NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers help students clearly understand one of the most important number systems in middle school mathematics. Class 8 Maths Chapter 1 solution guides explain rational numbers as numbers that can be written in the form pq\frac{p}{q}qp​, where p and q are whole numbers and q is not equal to zero. These solutions make learning easier by breaking concepts into simple, step-by-step explanations.

The chapter introduces key properties of rational numbers, such as closure, commutativity, and associativity for addition and multiplication. It also explains important ideas like how adding zero or multiplying by one does not change a rational number. Students learn about additive inverses, multiplicative inverses, and how rational numbers are represented on the number line.

NCERT Solutions also cover all types of examples, exercises, and word problems given in the textbook. The Class 8 Rational Numbers questions and answers help students practice comparing, simplifying, and performing operations on rational numbers with confidence. Clear solutions reduce confusion and strengthen problem-solving skills.

Overall, NCERT Solutions for Class 8 Maths Chapter 1 provide strong conceptual clarity, improve exam preparation, and help students build a solid foundation for advanced topics in algebra and higher mathematics.

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Class 8 Maths Chapter 1 Rational Numbers Questions Answers

Chapter 1 Exercise 1.1

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 = 3/5 ((–5)/6)+ 5/2 (by 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 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 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 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 (ii) -13/19 (iii) 1/5 (iv) -5/8 × (-3/7) (v) -1 × (-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 × (-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. (vi) -1 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 the multiplicative identity. (ii) -13/17 × (-2/7) = -2/7 × (-13/17) The property of commutativity is used in the equation. (iii) -19/29 × 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 × (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: 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. Is 8/9 the multiplication inverse of –? Why or why not?

Solution: –= -9/8 [Multiplicative inverse ⟹ product should be 1] According to the question, 8/9 × (-9/8) = -1 ≠ 1 Therefore, 8/9 is not the multiplicative inverse of –.

9. If 0.3 is the multiplicative inverse of

? Why or why not? Solution:= 10/3 0.3 = 3/10 [Multiplicative inverse ⟹ product should be 1] According to the question, 3/10 × 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 = – 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 – 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.

Chapter 1 Exercise 1.2

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 – 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 ½.

Solution: 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

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 × 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. (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 × 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. (iii) ¼ and ½ 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.

Solution: -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.

Solution: 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.

Read more: Adding and Subtracting of Rational Numbers

Best Way to Do Your Class 8 Maths Chapter 1 Math

Math doesn't have to be a headache if you know the basic "rules of the road." When you work through the class 8 maths chapter 1 solution, you're basically learning the logic of how fractions and integers interact. Whether you're using the class 8 maths chapter 1 solutions,, the goal is to make your calculations faster and more reliable.

Many students find that a class 8 maths chapter 1 solutions is great for quick practice on the go. 

Defining Rational Numbers

A rational number is simply any number that can be written as a fraction (p/q). The only rule is that the bottom number (q) cannot be zero. This group includes your regular counting numbers, zero, and all positive and negative fractions.

How Rational Numbers Behave (Properties)

Learning these properties is a vital part of the class 8 maths chapter 1 solution because they help you simplify big problems.

• Closure: If you add or multiply two rational numbers, the answer is always another rational number.

• Commutative: For addition, the order doesn't matter (A + B is the same as B + A). Don't forget, this doesn't work for subtraction!

• Associative: This is about how you group numbers. In addition, (A + B) + C gives the same result as A + (B + C).

• Distributive Property: This is a lifesaver for hard problems: A(B + C) = AB + AC.

Additive and Multiplicative Inverses

In the class 8 maths chapter 1 solutions, you’ll often be asked to find the "opposite" or "flip" of a number.

• Additive Inverse: This is just the number with the opposite sign. For 5, it is -5, because adding them gives you zero.

• Multiplicative Inverse (Reciprocal): This is the number flipped upside down. For ⅔  it is 3/2, because multiplying them gives you 1

Using a Number Line

Putting these numbers on a line helps you see which one is bigger. Positive numbers move to the right, and negative numbers move to the left. If you need to mark 3/4, you just split the space between 0 and 1 into four equal "rooms" and pick the third one.

Finding Numbers in Between

There’s a cool secret in math: there are actually millions of numbers hidden between any two fractions! To find them, you can either find the average of the two numbers or just change them to have a much larger common denominator so you can see the "steps" in between.

Read More: Difference Between Rational Numbers and Irrational Numbers?

Building Mathematical Fluency Through Logical Application

Understanding rational numbers isn't just about passing your next class test; it's about building a foundation for every complex equation you'll face in high school. When you master the distributive property, you aren't just moving numbers around; you're learning how to break down massive, intimidating problems into tiny, bite-sized pieces. We've found that students who spend time visualizing these numbers on a number line develop a much stronger "number sense" than those who simply memorize the formulas. You can't ignore the importance of the additive identity, which is zero, or the multiplicative identity, which is one, because they are the invisible anchors of algebra. At the end of the day, these properties ensure that the math remains consistent and logical, no matter how large the fractions become. Don't let the negative signs scare you off; they just tell a story about direction and debt on the number line. By practicing with the right solutions, you'll start to see patterns where others only see a chaotic mess of symbols. We've seen that the best way to handle the class 8 syllabus is to treat it like a puzzle where every property is a different piece you can use to finish the picture. Whether you are solving for x or just simplifying a long string of additions, having these rules in your back pocket makes you a faster and more confident learner. It's truly rewarding when you realize that every integer and whole number you've ever known is actually just a rational number in disguise, waiting to be used in a larger calculation.

Strategies for Long-Term Mastery and Academic Success

The quality of the materials you use every day and how consistently you practise maths can have a big impact on how well you do. We suggest that you not only read the class 8 maths chapter 1 solution, but also work through each issue by hand to really understand the concept. If you come across a hard word problem in class 8 maths chapter 1 solution, see if you can figure out which property, such as the associative or distributive law, is being tested before you look at the answer. This practice changes you from a passive reader to an active problem-solver, which is an important aspect of your growth as a learner. If you ever feel stuck, switching to class 8 maths chapter 1 solutions might help you see things in a new way and clear up any language problems you might be having. We made our tools to help with this process of learning by doing, so that each learner can discover a pace that works for them. You may improve your memory and get faster for competitive exams by going over the class 8 maths chapter 1 solutions ganita prakash activities every so often. Every excellent mathematician began out just like you, learning the same rules about fractions and signs until they became second nature. Your willingness to learn about these basic ideas with interest and perseverance will help you do well in higher-level physics and chemistry in the future

Also Read: Distance Formula in Maths - Derivation, Examples

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