NCERT Solutions For Class 8 Maths Chapter 1: Get Free PDF - Embibe
  • Written By Varsha
  • Last Modified 25-06-2022
  • Written By Varsha
  • Last Modified 25-06-2022

NCERT Solutions For Class 8 Maths Chapter 1: Get Free PDF

CBSE NCERT Solutions For Class 8 Maths Chapter 1: The NCERT solutions for Class 8 students is a very valuable and important resource. Students should refer to the solutions in order to perform well and ace their examinations. The NCERT Maths Class 8 Chapter 1 syllabus can often provide challenges and overwhelm students when preparing for the exam. NCERT solutions are designed by subject matter experts at Embibe so that students can easily understand complex topics in no time.

These NCERT Solutions for Class 8 Maths Chapter 1 exercises 1.1 and 1.2 are slightly more difficult than the other units. However, students do not need to worry about anything because the article has provided free downloadable PDFs for the exercises. Students can download them for free and study them offline and improve their preparation. Continue to read the article to know more.

CBSE NCERT Solutions for Class 8 Maths Chapter 1: Rational Numbers

Before getting into the details of NCERT Solutions For Class 8 Maths Chapter 1 PDF, let’s look at the list of topics and sub-topics covered in the Rational Numbers chapter. Click on any topic to download the solution as a PDF.

1.1Introduction
1.2Properties of Rational Numbers

Also, Check:

CBSE Class 8 English SyllabusNCERT Class 8 English Book
CBSE Class 8 Maths SyllabusNCERT Class 8 Maths Book
CBSE Class 8 Science SyllabusNCERT Class 8 Science Book
CBSE Class 8 Social Science SyllabusNCERT Class 8 Social Science Book
CBSE Class 8 Hindi SyllabusNCERT Class 8 Hindi Book
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NCERT Maths Class 8 Chapter 1: Solved Exercises and In-Text Questions

NCERT Solutions for Class 8 Maths Chapter 1 PDF provided here has been solved by the academic experts of Embibe. They have taken into consideration the CBSE guidelines and marking scheme while drafting these NCERT Solutions for Class 8 Maths. The language used in these solutions can be easily understandable by a Class 8 student.

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NCERT Solutions For Class 8 Chapter 1 Maths: Chapter Summary

In earlier classes, you must have studied various numbers like natural numbers, whole numbers, integers, fractions, etc. Here in this chapter, you will learn about Rational Numbers. A rational number is expressed in the form p/q, where p and q are integers and q≠0. The concept of rational numbers is pivotal in Class 8 Maths as well as for several important mathematical concepts that come after it.

Any fraction with a non-zero denominator is said to be a rational number. A rational number can be represented on a number line by simplifying them first. In this chapter you will also study various properties of rational numbers like closure, commutativity, associativity, the role of zero, the role of 1, negative of a number, reciprocal, distributivity of multiplication over addition for rational numbers, representation of rational numbers on the number line and rational numbers between two rational numbers.

Let us have an overview of some of the concepts that are being discussed in this chapter.

  • Rational numbers are closed during operations such as addition, subtraction and multiplication.
  • The operations of addition and multiplication are
    • commutative for rational numbers &
    • associative for rational numbers.
  • Zero (0) is the additive identity for rational numbers.
  • One (1) is the multiplicative identity for rational numbers.
  • The law of distributivity of rational numbers: For all rational numbers a, b and c, a(b + c) = ab + ac and a(b – c) = ab – ac
  • All rational numbers can be represented along a number line.
  • There are countless rational numbers between any two given rational numbers. We use the concept of mathematical mean to find rational numbers between two rational numbers.

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You get to know the difference between a rational number and a fraction. Are they similar or different?

Rational Number

Fraction

These are the numbers which are written in the form p/q, where p and q are integers and q≠0.

These are the numbers which are written in the form p/q, where p and q are whole numbers and q≠0.

They can be positive or negative numbers.

Fraction cannot be negative.

For example: 12/7, 2/-8, -22/-56/ -1/22

For example: 10/12, 15/76, 55/98, 12/9

So, we can say that a fractional number can always be a rational number, but a rational number may or may not be a fractional number.

Answers to NCERT Class 8 Maths Chapter 1 Exercises

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 law of commutativity)

= 3/5 (-2/3 – 1/6)+ 5/2 = 3/5 ((- 4 – 1)/6)+ 5/2

= 3/5 ((–5)/6)+ 5/2 (by law of distributivity)

= – 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  law of commutativity)

= 2/5 × (- 3/7 + 1/14) – 3/12

= 2/5 × ((- 6 + 1)/14) – 3/12

= 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

Solution:

(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

(ii) x = -13/17

Solution:

(i) x = 11/15

We know that, x = 11/15

The additive inverse of x is – x (because x + (-x) = 0)

and the additive inverse of 11/15 is – 11/15 (because 11/15 + (-11/15) = 0)

The same logic is applied to 11/15 + (-11/15) = 0, to conclude that the additive inverse of -11/15 is 11/15.

Or, – (-11/15) = 11/15

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

(ii) -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

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

Solution:

(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

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

(iii) -19/29 × 29/-19 = 1

Solution:

(i) -4/5 × 1 = 1 × (-4/5) = -4/5

Here 1 is used as the multiplicative identity.

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

The law of commutativity is used in this equation

(iii) -19/29 × 29/-19 = 1

The property of multiplicative inverse is used in this equation.

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

Solution:

1/3 × (6 × 4/3) = (1/3 × 6) × 4/3

The way in which numbers are grouped in a multiplication problem does not change the product. Therefore, the Associativity Property is used here.

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

Solution:

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

According to the question,

6/13 × (Reciprocal of -7/16)

6/13 × (-16/7) = -96/91

8. Is 8/9 the multiplication inverse ofNCERT Solution For Class 8 Maths Chapter 1 Image 1? Why or why not?

Solution:

NCERT Solution For Class 8 Maths Chapter 1 Image 2 = -7/8

[The product of the number with its multiplicative inverse should be 1]

Since,

8/9 × (-7/8) = -7/9 ≠ 1

Therefore, 8/9 is not the multiplicative inverse of

NCERT Solution For Class 8 Maths Chapter 1 Image 3.

9. If 0.3 the multiplicative inverse of

? Why or why not?

Solution:

NCERT Solution For Class 8 Maths Chapter 1 Image 5 = 10/3

0.3 = 3/10

[The product of the number with its multiplicative inverse should be 1]

Since,

3/10 × 10/3 = 1

Therefore, 0.3 is the multiplicative inverse of

NCERT Solution For Class 8 Maths Chapter 1 Image 6.

10. 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.

11. 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) 0 is the rational number that does not have a reciprocal. Reason is that

Reciprocal of 0 = 1/0, which is undefined.

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

Reciprocal of 1 = 1/1 = 1 Similarly, Reciprocal of -1 = – 1

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

The reason is that:

Negative of 0=-0=0


Exercise 1.2 Page: 20

1. Represent these numbers on the number line.

(i) 7/4

(ii) -5/6

Solution:

(i) 7/4

First, divide the line between the whole numbers into 4 sections. i.e., divide the line between 0 and 1 to 4 parts, and the part between 1 and 2 to 4 parts and so on.

So, the rational number 7/4 lies at a distance that is 7 points away from 0 towards the positive number line.

NCERT Solution For Class 8 Maths Chapter 1 Image 7

(ii) -5/6

Then, divide the line between the integers into 4 sections. 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 – 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

NCERT Solution For Class 8 Maths Chapter 1 Image 8

2. 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

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

Solution:

Divide the number line between the integers into 11 sections.

So, the rational numbers -2/11, -5/11, -9/11 lies at a distance of 2, 5, 9 points away from 0, towards the negative side of the number line respectively.

NCERT Solution For Class 8 Maths Chapter 1 Image 9

4. Find five rational numbers between.

(i) 2/3 and 4/5

(ii) -3/2 and 5/3

(iii) ¼ and ½

Solution:

(i) 2/3 and 4/5

Let us rewrite the rational numbers so that they have a common denominator, say 60

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

2/3 = (2 × 20)/(3 × 20) = 40/60

4/5 = (4 × 12)/(5 × 12) = 48/60

Five rational numbers between 2/3 and 4/5 are the same as 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 rewrite the rational numbers so that they have a common denominator, say 6

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

-3/2 = (-3 × 3)/(2× 3) = -9/6

5/3 = (5 × 2)/(3 × 2) = 10/6

Five rational numbers between -3/2 and 5/3 are the same as five rational numbers between -9/6 and 10/6

So, Five rational numbers between -9/6 and 10/6 = -1/6, 2/6, 3/6, 4/6, 5/6

(iii) ¼ and ½

Let us rewrite the rational numbers so that they have a common denominator, say 24.

i.e., ¼ and ½ can be written as:

¼ = (1 × 6)/(4 × 6) = 6/24

½ = (1 × 12)/(2 × 12) = 12/24

Five rational numbers between ¼ and ½ are the same as five rational numbers between 6/24 and 12/24

So, Five rational numbers between 6/24 and 12/24 = 7/24, 8/24, 9/24, 10/24, 11/24

5. Find the rational numbers between -2/5 and ½.

Solution:

Let us make the denominators same, say 50.

-2/5 = (-2 × 10)/(5 × 10) = -20/50

½ = (1 × 25)/(2 × 25) = 25/50

Ten rational numbers between -2/5 and ½ are the same as ten rational numbers between -20/50 and 25/50

So, 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

6. Write five rational numbers greater than -2.

Solution:

-2 can also be written as -20/10

So, five rational numbers greater than -2 can be easily found as

-10/10, -5/10, -1/10, 5/10, 7/10

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

Solution:

First, multiply the numerator and denominator of the fractions with the same number to make the denominators same, say 80.

3/5 = (3 × 16)/(5× 16) = 48/80

3/4 = (3 × 20)/(4 × 20) = 60/80

Now, ten rational numbers between 3/5 and 3/4  is the same as ten rational numbers between 48/80 and 60/80

Therefore, we can easily find 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

Important Topics in NCERT Class 8 Maths Chapter 1

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

Students can refer to this NCERT Solutions for Class 8 to learn relevant concepts in CBSE Class 8 Maths. They can also clear their doubts instantly by referring to these solutions.

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FAQs

Here we have provided some of the frequently asked questions related to CBSE Class 8 Maths Chapter 1.

Q1: What are the important topics in Class 8 Maths Chapter 1?
Ans: Read this article to find out all details regarding Class 8 Maths Chapter 1.

Q2: Is Embibe providing solutions for Class 8 Maths Chapter 1?
Ans: Yes, Embibe provides accurate and detailed solutions for all questions provided in the NCERT Class 8 Maths book. We bring you NCERT Solutions for Class 8 Maths, designed by our subject experts to facilitate an easy and clear understanding of the fundamental concepts. The solutions also contain detailed stepwise explanations of problems given in the NCERT Textbook. The NCERT Solutions for Class 8 Maths Chapter 1 is provided in a downloadable PDF format and students can use it as a reference tool to quickly review all the topics.

Q3: What are rational numbers according to NCERT Solutions for Class 8 Maths Chapter 1?
Ans: As per NCERT Solutions for Class 8 Maths Chapter 1, a number that is represented in p/q form is called rational numbers where q is not equal to zero. A rational number is also a type of real number. Any fraction with non-zero denominators is a rational number. Therefore, we can conclude that ‘0’ is also a rational number, as we can represent it in many forms such as 0/1, 0/2, 0/3, etc. However, 1/0, 2/0, 3/0, etc. are not rational numbers, as they give us infinite values.

Q4: List out the important concepts discussed in NCERT Solutions for Class 8 Maths Chapter 1.
Ans: 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.

Q5: How can I get in-depth information on NCERT Class 8 Maths Chapter 1?
Ans: Maths Chapter 1 – Rational Numbers is an important chapter in the CBSE syllabus for Class 8. Students can establish a firm knowledge of the chapter by practising a range of questions provided by Embibe. Still, it is advisable to first read the chapter properly from the NCERT textbook before referring to the Important Questions PDF. Students can avail of Important Questions for Class 8 Maths Chapter 1 provided by expert teachers at Embibe to practice different questions during the exam and concept building.

You can solve the free CBSE Class 8 practice questions on Embibe for Mathematics and Science. You can also learn & Practice CBSE Class 8 Questions for these subjects. These will be of great help to you.

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