Chapter 1 NCERT Solutions for Class 8 Maths The rational numbers are offered here to assist learners in understanding the principles from the start. The topics taught in Class 8 are important to learn because they are repeated in Classes 9 and 10. It is recommended that students solve the problems supplied at the end of each chapter in the NCERT book in order to achieve good results in the Class 8 Mathematics examination. These NCERT Solutions for Class 8 Maths assist students in better learning all subjects.
Rational Numbers are numbers that may be represented in the form p/q, where q is not equal to zero. It is one of the most important concepts in Maths in Class 8. In simplified terms, a rational number is any fraction with a non-zero denominator. We must first simplify rational numbers in order to express them on a number line. Does it appear to be difficult? That is no longer the case. Students can now consult the NCERT Solutions for Class 8 Maths Chapter 1 while working on the practice problems if they have any questions or need clarification on a concept. Try practicing these NCERT Solutions to gain a better understanding of the relevant concepts.
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
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
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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
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
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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
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
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 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
5
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
–
Solution:
[Multiplicative inverse ⟹ product should be 1]
According to the question,
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:
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.
2
(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.
Exercise 1.2 Page: 20
1. Represent these numbers on the number line.
(i) 7/4
(ii) -5/6
Solution:
(i) 7/4
The line between the complete numbers should be divided into four sections. Divide the line into portions ranging from 0 to 1 to 4 parts, 1 to 2 to 4 parts, and so on.
As a result, the rational number 7/4 is 7 points away from 0 on 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, -9/11 lies at a distance of 2, 5, 9 points away from 0, towards the negative number line respectively.
2
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 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
1
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 & 48/60
Therefore, Five rational numbers between 40/60 and 48/60 = 41/60, 42/60, 43/60, 44/60, 45/60
3
(ii) -3/2 and 5/3
Let us make the denominators 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
As a result, there are five rational numbers between -9/6 &10/6 = -1/6, 2/6, 3/6, 4/6, 5/6
(iii) ¼ & ½
Let’s suppose the denominators are the same. 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
As a result, we may conclude that the five rational integers bigger than are -2 are
-10/10, -5/10, -1/10, 5/10, 7/10
7. Find 10 rational numbers that are in the range of 3/5 and ¾,
Solution:
Let’s suppose the denominators are the same. 80.
3/5 = (3 × 16)/(5× 16) = 48/80
3/4 = (3 × 20)/(4 × 20) = 60/80
ten rational integers in the range of 3/5 & ¾ = ten rational numbers between 48/80 and 60/80
As a result, there are 10 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
There are two exercises in Chapter 1: Rational Numbers, and the NCERT Solutions for Class 8 Maths provided here offer precise answers to all of the questions in both exercises. Let’s take a closer look at some of the concepts covered in this chapter.
- The operations of addition, subtraction, and multiplication close rational numbers.
- Addition and multiplication are two procedures that can be used to solve problems.
- For rational numbers, it is commutative.
- For rational numbers, associative is used.
- The additive identity for rational numbers is 0 (rational number).
- The multiplicative identity for rational numbers is rational number 1.
- Distributivity of rational numbers: For all rational numbers a, b and c, a(b + c) = ab + ac and a(b – c) = ab – ac
- A number line can be used to represent rational numbers.
- There are many rational numbers between any two specified rational numbers. The concept of mean aids us in determining the rational number that is between two rational numbers.
The main topics covered in this chapter include:
1.1 An Overview
1.2 Rational Number Properties
1.2.1 Commutativity
1.2.1 Commutativity
1.2.3 Associativity
1.2.3 Associativity
1.2.5 The significance of 1
1.2.6 The inverse of a number
For rational numbers, 1.2.7 Reciprocal 1.2.8 Distributivity of multiplication over addition.
1.3 Number Line Representation of Rational Numbers
1.4 Rational Numbers in the Intersection of Two Rational Numbers
To understand essential topics in CBSE Class 8 Maths, students are strongly advised to use the NCERT Solutions for Class 8. When these solutions are referred to, unasked doubts can be clarified promptly.
NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers
The method of representing a rational number on a number line, as well as the method of obtaining rational numbers between two rational numbers, are also covered in Chapter 1 of NCERT Class 8 Maths. To discover more about Rational Numbers and the principles addressed in them, read Chapter 1 of the NCERT textbook. To do well in the board exam, learn the NCERT Solutions for Class 8 thoroughly.
The method of representing a rational number on a number line, as well as the method of obtaining rational numbers between two rational numbers, are also covered in Chapter 1 of NCERT Class 8 Maths. To discover more about Rational Numbers and the principles addressed in them, read Chapter 1 of the NCERT textbook. To do well in the board exam, learn the NCERT Solutions for Class 8 thoroughly.
Frequently Asked Questions on NCERT Solutions for Class 8 Math Chapter 1
What is the meaning of rational numbers according to NCERT Solutions for Class 8 Maths Chapter 1?
Rational numbers are represented in p/q form when q is not equal to zero, according to NCERT Solutions for Class 8 Maths Chapter 1. It’s a form of a real number as well. A rational number is any fraction having non-zero denominators. As a result, we may argue that ‘0’ is a rational number because it can be represented in a variety of ways, such as 0/1, 0/2, 0/3, and so on. However, 1/0, 2/0, 3/0, and so on are irrational because they give us unlimited values.
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.
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