Class 8 NCERT Mathematics Solutions Chapter 1 Rational Numbers Exercise 1.1 (original) (raw)
Last Updated : 23 Jul, 2025
The Chapter 1 " **Rational Numbers" of the Class 8 NCERT Mathematics textbook, which focuses on Rational Numbers. This chapter introduces students to the concept of rational numbers, their properties, and operations. Exercise 1.1 is designed to help students strengthen their understanding of rational numbers by solving various problems related to identifying, comparing, and performing basic arithmetic operations with them.
This section provides detailed **solutions for Exercise 1.1 from Chapter 1 of the Class 8 NCERT Mathematics textbook. These solutions are intended to guide students through the fundamental concepts of rational numbers, ensuring a solid foundation for more advanced topics in number theory.
The Exercise 1.1, of Chapter 1 Rational Numbers from NCERT Mathematics Class 8 covers the following topics :
- **Introduction to Rational Numbers
- **Properties of Rational Numbers
- Closure Property
- Commutative Property
- Associative Property
- Distributive Property
- Identity Elements
- Inverse Elements
- **Representation of Rational Numbers on the Number Line
- **Comparison of Rational Numbers
- **Finding Rational Numbers between Two Given Rational Numbers
- **Simplification of Rational Expressions
Class 8 NCERT Mathematics Solutions - Exercise 1.1
**Question 1: Using appropriate properties find.
****(i) -2/3 × 3/5 + 5/2 – 3/5 × 1/6**
****(ii) 2/5 × (- 3/7) – 1/6 × 3/2 + 1/14 × 2/5**
**Solution:
****(i) -2/3 × 3/5 + 5/2 – 3/5 × 1/6**
Given equation: -2/3 × 3/5 + 5/2 – 3/5 × 1/6
By regrouping we get,
= -2/3 × 3/5 - 3/5 × 1/6 + 5/2
= 3/5 (-2/3 - 1/6)+ 5/2 [taking 3/5 as common]
= 3/5 ((-2×2/3×2 -1×1/6×1 )+ 5/2 [by using distributive property]
= 3/5 ((-4-1)/6)+ 5/2
= 3/5 ((–5)/6)+ 5/2
= – 15/30 + 5/2 [Dividing -15 and 30 by 2 we get -1/2]
= – 1/2 + 5/2
= 4/2
= 2
Therefore,
-2/3 × 3/5 + 5/2 – 3/5 × 1/6 = 2
****(ii) 2/5 × (- 3/7) – 1/6 × 3/2 + 1/14 × 2/5**
Given equation: 2/5 × (- 3/7) – 1/6 × 3/2 + 1/14 × 2/5
By regrouping we get,
= 2/5 × (-3/7) + 1/14 × 2/5 – (1/6 × 3/2)
= 2/5 × (-3/7 + 1/14) – 3/12
= 2/5 × ((-6 + 1)/14) – 3/12 [by using distributive property]
= 2/5 × ((-5)/14)) – 1/4
= (-10/70) - 1/4 [Dividing -10 and 70 by 10 we get -1/7]
= -1/7 - 1/4
= (-4 -7)/28
= -11/28
Therefore,
2/5 × (- 3/7) – 1/6 × 3/2 + 1/14 × 2/5 = -11/28
**Question 2: Write the additive inverse of each of the following
****(i) 2/8**
****(ii) -5/9**
****(iii) -6/-5**
****(iv) 2/-9**
****(v) 19/-16**
**Solution:
We know that the additive inverse of x will be -x,
****(i) 2/8**
Given: 2/8
Additive inverse of 2/8 will be -2/8
****(ii) -5/9**
Given: -5/9
Additive inverse of -5/9 will be 5/9
****(iii) -6/-5**
Given: -6/-5
-6/-5 = 6/5 [Dividing both by -1 ]
Additive inverse of 6/5 will be -6/5
****(iv) 2/-9**
Given: 2/-9
2/-9 = -2/9
Additive inverse of -2/9 will be 2/9
****(v)** **19/-16
Given: 19/-16
19/-16 = -19/16
Additive inverse of -19/16 will be 19/16
**Question 3: Verify that: -(-x) = x for.
****(i) x = 11/15**
****(ii) x = -13/17**
**Solution:
****(i)** x = 11/15
Given, x = 11/15
Since, additive inverse of x will be -x
Therefore, the additive inverse of 11/15 will be -11/15 (as 11/15 + (-11/15) = 0)
We can also represent the following as 11/15 = -(-11/15)
Thus, -x = -11/15
-(-x) = -(-11/15) = (11/15) = x
Hence, verified: -(-x) = x
****(ii)** -13/17
Given, x = -13/17
Since, additive inverse of x will be -x as x + (-x) = 0
Therefore, the additive inverse of -13/17 will be 13/17 as 13/17 + (-13/17) = 0
We can also represent the following as 13/17 = -(-13/17)
Thus, -x = -13/17
-(-x) = -(-13/17) = (13/17) = x
Hence, verified: -(-x) = x
**Question 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:
We know that the multiplicative inverse of x will be 1/x as a × 1/a = 1
****(i)** **-13
Given: -13
The multiplicative inverse of -13 will be -1/13
****(ii) -13/19**
Given: -13/19
The multiplicative inverse of -13/19 will be -19/13
****(iii) 1/5**
Given: 1/5
The multiplicative inverse of 1/5 will be 5
****(iv)** **-5/8 × (-3/7)
Given: -5/8 × (-3/7)
-5/8 × (-3/7) = 15/56
The multiplicative inverse of 15/56 will be 56/15
****(v)** **-1 × (-2/5)
Given: -1 × (-2/5)
-1 × (-2/5) = 2/5
The multiplicative inverse of 2/5 will be 5/2
****(vi)** **-1
Given: -1
The multiplicative inverse of -1 will be -1
**Question 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**
Given: -4/5 × 1 = 1 × (-4/5) = -4/5
It is representing the property of multiplicative identity.
****(ii) -13/17 × (-2/7) = -2/7 × (-13/17)**
Given: -13/17 × (-2/7) = -2/7 × (-13/17)
It is representing the property of commutativity.
****(iii)** **-19/29 × 29/-19 = 1
Given: -19/29 × 29/-19 = 1
It is representing the property of multiplicative inverse
**Question 6: Multiply 6/13 by the reciprocal of -7/16
**Solution:
Given: 6/13 × (Reciprocal of -7/16)
Since, reciprocal of -7/16 = 16/-7 = -16/7
Therefore,
6/13 × (-16/7) = -96/91
**Question 7: Tell what property allows you to compute 1/3 × (6 × 4/3) as (1/3 × 6) × 4/3
**Solution:
Given: 1/3 × (6 × 4/3) = (1/3 × 6) × 4/3
Here, the product of their multiplication does not change. Hence, Associativity Property is used in the given equation.
**Question 8: Is 8/9 the multiplication inverse of -1 1/8? Why or why not?
**Solution:
Given: -1 1/8 which is equal to -9/8
Since it is the multiplication inverse, therefore the product should be 1.
8/9 × (-9/8) = -1 ≠ 1
Hence, 8/9 is not the multiplication inverse of -1 1/8
**Question 9: If 0.3 the multiplicative inverse of 3 1/3? Why or why not?
**Solution:
Give: 3 1/3 = 10/3
Since it is the multiplication inverse, therefore the product should be 1.
0.3 × 10/3 = 3/3 = 1
Hence, 0.3 is the multiplicative inverse of 3 1/3.
**Question 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.
Since, 0 = 0/1
Therefore, the reciprocal of 0 = 1/0, which is not defined.
Hence, the rational number that does not have a reciprocal is 0.
****(ii)** The rational numbers that are equal to their reciprocals.
Since, 1 = 1/1
Therefore, the reciprocal of 1 = 1/1 = 1
Similarly,
-1 = -1/1
Therefore, the reciprocal of -1 = -1/1 = -1
Hence, the rational numbers that are equal to their reciprocals are 1 and -1
****(iii)** The rational number that is equal to its negative.
Since negative of 0 = -0 = 0
Therefore, the rational number that is equal to its negative is 0.
**Question 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.
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Conclusion
Mastering the concept of **Rational numbers as presented in **Chapter 1 of the Class 8 NCERT Mathematics textbook is essential for building a strong mathematical foundation. The **solutions to Exercise 1.1 provided here not only reinforce the understanding of rational numbers but also equip students with the skills needed to tackle more complex problems in the future.