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NCERT Solutions for Class 8 Maths Exercises - PDF Solutions

Class 8 Maths covers a vivid CBSE syllabus that includes the basic and advanced concepts of arithmetic, geometry and algebra topics. The preparation of these chapters needs the assistance of the complete study material. Add the NCERT Class 8 Maths All Exercises Solutions developed by the experts of Vedantu to learn how the formulas and concepts of these chapters are skillfully used to frame the accurate answers to the exercise questions.

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Ncert solutions for class 8 maths - chapter-wise list.

Given below are the chapter-wise NCERT Solutions for Class 8 Maths . These solutions are provided by the Maths experts at Vedantu in a detailed manner. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.

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NCERT Solutions for Class 8 Maths Chapter 1

NCERT Solutions for Class 8 Maths Chapter 2

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Importance of NCERT Maths All Exercises book Class 8 Solutions

The solutions have been framed chapter-wise so that students can find them right away on the list. These solutions contain precise answers to all the fundamental questions asked in the exercises. All these answers have been framed following the latest CBSE syllabus so that students can get accustomed to the best methods of answering such questions to score more.

The exercise solutions will act as the perfect guide for the students to follow during preparing an exam syllabus. They will learn the specific techniques and can even assess their preparation level by comparing their answers to the solutions.

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The 8th Maths All Exercises Book Answers will help you find the right answers and answering methods in no time. You can make your preparation sessions more efficient.

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Follow the classes and focus on the theoretical explanations. Solve exercise problems by seeking help from the solutions to understand the application of the concepts.

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Assignments For Class 8 Mathematics

Assignments for Class 8 Mathematics have been developed for Standard 8 students based on the latest syllabus and textbooks applicable in CBSE, NCERT and KVS schools. Parents and students can download the full collection of class assignments for class 8 Mathematics from our website as we have provided all topic wise assignments free in PDF format which can be downloaded easily. Students are recommended to do these assignments daily by taking printouts and going through the questions and answers for Grade 8 Mathematics. You should try to do these test assignments on a daily basis so that you are able to understand the concepts and details of each chapter in your Mathematics book and get good marks in class 8 exams.

Assignments for Class 8 Mathematics as per CBSE and NCERT pattern

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All students studying in Grade 8 Mathematics should download the assignments provided here and use them for their daily routine practice. This will help them to get better grades in Mathematics exam for standard 8. We have made sure that all topics given in your textbook for Mathematics which is suggested in Class 8 have been covered ad we have made assignments and test papers for all topics which your teacher has been teaching in your class. All chapter wise assignments have been made by our teachers after full research of each important topic in the textbooks so that you have enough questions and their solutions to help them practice so that they are able to get full practice and understanding of all important topics. Our teachers at https://www.assignmentsbag.com have made sure that all test papers have been designed as per CBSE, NCERT and KVS syllabus and examination pattern. These question banks have been recommended in various schools and have supported many students to practice and further enhance their scores in school and have also assisted them to appear in other school level tests and examinations. Its easy to take print of thee assignments as all are available in PDF format.

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All free Printable practice assignments are in PDF single lick download format and have been prepared by Class 8 Mathematics teachers after full study of all topics which have been given in each chapter so that the students are able to take complete benefit from the worksheets. The Chapter wise question bank and revision assignments can be accessed free and anywhere. Go ahead and click on the links above to download free CBSE Class 8 Mathematics Assignments PDF.

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• Chapter 1: Rational Numbers

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers

Ncert solutions class 8 maths chapter 1 – 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.

Download Exclusively Curated Chapter Notes for Class 8 Maths Chapter – 1 Rational Numbers

Download most important questions for class 8 maths chapter – 1 rational numbers.

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.

NCERT Solutions for Class 8 Maths – Rational Numbers

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Access Answers to NCERT Class 8 Maths Chapter 1 Rational Numbers

Exercise 1.1 page: 14, 1. using appropriate properties, find:.

(i) -2/3 × 3/5 + 5/2 – 3/5 × 1/6

-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

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

= – 15 /30 + 5/2

= – 1 /2 + 5/2

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

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) – 3/12

= 2/5 × ((- 5)/14)) – 1/4

= (-10/70) – 1/4

= – 1/7 – 1/4

= (– 4– 7)/28

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

The Additive inverse of 2/8 is – 2/8

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

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) -13/17

We have, x = -13/17

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,

4. Find the multiplicative inverse of the following:

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

Multiplicative inverse of -13 is -1/13.

(ii) -13/19

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

Multiplicative inverse of 1/5 is 5.

(iv) -5/8 × (-3/7) = 15/56

Multiplicative inverse of 15/56 is 56/15.

(v) -1 × (-2/5) = 2/5

Multiplicative inverse of 2/5 is 5/2.

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

Here 1 is the multiplicative identity.

The property of commutativity is used in the equation.

The multiplicative inverse is the property used in this equation.

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

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

According to the question,

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.

1/3 × (6 × 4/3) = (1/3 × 6) × 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.

8/9 × (-9/8) = -1 ≠ 1

3/10 × 10/3 = 1

(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.

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

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

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

Reciprocal of 1 = 1/1 = 1, similarly, reciprocal of -1 = – 1

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

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 – 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 _________.

(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.

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.

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 – 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.

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.

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 ½.

Let us make the denominators the 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 = 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) ¼ and ½

Let us make the denominators the same, 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 = 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.

Let us make the denominators the same, 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 = 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.

Let us make the denominators the same, 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 = 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.

-2 can be written as – 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 ¾.

Let us make the denominators the same, say 80.

3/5 = (3 × 16)/(5× 16) = 48/80

3/4 = (3 × 20)/(4 × 20) = 60/80

Ten rational numbers between 3/5 and ¾ = 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 Solutions 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.
• 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 – 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.

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

It’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)

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 Maths  NCERT Solutions takes the students to another set of numbers, the rational numbers.  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 the  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 – 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.

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Assignments Class 8 Mathematics PDF Download

Students can refer to Assignments for Class 8 Mathematics available for download in Pdf. We have given below links to subject-wise free printable Assignments for Mathematics Class 8 which you can download easily. All assignments have a collection of questions and answers designed for all topics given in your latest NCERT Books for Class 8 Mathematics for the current academic session. All Assignments for Mathematics Grade 8 have been designed by expert faculty members and have been designed based on the type of questions asked in standard 8 class tests and exams. All Free printable Assignments for NCERT CBSE Class 8, practice worksheets, and question banks have been designed to help you understand all concepts properly. Practicing questions given in CBSE NCERT printable assignments for Class 8 with solutions and answers will help you to further improve your understanding. Our faculty have used the latest syllabus for Class 8. You can click on the links below to download all Pdf assignments for class 8 for free. You can get the best collection of Kendriya Vidyalaya Class 8 Mathematics assignments and questions workbooks below.

Class 8 Mathematics Assignments Pdf Download

CBSE NCERT KVS Assignments for Mathematics Class 8 have been provided below covering all chapters given in your CBSE NCERT books. We have provided below a good collection of assignments in Pdf for Mathematics standard 8th covering Class 8 questions and answers for Mathematics. These practice test papers and workbooks with question banks for Class 8 Mathematics Pdf Download and free CBSE Assignments for Class 8 are really beneficial for you and will support in preparing for class tests and exams. Standard 8th students can download in Pdf by clicking on the links below.

Subjectwise Assignments for Class 8 Mathematics

More assignments for class 8 mathematics.

Benefits of Solving Class 8 Mathematics Assignments

• The best collection of Grade 8 assignments for Mathematics have been provided below which will help you in getting better marks in class tests and exams.
• The solved question for Class 8 Mathematics will help you to gain more confidence to attempt all types of problems in exams
• Latest NCERT Books for Class 8 Mathematics have been referred to for designing these assignments
• We have provided step by step solutions for all questions in the Class 8 assignments so that you can understand the solutions in detail.
• We have provided single click download links to all chapterwise worksheets and assignments in Pdf.
• Class 8 practice question banks will support to enhance subject knowledge and therefore help to get better marks in exam

FAQs by Mathematics Students in Class 8

At https://www.ncertbooksolutions.com is the best website that has the biggest collection of free printable assignments for Class 8 Mathematics .

We provide here Standard 8 subject-wise assignments which can be easily downloaded in Pdf format for free. Our teachers have provided these Grade 8 Mathematics test sheets for Mathematics given in your books.

You can click on the links above and get assignments for Mathematics in Grade 8, all chapters and topic-wise question banks with solutions have been provided here. You can click on the links to download in Pdf.

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We have provided the best quality question bank for Class 8 for Mathematics available for Pdf Download. You can download them all and use them offline without the internet.

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Unit 1: Numbers and operations

Unit 2: solving equations with one unknown, unit 3: linear equations and functions, unit 4: systems of equations, unit 5: geometry, unit 6: geometric transformations, unit 7: data and modeling, review articles.

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‘It Feels Like I Am Screaming Into the Void With Each Application’

By Peter Coy

Opinion Writer

When I asked new college graduates last month to tell me about their job searches, I got back a ton of heartache. Unanswered applications. Lowered expectations. For some, a sense that college was a waste of time and money.

John York wrote that he was about to earn a master’s degree in mathematics from New York University. “I have submitted close to 400 applications. I have heard back from less than 40, all rejections,” he wrote. “I essentially cannot get any job, because there are no entry-level positions anywhere at all.” He has a patent, he passed the first-level exam for Chartered Financial Analysts and he’s getting his Series 3 license, another financial credential. Nevertheless, he wrote, “It is just so silent, it feels like I am screaming into the void with each application I am filling out.”

Mauricio Naranjo, who is seeking work as a graphic designer, wrote, “Over the past year, I have submitted more than 400 applications and consistently receive a response that appears to be A.I.-generated, stating that unfortunately, they have moved forward with another candidate who better fits their expectations. This is the exact phrasing every time. Very few respond, as most do not reply at all.”

“Exhausting. Utterly demoralizing,” wrote Beth Donnelly, who is graduating this month with a major in linguistics and minors in German and teaching English as a second language. “I’ve been searching since early August for full-time, part-time or internship positions after I graduate. I’ve started putting my ‘desired salary’ at $35,000 in hope that just one person will think, ‘Oh, I won’t have to pay this person a large wage, so they get a leg up in the hiring process.’”

I got some positive responses, too. Lucinda Warnke, who landed a job in journalism as a general assignment reporter, wrote: “I am optimistic and excited! I feel confident in my career trajectory and my ability to build a stable, satisfying career. The job I got out of school comes with a livable wage and benefits, so I can build savings in the event that I am laid off or have some other financially demanding emergency. I feel like I made a good investment in my education because I went to a school that was affordable and studied subjects that balanced my interests with my professional needs.”

A majority of responses were grim, though. That’s not too surprising, given that half of college graduates are underemployed a year after graduation, meaning that they are working in jobs that don’t require the degrees they earned, as I wrote in my April 29 newsletter.

There’s clearly something wrong when young graduates can’t find jobs at the same time that employers complain of not being able to find qualified workers. As of March, there were still fewer unemployed people than job openings, according to the Bureau of Labor Statistics. In April the unemployment rate remained below average at 3.9 percent.

The responses I got aren’t a representative sample of all college graduates. It’s possible that unhappy people were more likely to write in. (I had to leave out some of the angriest and most dejected people because they didn’t want their names to appear.) Separately, my informal impression is that the people who wrote — happy or sad — were more likely to have attended a highly ranked school and to have graduated without student loans than the general student population.

Many students wrote that the jobs they were seeking or secured didn’t draw on what they learned in the classroom. “I will be using the skills I picked up in my data science minor, but nothing from my major (international relations),” Rain Orsi, a 2024 graduate, wrote. “A lot of the educational stuff could’ve been condensed to a 20-page PDF and I probably would be at the same knowledge level,” another student wrote. Jackeline Arcara wrote that if she had it to do over again, “I wouldn’t go to a four-year, fancy-pants school. I would take classes at a local college part-time and see where that takes me.”

Some students said that classroom learning was only part of what made college worthwhile to them. “College gives you four years to grow up — I have the maturity now to handle a full-time job. Before college, not so much,” wrote Caroline Lidz, who got a job in public relations after graduating in December with a degree in media studies and communications and a minor in art history.

Several said internships matter, a lot. “I wish I interned for a company outside of the school instead of being a research/lab assistant,” wrote Roger Vitek, who is graduating in June with a degree in product design and is still job hunting.

Economists have found that what you study in college is at least as important as where you study. As I wrote in my April 29 piece, there’s relatively strong demand for computer science, engineering, mathematics and math-intensive business fields such as finance and accounting.

But as I found out from the people who wrote in, that’s not always the case. Robert Vermeulen, a computer science major, wrote, “Out of the ~155 applications I haven’t had a reference on, I have gotten zero interviews.” Morgan Steckler wrote that he is looking for a software engineering or I.T. administration role paying at least $70,000 a year, but has had no luck so far. He said he’s thinking of bartending while continuing to send out applications. On the positive side, there are people like Warnke, who got a job as a reporter — not exactly a fast-growing profession.

As I read students’ responses, I had to remind myself that this is actually a relatively good year for finding a job. To a lot of members of the class of ’24, it doesn’t feel that way. Julia Brukx, who is graduating with a degree in history and art history, wrote, “I think I hit a new low just this morning when asked to write a cover letter for a retail position.”

Donnelly, the woman who described her job search as demoralizing, wrote: “I was told that if I was involved, active, kind, ready to learn, driven and intelligent, I would end up with a job out of college. This is evidently not true, and few older people seem to understand this.” She added, “I don’t have a backup plan besides working in the service industry.”

Elsewhere: Caps, Not Bans, for Short-Term Rentals

New York City’s Local Law 18, which was passed with the support of the hotel industry, tightens the rules on renting out rooms for less than 30 days. Supporters say renting rooms to tourists raises rents for New Yorkers. But an article published in Harvard Business Review by three scholars — one of whom used to work for Airbnb — calculates that Airbnb caused only about 1 percent of the aggregate increase in rents over the past decade or so. Hosts, guests and the businesses that serve them benefit. To keep certain neighborhoods from being overwhelmed by tourists, the authors recommend caps on how many nights per year a place may be rented out.

Quote of the Day

“The hedonistic conception of man is that of a lightning calculator of pleasures and pains who oscillates like a homogeneous globule of desire of happiness under the impulse of stimuli that shift him about the area, but leave him intact. He has neither antecedent nor consequent.”

— Thorstein Veblen, “Why Is Economics Not an Evolutionary Science?” (1898)

Peter Coy is a writer for the Opinion section of The Times, covering economics and business. Email him at [email protected] . @ petercoy

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