frac{29}{-19}=1\\ \text{Here,multiplicative inverse is being used.}\end{array}$

$\mathrm{Q.8 Is}\frac{8}{9}\mathrm{the}\mathrm{multiplicative}\mathrm{inverse}\mathrm{of}-1\frac{1}{8}?\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{Why}\mathrm{or}\mathrm{why}\mathrm{not}?$

Ans-

$\begin{array}{l}\mathrm{If}\frac{8}{9}\mathrm{is}\mathrm{the}\mathrm{multiplicative}\mathrm{inverse}\mathrm{of}-1\frac{1}{8},\mathrm{then}\mathrm{their}\\ \mathrm{product}\mathrm{should}\mathrm{b}\mathrm{eequal}\mathrm{to}1.\\ \\ \mathrm{However},\mathrm{their}\mathrm{product}\mathrm{is}\mathrm{not}\mathrm{equal}\mathrm{to}1\mathrm{as}\\ \frac{8}{9}×\left(-1\frac{1}{8}\right)=\frac{8}{9}×\frac{-9}{8}=-1\ne 1\\ \therefore \frac{8}{9}\mathrm{is}\text{}\mathrm{not}\mathrm{the}\mathrm{multiplicative}\mathrm{inverse}\mathrm{of}-1\frac{1}{8}.\end{array}$ $\mathrm{Q.9 Is}0.3\mathrm{the}\mathrm{multiplicative}\mathrm{inverse}\mathrm{of}3\frac{1}{3}?\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{Why}\mathrm{or}\mathrm{why}\mathrm{not}?$

Ans-

$\begin{array}{l}\mathrm{If}0.3\mathrm{is}\mathrm{the}\mathrm{multiplicative}\mathrm{inverse}\mathrm{of}\text{}3\frac{1}{3},\mathrm{then}\mathrm{their}\\ \mathrm{product}\mathrm{should}\mathrm{be}\mathrm{equal}\mathrm{to}1.\\ \\ ⇒0.3×\left(3\frac{1}{3}\right)=0.3×\frac{10}{3}=\frac{3}{10}×\frac{10}{3}=1\\ \therefore 0.3\mathrm{is}\text{}\mathrm{the}\mathrm{multiplicative}\mathrm{inverse}\mathrm{of}\text{}3\frac{1}{3}.\end{array}$

Q.10 Write.
(i) The rational number that does not have a reciprocal.
(ii) The rational numbers that are equal to their reciprocals.
(iii) The rational number that is equal to its negative.

Ans-

(i) The rational number that does not have a reciprocal is 0 because its reciprocal is not defined.
(ii) The rational numbers that are equal to their reciprocals are 1 and −1.
(iii) The rational number that is equal to its negative is 0.

Q.11 Fill in the blanks.

(i) Zero has ________ reciprocal.
(ii) The numbers ________ and ________ are their own reciprocals.
(iii) The reciprocal of – 5 is ________.

$\left(\mathrm{iv}\right)\mathrm{Reciprocal}\mathrm{of}\frac{1}{\mathrm{x}},\mathrm{where}\mathrm{x}\ne 0\mathrm{is}__________.$

(v) The product of two rational numbers is always a _______.
(vi) The reciprocal of a positive rational number is ________.

Ans-

$\begin{array}{l}\left(\text{i}\right)\text{No}\\ \left(\text{ii}\right)\text{1},\text{}-\text{1}\\ \left(\text{iii}\right)\frac{-1}{5}\\ \left(\text{iv}\right)x\\ \left(\text{v}\right)\text{Rational number}\\ \left(\text{vi}\right)\text{Positive rational number}\end{array}$

Represent these numbers on the number line.

$\left(\mathrm{i}\right)\frac{7}{4}\left(\mathrm{ii}\right)\frac{-5}{6}$

(i) (ii) $\mathrm{Q.12 Represent}\frac{-2}{11},\frac{-5}{11},\frac{-9}{11}\mathrm{on}\mathrm{the}\mathrm{number}\mathrm{line}.$

Ans- Q.13 Write five rational numbers which are smaller than 2.

Ans-

$\begin{array}{l}\text{We can write 2 as}\frac{10}{5}.\\ \therefore \mathrm{F}\text{ive rational numbers which are smaller than 2 are}\\ \frac{9}{5},\frac{8}{5},\frac{7}{5},\frac{6}{5}\mathrm{and}\frac{5}{5}\text{.}\end{array}$ $\mathrm{Q.14 Find}\mathrm{ten}\mathrm{rational}\mathrm{numbers}\mathrm{between}\frac{-2}{5}\mathrm{and}\frac{1}{2}.$

Ans-

$\begin{array}{l}\mathrm{We}\text{}\mathrm{can}\mathrm{write}\frac{-2}{5}\mathrm{and}\text{}\frac{1}{2}\text{as}\frac{-8}{20}\text{}\mathrm{and}\text{}\frac{10}{20}.\\ \\ \therefore \mathrm{ten}\mathrm{rational}\mathrm{numbers}\mathrm{between}\frac{-2}{5}\mathrm{and}\frac{1}{2}\text{are}\\ \frac{9}{20},\frac{8}{20},\frac{7}{20},\frac{6}{20},\frac{5}{20},\frac{4}{20}\text{,}\frac{3}{20},\frac{2}{20}\text{,}\frac{1}{20}\text{,}0\dots \dots .\frac{-8}{20}\end{array}$

$\mathrm{Q.15 Find}\mathrm{ten}\mathrm{rational}\mathrm{numbers}\mathrm{between}\frac{-2}{5}\mathrm{and}\frac{1}{2}.$

Ans-

$\begin{array}{l}\mathrm{We}\text{}\mathrm{can}\mathrm{write}\frac{-2}{5}\mathrm{and}\text{}\frac{1}{2}\text{as}\frac{-8}{20}\text{}\mathrm{and}\text{}\frac{10}{20}.\\ \\ \therefore \mathrm{Ten}\mathrm{rational}\mathrm{numbers}\mathrm{between}\frac{-2}{5}\mathrm{and}\frac{1}{2}\text{are}\\ \frac{9}{20},\frac{8}{20},\frac{7}{20},\frac{6}{20},\frac{5}{20},\frac{4}{20}\text{,}\frac{3}{20},\frac{2}{20}\text{,}\frac{1}{20}\text{,}0\dots \dots .\frac{-7}{20}.\end{array}$

$\begin{array}{l}\text{Q.16 Find five rational numbers between}\\ \left(\mathrm{i}\right)\frac{2}{3}\mathrm{and}\frac{4}{5}\left(\mathrm{ii}\right)\frac{-3}{2}\mathrm{and}\frac{5}{3}\left(\mathrm{iii}\right)\frac{1}{4}\mathrm{and}\frac{1}{2}\end{array}$

Ans-

$\begin{array}{l}\left(\mathrm{i}\right)\frac{2}{3}\mathrm{and}\frac{4}{5}\\ \\ \frac{2}{3}\mathrm{and}\frac{4}{5}\mathrm{can}\text{}\mathrm{be}\text{}\mathrm{represented}\text{}\mathrm{as}\text{}\frac{30}{45}\mathrm{and}\frac{36}{45}.\\ \\ \therefore \mathrm{Five}\mathrm{rational}\mathrm{numbers}\mathrm{between}\frac{2}{3}\mathrm{and}\frac{4}{5}\text{}\mathrm{are}\\ \frac{31}{45},\frac{32}{45},\frac{33}{45},\frac{34}{45}\mathrm{and}\frac{35}{45}.\\ \\ \left(\mathrm{ii}\right)\frac{-3}{2}\mathrm{and}\frac{5}{3}\\ \\ \frac{-3}{2}\mathrm{and}\frac{5}{3}\mathrm{can}\text{}\mathrm{be}\text{}\mathrm{represented}\text{}\mathrm{as}\text{}\frac{-9}{6}\mathrm{and}\frac{10}{6}.\\ \\ \therefore \mathrm{Five}\mathrm{rational}\mathrm{numbers}\mathrm{between}\frac{-3}{2}\mathrm{and}\frac{5}{3}\text{}\mathrm{are}\\ \frac{-8}{6},\frac{-7}{6},\frac{-6}{6},\frac{-5}{6}\mathrm{and}\frac{-4}{6}.\\ \\ \left(\mathrm{iii}\right)\frac{1}{4}\mathrm{and}\frac{1}{2}\\ \\ \frac{1}{4}\mathrm{and}\frac{1}{2}\mathrm{can}\text{}\mathrm{be}\text{}\mathrm{represented}\text{}\mathrm{as}\text{}\frac{6}{24}\mathrm{and}\frac{12}{24}.\\ \\ \therefore \mathrm{Five}\mathrm{rational}\mathrm{numbers}\mathrm{between}\frac{1}{4}\mathrm{and}\frac{1}{2}\text{}\mathrm{are}\\ \frac{7}{24},\frac{8}{24},\frac{9}{24},\frac{10}{24}\mathrm{and}\frac{11}{24}.\end{array}$

Q.17 Write five rational numbers greater than –2.

Ans-

$\begin{array}{l}\text{We can write}-\text{2 as}\frac{-10}{5}.\\ \\ \therefore \mathrm{Five}\mathrm{rational}\mathrm{numbers}\mathrm{greater}\text{}\mathrm{than}-2\text{}\mathrm{are}\\ \frac{-9}{5},\frac{-8}{5},\frac{-7}{5},\frac{-6}{5}\mathrm{and}\frac{-5}{5}.\end{array}$

$\mathrm{Q.18 Find}\mathrm{ten}\mathrm{rational}\mathrm{numbers}\mathrm{between}\frac{3}{5}\mathrm{and}\frac{3}{4}.$

Ans-

$\begin{array}{l}\frac{3}{5}\mathrm{and}\frac{3}{4}\mathrm{can}\text{}\mathrm{be}\text{}\mathrm{represented}\text{}\mathrm{as}\text{}\frac{48}{80}\mathrm{and}\frac{60}{80}.\\ \\ \therefore \mathrm{Ten}\mathrm{rational}\mathrm{numbers}\mathrm{between}\frac{3}{5}\mathrm{and}\frac{3}{4}\text{}\mathrm{are}\\ \frac{49}{80},\frac{50}{80},\frac{51}{80},\frac{52}{80},\frac{53}{80},\frac{54}{80},\frac{55}{80},\frac{56}{80},\frac{57}{80}\mathrm{and}\frac{58}{80}.\end{array}$