# NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers

## NCERT Solutions Class 8 Maths Chapter 1 â€“ Free PDF Download

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers are provided here to help the students understand the concepts right from the beginning. The concepts taught in Class 8 are important to be understood as these concepts are continued in Classes 9 and 10. To score good marks in the Class 8 Mathematics examination, it is advised to solve questions provided at the end of each chapter in the NCERT textbook. These NCERT Solutions for Class 8 Maths help the students understand all the concepts in a better way.

Numbers that can be represented in the form of p/q, where q is not equal to zero, are known as Rational Numbers. It is one of the most critical topics in Class 8 Maths. In simpler words, any fraction with a non-zero denominator is said to be a rational number. To represent rational numbers on a number line, we need to simplify them first. Does it sound hard? Not anymore. Students can now access the NCERT Solutions for Class 8 Maths Chapter 1 while solving the exercise problems for any concept clarity and doubt clearance. Try practising these NCERT Solutions to grasp important topics with ease.

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## Exercise 1.1 Page: 14

#### 1. Using appropriate properties, find:

(i) -2/3 Ã— 3/5 + 5/2 â€“ 3/5 Ã— 1/6

##### Solution:

-2/3 Ã— 3/5 + 5/2 â€“ 3/5 Ã— 1/6

= -2/3 Ã— 3/5â€“ 3/5 Ã— 1/6+ 5/2 (by commutativity)

= 3/5 (-2/3 â€“ 1/6)+ 5/2

= 3/5 ((- 4 â€“ 1)/6)+ 5/2

= 3/5 ((â€“5)/6)+ 5/2 (by distributivity)

= â€“ 15 /30 + 5/2

= â€“ 1 /2 + 5/2

= 4/2

= 2

(ii) 2/5 Ã— (- 3/7) â€“ 1/6 Ã— 3/2 + 1/14 Ã— 2/5

Solution:

##### 2/5 Ã— (- 3/7) â€“ 1/6 Ã— 3/2 + 1/14 Ã— 2/5

= 2/5 Ã— (- 3/7) + 1/14 Ã— 2/5 â€“ (1/6 Ã— 3/2) (by commutativity)

= 2/5 Ã— (- 3/7 + 1/14) â€“ 3/12

= 2/5 Ã— ((- 6 + 1)/14) â€“ 3/12

= 2/5 Ã— ((- 5)/14)) â€“ 1/4

= (-10/70) â€“ 1/4

= â€“ 1/7 â€“ 1/4

= (â€“ 4â€“ 7)/28

= â€“ 11/28

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

Solution:

(i) 2/8

The Additive inverse of 2/8 is â€“ 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

(ii) x = -13/17

Solution:

(i) x = 11/15

We have, x = 11/15

The additive inverse of x is â€“ x (as x + (-x) = 0).

Then, the additive inverse of 11/15 is â€“ 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, â€“ (-11/15) = 11/15

i.e., -(-x) = x

(ii) -13/17

We have, x = -13/17

The additive inverse of x is â€“ 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, â€“ (13/17) = -13/17,

i.e., -(-x) = x

4. Find the multiplicative inverse of the following:

(i) -13 (ii) -13/19 (iii) 1/5 (iv) -5/8 Ã— (-3/7) (v) -1 Ã— (-2/5) (vi) -1

Solution:

(i) -13

Multiplicative inverse of -13 is -1/13.

(ii) -13/19

Multiplicative inverse of -13/19 is -19/13.

(iii) 1/5

Multiplicative inverse of 1/5 is 5.

(iv) -5/8 Ã— (-3/7) = 15/56

Multiplicative inverse of 15/56 is 56/15.

(v) -1 Ã— (-2/5) = 2/5

Multiplicative inverse of 2/5 is 5/2.

(vi) -1

Multiplicative inverse of -1 is -1.

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

(i) -4/5 Ã— 1 = 1 Ã— (-4/5) = -4/5

(ii) -13/17 Ã— (-2/7) = -2/7 Ã— (-13/17)

(iii) -19/29 Ã— 29/-19 = 1

Solution:

(i) -4/5 Ã— 1 = 1 Ã— (-4/5) = -4/5

Here 1 is the multiplicative identity.

(ii) -13/17 Ã— (-2/7) = -2/7 Ã— (-13/17)

The property of commutativity is used in the equation.

(iii) -19/29 Ã— 29/-19 = 1

The multiplicative inverse is the property used in this equation.

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

Solution:

Reciprocal of -7/16 = 16/-7 = -16/7

According to the question,

6/13 Ã— (Reciprocal of -7/16)

6/13 Ã— (-16/7) = -96/91

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

Solution:

1/3 Ã— (6 Ã— 4/3) = (1/3 Ã— 6) Ã— 4/3

Here, the way in which factors are grouped in a multiplication problem supposedly does not change the product. Hence, the Associativity Property is used here.

8. Is 8/9 the multiplication inverse ofÂ â€“? Why or why not?

Solution:

â€“ = -9/8

[Multiplicative inverse âŸ¹ product should be 1]

According to the question,

8/9 Ã— (-9/8) = -1 â‰  1

Therefore, 8/9 is not the multiplicative inverse of â€“.

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

Solution:

= 10/3

0.3 = 3/10

[Multiplicative inverse âŸ¹ product should be 1]

According to the question,

3/10 Ã— 10/3 = 1

Therefore, 0.3 is the multiplicative inverse of
.

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:

0 = 0/1

Reciprocal of 0 = 1/0, which is not defined.

(ii) The rational numbers that are equal to their reciprocals are 1 and -1.

Reason:

1 = 1/1

Reciprocal of 1 = 1/1 = 1, similarly, reciprocal of -1 = â€“ 1

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

Reason:

Negative of 0=-0=0

11. Fill in the blanks.

(i) Zero has _______reciprocal.

(ii) The numbers ______and _______are their own reciprocals

(iii) The reciprocal of â€“ 5 is ________.

(iv) Reciprocal of 1/x, where x â‰  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 â€“ 5 is -1/5.

(iv) Reciprocal of 1/x, where x â‰  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, i.e. divide the line between 0 and 1 to 4 parts, 1 and 2 to 4 parts, and so on.

Thus, the rational number 7/4 lies at a distance of 7 points away from 0 towards the positive number line.

(ii) -5/6

Divide the line between the integers into 4 parts, i.e. divide the line between 0 and -1 to 6 parts, -1 and -2 to 6 parts, and so on. Here, since the numerator is less than the denominator, dividing 0 to â€“ 1 into 6 parts is sufficient.

Thus, the rational number -5/6 lies at a distance of 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.

Thus, 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:

The number 2 can be written as 20/10

Hence, we can say that the five rational numbers which are smaller than 2 are:

2/10, 5/10, 10/10, 15/10, 19/10

4. Find the rational numbers between -2/5 and Â½.

Solution:

Let us make the denominators the same, say 50.

-2/5 = (-2 Ã— 10)/(5 Ã— 10) = -20/50

Â½ = (1 Ã— 25)/(2 Ã— 25) = 25/50

Ten rational numbers between -2/5 and Â½ = ten rational numbers between -20/50 and 25/50.

Therefore, ten rational numbers between -20/50 and 25/50 = -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) Â¼ and Â½

Solution:

(i) 2/3 and 4/5

Let us make the denominators the same, say 60

i.e., 2/3 and 4/5 can be written as:

2/3 = (2 Ã— 20)/(3 Ã— 20) = 40/60

4/5 = (4 Ã— 12)/(5 Ã— 12) = 48/60

Five rational numbers between 2/3 and 4/5 = five rational numbers between 40/60 and 48/60.

Therefore, five rational numbers between 40/60 and 48/60 = 41/60, 42/60, 43/60, 44/60, 45/60.

(ii) -3/2 and 5/3

Let us make the denominators the same, say 6

i.e., -3/2 and 5/3 can be written as:

-3/2 = (-3 Ã— 3)/(2Ã— 3) = -9/6

5/3 = (5 Ã— 2)/(3 Ã— 2) = 10/6

Five rational numbers between -3/2 and 5/3 = five rational numbers between -9/6 and 10/6.

Therefore, five rational numbers between -9/6 and 10/6 = -1/6, 2/6, 3/6, 4/6, 5/6.

(iii) Â¼ and Â½

Let us make the denominators the same, say 24

i.e., Â¼ and Â½ can be written as:

Â¼ = (1 Ã— 6)/(4 Ã— 6) = 6/24

Â½ = (1 Ã— 12)/(2 Ã— 12) = 12/24

Five rational numbers between Â¼ and Â½ = five rational numbers between 6/24 and 12/24.

Therefore, five rational numbers between 6/24 and 12/24 = 7/24, 8/24, 9/24, 10/24, 11/24.

6. Write five rational numbers greater than -2.

Solution:

-2 can be written as â€“ 20/10

Hence, we can say that 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 Â¾.

Solution:

Let us make the denominators the same, say 80.

3/5 = (3 Ã— 16)/(5Ã— 16) = 48/80

3/4 = (3 Ã— 20)/(4 Ã— 20) = 60/80

Ten rational numbers between 3/5 and Â¾ = ten rational numbers between 48/80 and 60/80.

Therefore, ten rational numbers between 48/80 and 60/80 = 49/80, 50/80, 51/80, 52/80, 54/80, 55/80, 56/80, 57/80, 58/80, 59/80.

## NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Summary

Chapter 1, Rational Numbers, contains 2 exercises, and the NCERT SolutionsÂ for Class 8 Maths given here contains precise answers for all the questions present in these exercises. Let us have a look at some of the concepts that are being discussed in this chapter.

• Rational numbers are closed under the operations of addition, subtraction and multiplication.
• The operations of addition and multiplication are as follows:
• Commutative for rational numbers
• Associative for rational numbers
• The 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 â€“ c) = ab â€“ 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.

### 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

Itâ€™s highly recommended that students utilise the NCERT Solutions for Class 8 to comprehend key concepts in CBSE Class 8 Maths. Unasked doubts can also be clearly instantly when referred to these solutions.

### Access Exercise-wise NCERT Solutions Class 8 Maths of Chapter 1:

Exercise 1.1 Solutions 11 Questions (11 Short Answer Questions)
Exercise 1.2 Solutions 7 Questions (7 Short Answer Questions)

### NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers

Numbers are considered to be the most basic block of Mathematics. In lower classes, the students would have learnt the different types of numbers, including natural numbers, whole numbers, integers etc. Chapter 1 of Class 8 MathsÂ NCERT Solutions takes the students to another set of numbers, the rational numbers.Â  This chapter explains almost all the concepts that a student of Class 8 has to learn about rational numbers.

Chapter 1 of NCERT Class 8 Maths also describes the method of representing a rational number on a number line as well as the method of finding rational numbers between 2 rational numbers. Read and learn Chapter 1 of theÂ NCERT textbook to learn more about Rational Numbers and the concepts covered in them. Learn the NCERT Solutions for Class 8 effectively to score high in the annual examination.

Disclaimer:

Dropped Topics â€“ 1.2.6 Negative of a number, 1.2.7 Reciprocal, 1.3 Representation of rational numbers on the number line and 1.4 Rational numbers between two rational numbers.

## Frequently Asked Questions on NCERT Solutions for Class 8 Maths Chapter 1

Q1

### What is the meaning of rational numbers according to NCERT Solutions for Class 8 Maths Chapter 1?

According to NCERT Solutions for Class 8 Maths Chapter 1, rational numbers are represented in p/q form where q is not equal to zero. It is also a type of real number. Any fraction with non-zero denominators is a rational number. Hence, we can say that â€˜0â€™ 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.
Q2

### List out the important concepts discussed in NCERT Solutions for Class 8 Maths Chapter 1.

The main concepts covered in NCERT Solutions for Class 8 Maths Chapter 1 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.
Q3

### Is BYJUâ€™S providing solutions for Class 8 Maths Chapter 1?

Yes, BYJUâ€™S website provides accurate and detailed solutions for all questions provided in the NCERT Textbook. BYJUâ€™S brings you NCERT Solutions for Class 8 Maths, designed by the subject experts to facilitate smooth and precise understanding of concepts. These solutions have detailed step-by-step explanations of problems given in the NCERT Textbook. The NCERT Solutions for Class 8 Maths Chapter 1 can be downloaded in the form of a PDF, and students can use it as a reference tool to quickly review all the topics.