NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers

Subject specialists have designed NCERT solutions for Maths Class 8 Chapter 1 which includes thorough solutions for reference. These solutions are updated according to the latest CBSE syllabus for class 8 2024-25 and are provided in easy language for understanding. Tips and tricks are also provided.

These solutions are provided so a student can clear his doubts and get help with deep understanding of the concept. Also you can refer them to make the chapter notes and revisions notes. PDF of this can also be downloaded from website.

Chapter 1 Rational Numbers Class 8 PDF

1. Rational Numbers

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Summary

This chapter includes 2 exercises and the NCERT Solutions for Class 8 Maths provided here contains to the point answers for all the questions present in these exercises. The concepts present in the chapter are given below.

• Rational numbers are closed under the operations of addition, subtraction and multiplication.
• The operations addition and multiplication are
• Commutative for rational numbers.
• Associative for rational numbers.
• Rational number 0 is the additive identity for rational numbers.
• Rational number 1 is the multiplicative identity for rational numbers.
• Distributivity of rational numbers: For all rational numbers a, b and c, a(b + c) = ab + ac and a(b – c) = ab – ac
• Rational numbers can be represented on a number line.
• Between any two given rational numbers, there are countless rational numbers. The idea of mean helps us to find rational numbers between two rational numbers.

Access Question and Answers to NCERT Class 8 Maths Chapter 1 Rational Numbers

1. Using appropriate properties, find: (i) -2/3 × 3/5 + 5/2 – 3/5 × 1/6

Solution:

• -2/3 × 3/5 + 5/2 – 3/5 × 1/6
• = -2/3 × 3/5 – 3/5 × 1/6 + 5/2 (by commutativity)
• = 3/5 (-2/3 – 1/6) + 5/2
• = 3/5 ((-4 – 1)/6) + 5/2 (by distributivity)
• = 3/5 ((-5)/6) + 5/2
• = -15/30 + 5/2
• = -1/2 + 5/2
• = 4/2
• = 2

(ii) 2/5 × (- 3/7) – 1/6 × 3/2 + 1/14 × 2/5

Solution:

• 2/5 × (- 3/7) – 1/6 × 3/2 + 1/14 × 2/5
• = 2/5 × (- 3/7) + 1/14 × 2/5 – (1/6 × 3/2) (by commutativity)
• = 2/5 × (- 3/7 + 1/14) – 3/12
• = 2/5 × ((-6 + 1)/14) – 1/4
• = 2/5 × ((-5)/14) – 1/4
• = -10/70 – 1/4
• = -1/7 – 1/4
• = (-4 – 7)/28
• = -11/28

2. Write the additive inverse of each of the following: (i) 2/8

The additive inverse of 2/8 is -2/8.

(ii) -5/9

The additive inverse of -5/9 is 5/9.

(iii) -6/-5 = 6/5

The additive inverse of 6/5 is -6/5.

(iv) 2/-9 = -2/9

The additive inverse of -2/9 is 2/9.

(v) 19/-16 = -19/16

The additive inverse of -19/16 is 19/16.

3. Verify that: -(-x) = x for:

(i) x = 11/15

We have, x = 11/15.
The additive inverse of x is -x (as x + (-x) = 0).
Then, the additive inverse of 11/15 is -11/15 (as 11/15 + (-11/15) = 0).
The same equality, 11/15 + (-11/15) = 0, shows that the additive inverse of -11/15 is 11/15.
Or, -(-11/15) = 11/15, i.e., -(-x) = x.

(ii) x = -13/17

We have, x = -13/17.
The additive inverse of x is -x (as x + (-x) = 0).
Then, the additive inverse of -13/17 is 13/17 (as 13/17 + (-13/17) = 0).
The same equality (-13/17 + 13/17) = 0 shows that the additive inverse of 13/17 is -13/17.
Or, -(-13/17) = -13/17, i.e., -(-x) = x.

4. Find the multiplicative inverse of the following: (i) -13

The multiplicative inverse of -13 is -1/13.

(ii) -13/19

The multiplicative inverse of -13/19 is -19/13.

(iii) 1/5

The multiplicative inverse of 1/5 is 5.

(iv) -5/8 × (-3/7) = 15/56

The multiplicative inverse of 15/56 is 56/15.

(v) -1 × (-2/5) = 2/5

The multiplicative inverse of 2/5 is 5/2.

(vi) -1

The multiplicative inverse of -1 is -1.

5. Name the property under multiplication used in each of the following: (i) -4/5 × 1 = 1 × (-4/5) = -4/5

The property of multiplicative identity is used here.

(ii) -13/17 × (-2/7) = -2/7 × (-13/17)

The property of commutativity is used here.

(iii) -19/29 × 29/-19 = 1

The property of multiplicative inverse is used here.

6. Multiply 6/13 by the reciprocal of -7/16.

Solution:
Reciprocal of -7/16 = 16/-7 = -16/7
6/13 × (Reciprocal of -7/16) = 6/13 × (-16/7) = -96/91

7. Tell what property allows you to compute 1/3 × (6 × 4/3) as (1/3 × 6) × 4/3.

Solution: The Associativity Property allows you to compute 1/3 × (6 × 4/3) as (1/3 × 6) × 4/3. This property states that the way in which factors are grouped in a multiplication problem does not change the product.

8. Is 8/9 the multiplication inverse of -9/8? Why or why not?

Solution: No, 8/9 is not the multiplicative inverse of -9/8.The multiplicative inverse of a number x is a number y such that x × y = 1.In this case, if 8/9 were the multiplicative inverse of -9/8, then we should have:
8/9 × (-9/8) = 1However, the actual calculation shows:
8/9 × (-9/8) = -1 ≠ 1Therefore, 8/9 is not the multiplicative inverse of -9/8.

9. If 0.3 is the multiplicative inverse of x, why or why not?

Solution

• 0.3 = 3/10
• Multiplicative inverse means the product of the number and its inverse should be 1.
• 3/10 × 10/3 = 1
• Therefore, 0.3 is the multiplicative inverse of 10/3.

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.

Solution

(i) The rational number that does not have a reciprocal is 0.

Reason: The reciprocal of 0 is 1/0, which is undefined.(ii) The rational numbers that are equal to their reciprocals are 1 and -1.
Reason:

• Reciprocal of 1 is 1/1 = 1
• Reciprocal of -1 is -1/(-1) = 1

(iii) The rational number that is equal to its negative is 0.

Reason: The negative of 0 is -0, which is still 0.

11. Fill in the blanks.

(i) Zero has _______reciprocal.

(ii) The numbers ______and _______are their own reciprocals

(iii) The reciprocal of – 5 is ________.

(iv) Reciprocal of 1/x, where x ≠ 0 is _________.

(v) The product of two rational numbers is always a ________.

(vi) The reciprocal of a positive rational number is _________.

Solution:

(i) Zero has no reciprocal.
(ii) The numbers -1 and 1 are their own reciprocals.
(iii) The reciprocal of -5 is -1/5.
(iv) Reciprocal of 1/x, where x ≠ 0, is x.
(v) The product of two rational numbers is always a rational number.
(vi) The reciprocal of a positive rational number is positive.

Exercise 1.2 Page: 20

1. Represent these numbers on the number line:
   (i) 7/4
   (ii) -5/6

Solution

(i) 7/4

• Divide the line between the whole numbers into 4 parts.
• The rational number 7/4 lies 7 points away from 0 towards the positive number line.

(ii) -5/6

• Divide the line between the integers into 6 parts.
• The rational number -5/6 lies 5 points away from 0 towards the negative number line.

2. Represent -2/11, -5/11, -9/11 on a number line.

Solution

• Divide the line between the integers into 11 parts.
• The rational numbers -2/11, -5/11, and -9/11 lie at a distance of 2, 5, and 9 points away from 0 towards the negative number line, respectively.

3. Write five rational numbers which are smaller than 2.

Solution

• 2 can be written as 20/10
• The five rational numbers smaller than 2 are: 2/10, 5/10, 10/10, 15/10, 19/10

4. Find the rational numbers between -2/5 and 1/2.

Solution

• -2/5 = -20/50
• 1/2 = 25/50
• The rational numbers between -20/50 and 25/50 are: -18/50, -15/50, -5/50, -2/50, 4/50, 5/50, 8/50, 12/50, 15/50, 20/50

5. Find five rational numbers between:
   (i) 2/3 and 4/5
   (ii) -3/2 and 5/3
   (iii) 1/4 and 1/2

Solution

(i) 2/3 and 4/5

• 2/3 = 40/60, 4/5 = 48/60
• The five rational numbers between 40/60 and 48/60 are: 41/60, 42/60, 43/60, 44/60, 45/60

(ii) -3/2 and 5/3

• -3/2 = -9/6, 5/3 = 10/6
• The five rational numbers between -9/6 and 10/6 are: -1/6, 2/6, 3/6, 4/6, 5/6

(iii) 1/4 and 1/2

• 1/4 = 6/24, 1/2 = 12/24
• The five rational numbers between 6/24 and 12/24 are: 7/24, 8/24, 9/24, 10/24, 11/24

6. Write five rational numbers greater than -2.

Solution

• -2 can be written as -20/10
• The five rational numbers greater than -2 are: -10/10, -5/10, -1/10, 5/10, 7/10

7. Find ten rational numbers between 3/5 and 3/4.

Solution

• 3/5 = 48/80, 3/4 = 60/80
• The ten rational numbers between 48/80 and 60/80 are: 49/80, 50/80, 51/80, 52/80, 54/80, 55/80, 56/80, 57/80, 58/80, 59/80

The main topics covered in this chapter include:

• 1.1 Introduction
• 1.2 Properties of Rational Numbers
• 1.2.1 Closure
• 1.2.2 Commutativity
• 1.2.3 Associativity
• 1.2.4 The role of zero
• 1.2.5 The role of 1
• 1.2.6 Negative of a number
• 1.2.7 Reciprocal
• 1.2.8 Distributivity of multiplication over addition for rational numbers.
• 1.3 Representation of Rational Numbers on the Number Line
• 1.4 Rational Numbers between Two Rational Numbers

Access exercise-wise NCERT Solutions Class 8 Maths of this chapter here:

• 11 Questions (11 Short Answer Questions)
• 7 Questions (7 Short Answer Questions)

NCERT Rational Numbers Class 8 Solutions

Numbers are the most basic block of Mathematics. In middle school, the students may have learned the different types of numbers including natural numbers, whole numbers, integers etc. this chapter deals with another set of numbers, the rational numbers.  This chapter includes almost all the concepts that a student of Class 8 has to learn about rational numbers.

This chapter also provide the method of representing a rational number on a number line as well as the method of finding rational numbers between 2 rational numbers. Students must study this chapter to learn more about Rational Numbers and the concepts coming under them. By learning these solution students can easily score good marks in this chapter.

NCERT Solutions for Class 8 Maths Chapter 1 FAQs

What is the meaning of rational numbers according to these NCERT Solutions?

According to these NCERT Solutions, rational numbers can be represented in p/q form where q is not equal to zero. It comes under the type of real number. Any fraction with non-zero denominators is a rational number. Hence, we should say that ‘0’ is also a rational number, as we can represent it in many forms such as 0/1, 0/2, 0/3, etc. But, 1/0, 2/0, 3/0, etc. are not rational, since they give us infinite values.

List out the important concepts discussed in these NCERT Solutions.

Main concepts including in this chapter are listed below: 1.1 Introduction 1.2 Properties of Rational Numbers 1.2.1 Closure 1.2.2 Commutativity 1.2.3 Associativity 1.2.4 The role of zero 1.2.5 The role of 1 1.2.6 Negative of a number 1.2.7 Reciprocal 1.2.8 Distributivity of multiplication over addition for rational numbers.

Is INFINITE LEARN providing Solutions for Class 8 Maths Chapter 1?

Yes, INFINITE LEARN website comes with accurate and detailed solutions for all questions provided in the NCERT Textbook. INFINITE LEARN brings you NCERT Solutions for Class 8 Maths, made by our subject matter experts for smooth and easy understanding of concepts. These solutions comes with detailed step-by-step explanations of problems given in the NCERT Textbook. The NCERT Solutions of this chapter can be downloaded in the form of a PDF and it can be used as a quick revision tool.

What is a rational number in Class 8?

In Class 8, a rational number is a number that can be expressed as a fraction where the numerator and denominator are integers, and the denominator is not zero. Rational numbers include integers, fractions, and terminating or repeating decimals.

What is a short answer for a rational number?

A rational number is a number that can be expressed as a fraction where the numerator and denominator are integers, and the denominator is not zero. It can be written in the form a/b, where a and b are integers and b is not equal to zero.

What are 8 rational numbers?

Eight rational numbers can be any numbers that can be expressed as fractions. For example: 1/2 -3/4 5 -7/3 2.5 (which is 5/2) -1 0 3/7

What is an irrational number in Class 8?

In Class 8, an irrational number is a number that cannot be expressed as a fraction of two integers. These numbers have non-repeating and non-terminating decimal expansions. Examples of irrational numbers include √2, π (pi), and e (Euler's number).

How to solve rational numbers in Class 8?

To solve rational numbers in Class 8, you can perform operations like addition, subtraction, multiplication, and division on fractions. Simplify fractions by finding the common factors between the numerator and denominator. Convert mixed numbers to improper fractions for easier calculations. Practice converting between fractions and decimals to enhance your understanding of rational numbers.