ncert solutions for class 8 maths chapter 1 deals with Rational Numbers.

chapter 1 maths class 8

Ncert solutions for class 8 maths chapter 1 exercise 1.1

Exercise 1.1

class 8 maths chapter 1 exercise 1.1 (1)

1  Using appropriate properties find:

(i) –( ⅔) × (⅗) +(⁵/₂) –(⅗)×(⅙)

(ii) ( ⅖) × (³/₋₇) –(⅙) ×(³/₂)+(¹/₁₄)×(²/₅)

Solution to class 8 maths chapter 1 exercise 1.1(1)

Solution : (i) – ( ⅔) × (⅗) +(⁵/₂) –(⅗)×(⅙) 

= – ( ⅔) × (⅗) –(⅗)×(⅙)+(⁵/₂) [∵Associative property]

= ⅗(– ⅔ –⅙)+ ⁵/₂   [∵ Distributive property]

= ⅗[⁽⁻⁴⁻¹⁾/₆]+⁵/₂ = [³/₅ × (⁻⁵₆)]+(⁵₂)

= ⁻¹/₂ + ⁵/₂ = ⁽⁻¹⁺⁵⁾/ ₂ = ⁴ /₂ =2

(ii) ( ⅖) × (³/₋₇) –(⅙) ×(³/₂)+(¹/₁₄)×(²/₅)

= ( ⅖) × (⁻³/₇) +(¹/₁₄)×(²/₅)–(⅙) ×(³/₂)  [∵Associative property]

= ⅖ ×[⁻³/₇+¹/₁₄]–(¹/₄)  [∵Distributive property]

=⅖ ×[⁽⁻⁶⁺¹⁾/₁₄] –¼ = ⅖ × (⁻⁵/₁₄) – ¼

= – ⅐ – ¼ =⁽⁻⁴⁻⁷⁾/₂₈ = – ¹¹/₂₈

 

class 8 maths chapter 1 exercise 1.1 (2)

2  Write the additive inverse of each of the following.

(i) ²/₈  (ii) –⁵/₉ (iii) ⁻⁶/₋₅ (iv) ²/₋₉ (v) ¹⁹/₋₆

Solution to class 8 maths chapter 1 exercise 1.1(2)

We know that additive inverse of a rational number a/b  is (–a/b).

Such that (a/b)+(–a/b)=0

(i) Additive inverse of ²/₈ is  – ²/₈ 

(ii) Additive inverse of –⁵/₉ is  ⁵/₉

(iii) Additive inverse of ⁻⁶/₋₅  is   – ⁶/₅

(iv) Additive inverse of ²/₋₉is  ²/₉

(v) Additive inverse of ¹⁹/₋₆ is  ¹⁹/₆

class 8 maths chapter 1 exercise 1.1 (3)

3  Verify – (–x)=x  for :

(i)x=¹¹/₁₅  (ii) x=– ¹³/₁₇

Solution to class 8 maths chapter 1 exercise 1.1(3)

Answer: (i) Putting x=¹¹/₁₅ in – (–x)=x 

– (– ¹¹/₁₅)=¹¹/₁₅  ⇒ ¹¹/₁₅ = ¹¹/₁₅

LHS=RHS 

Hence Verified.

(ii) Putting x= – ¹³/₁₇  in – (–x)=x 

– {–(– ¹³/₁₇)}= – ¹³/₁₇  ⇒ – ¹³/₁₇= – ¹³/₁₇ 

 ∴ LHS=RHS. Hence Verified.

class 8 maths chapter 1 exercise 1.1 (4)

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 to class 8 maths chapter 1 exercise 1.1(4)

Answer : We know a multiplicative inverse of a is 1/a, such that a×(1/a)=1

(i) Multiplicative inverse of –13 is  – ¹/₁₃

(ii) Multiplicative inverse of – ¹³/₁₉ is  – ¹⁹/₁₃

(iii) Multiplicative inverse of   ¹/₅ is 5

(iv) Multiplicative inverse of – ⁵/₈ × – ³/₇ =

¹⁵/₅₆ is  ⁵⁶/₁₅

(v) Multiplicative inverse of –1× – ²/₅= ²/₅ is   ⁵/₂

(vi) Multiplicative inverse of –1 is ¹/ ₋₁= –1

class 8 maths chapter 1 exercise 1.1 (5)

5  Name the property under multiplication used in each of the following:

(i) – ⁴/₅×1=1× –⁴/₅

(ii) –¹³/₁₇ ×– ²/₇ = – ²/₇ × –¹³/₁₇

(iii) –¹⁹/₂₉ × ²⁹/₋₁₉=1

Solution to class 8 maths chapter 1 exercise 1.1(5)

Answer :

(i) 1 is the multiplicative identity.

(ii)Commutative property.

(iii)Multiplicative Inverse property.

class 8 maths chapter 1 exercise 1.1 (6)

6  Multiply ⁶/₁₃ by the reciprocal of ⁻⁷/₁₆

Solution to class 8 maths chapter 1 exercise 1.1(6)

Answer : Reciprocal of ⁻⁷/₁₆ is ¹⁶/₋₇

According to question

⁶/₁₃ × (¹⁶/₋₇)= – ⁹⁶/₉₁

class 8 maths chapter 1 exercise 1.1 (7)

7  Tell what property allows you to compute

⅓ ×(6 × ⁴/₃)  as (⅓ × 6)× ⁴/₃

Solution to class 8 maths chapter 1 exercise 1.1(7)

Answer :

By using associative property of multiplication, a×(b ×c) = (a ×b)×c.

class 8 maths chapter 1 exercise 1.1 (8)

8  Is ⁸/₉ the multiplicative inverse of  –1¹/₈? Why or why not? 

Solution to class 8 maths chapter 1 exercise 1.1(8)

Answer :

Since multiplicative inverse of a is 1/a, such that a×(1/a)=1

(⁸/₉) ×(–1¹/₈) =(⁸/₉) × (⁻⁹₈)= – 1≠1

∴ ⁸/₉ is not the multiplicative inverse of –1¹/₈

class 8 maths chapter 1 exercise 1.1 (9)

9  Is 0.3 is multiplicative inverse of 3⅓? Why or why not?

Solution to class 8 maths chapter 1 exercise 1.1(9)

Answer: Since multiplicative inverse of a is 1/a, such that a×(1/a)=1

∴ 0.3 × 3⅓ = ³/₁₀×¹⁰/₃ =1

∴ Yes, 0.3 is multiplicative inverse of 3⅓

class 8 maths chapter 1 exercise 1.1 (10)

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 to class 8 maths chapter 1 exercise 1.1(10)

Answer : (i) 0

(ii) 1 and –1

(iii) 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 to class 8 maths chapter 1 exercise 1.1(11)

Answer : (i)No

(ii)1 and –1

(iii) – ⅕

(iv) x

(v)Rational

(vi) positive

Exercise 2

class 8 maths chapter 1 exercise 1.2 (1)

1 Represent these numbers on number line.

(i) ⁷/₄   (ii) – ⁵/₆

Solution to class 8 maths chapter 1 exercise 1.2(1)

(i) ⁷/₄=1¾

Here, P is 1¾=⁷/₄.

(ii) – ⁵/₆

Here, M is  – ⁵/₆

solution to class 8 maths chapter 1 exercise 1.2 (2)

2  Represent  – ²/₁₁ , – ⁵/₁₁ ,  –⁹/₁₁ on the number line.

Solution to class 8 maths chapter 1 exercise 1.2(2)

Answer : B= – ²/₁₁ , C= – ⁵/₁₁ , D= – ⁹/₁₁

solution to class 8 maths chapter 1 exercise 1.2 (3)

3 Write five rational numbers which are smaller than 2.

Solution to class 8 maths chapter 1 exercise 1.2(3)

Answer :  ⅙ , ⅓ , ½ , – ¼,  – ⅕ and so on.

solution to class 8 maths chapter 1 exercise 1.2 (4)

4  Find ten rational numbers between – ⅖ and ½.

Solution to class 8 maths chapter 1 exercise 1.2(4)

Answer : Given rational numbers are –⅖ and ½

Here LCM of 5 and 2 is 10.

∴  – ⅖ × ²/₂= – ⁴/₁₀ and ¹/₂ ×⁵/₅= ⁵/₁₀

Again, – ⁴/₁₀ ײ/₂ = – ⁸/₂₀  and ⁵/₁₀ × ²/₂ =¹⁰/₂₀

∴ Ten rational numbers between – ⅖ and ½

are –⁷/₂₀ , – ⁶/₂₀ , – ⁵/₂₀ , – ⁴/₂₀ , – ³/₂₀ , – ²/₂₀ , – ¹/₂₀ , 0, ¹/₂₀ , ²/₂₀ 

solution to class 8 maths chapter 1 exercise 1.2 (5)

5  Find five rational numbers between (i) ⅔ and ⅘ (ii) – ³/₂ and ⁵/₃ (iii) ¼ and ½

Solution to class 8 maths chapter 1 exercise 1.2(5)

Answer : (i) ⅔ and ⅘

L.C.M of 3 and 5 is 15

∴ ⅔ × ⁵/₅=¹⁰/₁₅ and ⁴/₅׳/₃=¹²/₁₅

Again ¹⁰/₁₅ × ⁴/₄ = ⁴⁰/₆₀ and ¹²/₁₅ × ⁴/₄ = ⁴⁸/₆₀

 ∴ Five rational numbers between ⅔ and ⅘ are  ⁴¹/₆₀ ,⁴²/₆₀ , ⁴³/₆₀ , ⁴⁴/₆₀ , ⁴⁵/₆₀ 

(ii) – ³/₂ and ⁵/₃

L.C.M of 2 and 3 is 6.

∴ – ³/₂ × ³/₃= –⁹/₆ and ⁵/₃ ײ/₂=¹⁰/₆

Five rational numbers between –³/₂ and ⁵/₃ are – ⁸/₆ , – ⁷/₆ ,– ⁸/₆ , 0 , ⅙, 

(iii) ¼ and ½

L.C.M of 2 and 4 is 4.

∴ ¼ ×¹/₁=¼ and ½ × ²/₂=²/₄

Again, ¼ ×⁸/₈=⁸/₃₂ and ²/₄ ×⁸/₈=¹⁶/₃₂

∴ Five rational numbers between ¼ and ½ are ⁹/₃₂ ,¹⁰/₃₂ ,¹¹/₃₂ ,¹²/₃₂ ,¹³/₃₂ 

solution to class 8 maths chapter 1 exercise 1.2 (6)

6 Write five rational numbers greater than       –2

Solution to class 8 maths chapter 1 exercise 1.2(6)

Answer : Five rational numbers greater than –2 are – ³/₂ ,–1, – ½ , 0, 1 

solution to class 8 maths chapter 1 exercise 1.2 (7)

7 Find ten rational numbers between ⅗ and ¾

Solution to class 8 maths chapter 1 exercise 1.2(7)

Answer : L.C.M of 5 and 4 is 20.

∴ ⅗ ×⁴/₄=¹²/₂₀ and ¾ × ⁵/₅=¹⁵/₂₀

Again, ¹²/₂₀ ×⁸/₈ =⁹⁶/₁₆₀ and ¹⁵/₂₀ ×⁸/₈=¹²⁰/₁₆₀

∴ Ten rational numbers between ¼ and ½ are    ⁹⁷/₁₆₀ , ⁹⁸/₁₆₀ ,⁹⁹/₁₆₀ ,¹⁰⁰/₁₆₀ ,¹⁰¹/₁₆₀ , ¹⁰²/₁₆₀ ,      ¹⁰³/₁₆₀ ,¹⁰⁴/₁₆₀ ,¹⁰⁵/₁₆₀ ,¹⁰⁶/₁₆₀