Embibe offers NCERT Maths Class 8 Chapter 1 Solutions PDF to help CBSE Class 8 students excel in their Mathematics term and final exams. We have provided NCERT Solutions For Class 8 Maths Chapter 1 PDF for the topic Rational Numbers here. The solutions are precise, simple and as per the CBSE curriculum so that students can not only complete their homework on time but also score maximum marks in the final exam.
The concepts taught in Class 8 are continued in classes 9 and 10. Candidates should solve questions provided at the end of each chapter in the NCERT book to get a good grip of the topics in the chapter and score good marks in the Class 8 Mathematics examination. These NCERT Solutions for Class 8 Maths Chapter 1 PDF provided here act as a good source of reference for the students.
In this article, we have provided complete solutions for NCERT Class 8th Maths Chapter 1 on Rational Numbers.
Table of Contents
- Class 8 Maths Chapter 1 Solutions: Rational Numbers
- Class 8 Maths Chapter 1: Solved Exercises and In-Text Questions
- NCERT Solutions For Class 8 Chapter 1 Maths: Chapter Summary
- Answers to NCERT Class 8 Maths Chapter 1 Exercises
- Important Topics in NCERT Class 8 Maths Chapter 1
- FAQs Related To Class 8 Maths Chapter 1 Solutions
Class 8 Maths Chapter 1 Solutions: 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.
1.1 | 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 |
Also, Check:
Practice 8th CBSE Exam Questions
Class 8 Maths 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.
DOWNLOAD SOLUTIONS FOR CLASS 8 MATHS CHAPTER 1 PDF
Download CBSE Class 8 Solutions for Maths for other chapters from the table below:
- 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 – Visualizing Solid Shapes
- Chapter 11 – Mensuration
- Chapter 12 – Exponents & Powers
- Chapter 13 – Direct and Inverse Proportions
- Chapter 14 – Factorization
- Chapter 15 – Introduction to Graphs
- Chapter 16 – Playing with Numbers
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.
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 of – ? Why or why not?
Solution:
= -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
.
9. If 0.3 the multiplicative inverse of
? Why or why not?
Solution:
= 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
.
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.
(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
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.
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
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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. The can also clear their doubts instantly by referring to these solutions.
FAQs Related To Class 8 Maths Chapter 1 Solutions
Here we have provided some of the frequently asked questions related to CBSE Class 8 Maths Chapter 1.
Q1. 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.
Q2. 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.
Q3. 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.
Q4. 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.
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