Rational Numbers

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NCERT Solutions for Chapter 1 Rational Numbers Class 8 Maths

Book Solutions

1

Using appropriate properties to find:

(i) 

(ii) 

Answer

(i)
= [Using Associative property]
=  [Using distributive property]
= 
=
=
=

(ii)
=  [Using Associative property]
= [Using distributive property]
=  
= 
= -1/7 - 1/4
= (-4-7)/28
= -11/28
Exercise 1.1 Page Number 14

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

Answer

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/-9 IS 2/9   
(v) Additive inverse of 19/-6 IS 19/6  
Exercise 1.1 Page Number 14

3

Verify that -(x)=x for:
(i) x= 11/15 
(ii) x= - 13/17

Answer

(i) Putting x= 11/15 in -(-x) =x 
 -(-11/15) = 11/15
⇒ 11/15 = 11/15   
 L.H.S. = R.H.S.
Hence, verified.

(ii) Putting x= -13/17 in -(-x) =x,   
-{-(-13/17)} = -13/17 ⇒ -13/17 = -13/17
 L.H.S. = R.H.S.
Hence, verified.
Exercise 1.1 Page Number 14

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

Answer

We know that multiplicative inverse of a rational number 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 
Exercise 1.1 Page Number 14

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  

Answer

(i) 1 is the multiplicative identity.
(ii) commutativity property.
(iii) Multiplicative Inverse property.
Exercise 1.1 Page Number 14

6

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

Answer

The reciprocal of -7/16 is -16/7 
According to the question,
(6/13)× (-16/7)
= -96/91  
Exercise 1.1 Page Number 14

7

Tell what property allows you to compute
1/3 × (6×4/3) as (1/3 × 6) × 4/3  

Answer

By using associative property of multiplication,
we will compute as:
a ×(b×c) = (a×b)× c
Exercise 1.1 Page Number 14

8

Is 8/9 the multiplicative inverse of -1.1/8? Why or why not?

Answer

Since multiplicative inverse of a rational number a is (1/a), if a× 1/a = 1
Therefore, (8/9) × (-1.1/8) = 8/9 × -9/8 = -1  
But its product must be positive 1. 
Therefore, 8/9 is not the multiplicative inverse of (-1.1/8) 
Exercise 1.1 Page Number 14

9

Is 0.3 the multiplicative inverse of 3.1/3? Why or why not?

Answer

Since multiplicative inverse of a rational number a   is (1/a),  if a×1/a = 1 
Therefore,
0.3 × 3.1/3
= 3/10 ×10/3
= 1 
Therefore, Yes 0.3 is the multiplicative inverse of  3.1/3 
Exercise 1.1 Page Number 14

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.

Answer

(i) 0
(ii) 1 and
(iii) 0
Exercise 1.1 Page Number 15

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 _______________

Answer

(i) No
(ii) 1,  -1
(iii) -1/5  
(iv) x
(v) Rational Number
(vi) Positive
Exercise 1.1 Page Number 15

1

Represent these numbers on the number line:
(i)  7/4
(ii) -5/6

Answer

(i)  7/4 = 1.3/4 

Here, P 1.3/4 = 7/4   
(ii)  -5/6
 
Here, M =  -5/6
Exercise 1.2 Page Number 20

2

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

Answer

Here, B = -2/11   C = -5/11  and D = -9/11 
Exercise 1.2 Page Number 20

3

Write five rational numbers which are smaller than 2.

Answer

1/3 , 1/4, 1/2, -1/2, -1/5  and so on.
Exercise 1.2 Page Number 20

4

Find ten rational numbers between -2/5 and 1/2.

Answer

-2/5 and 1/2
Here, L.C.M. of 5 and 2 is 10.
∴ (-2/5)×(2/2) = -4/10 and (1/2) ×(5/5) = 5/10  
Again, (-4/10) × (2/2) = -8/20 and 5/10 ×2/2 = 10/20  
∴ Ten rational number between -2/5 and 1/2 are -7/20, -6/20, -5/20, -4/20, -3/20, -2/10, -1/20, 0, 1/20, 2/20
Exercise 1.2 Page Number 20

5

Find five rational numbers between:
(i) 2/3 and 4/5 
(ii) -3/2 and 5/3  
(iii) 1/4 and 1/2

Answer

(i) 2/3 and 4/5
L.C.M. of 3 and 5 is 15.
(2/3) × (5/5) = 10/15 and (4/5)×(3/3) = 12/15  
Again (10/15) × (4×4) = 40/60 and (12/15)×(4/4) = 48/60   
∴ Five rational numbers between 2/3 and 4/5 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) × (2/2) = 10/6 
Five rational numbers between -3/2 and 5/3 are -8/6, -7/6, 0, 1/6, 2/6

(iii) 1/4 and 1/2 
L.C.M. of 4 and 2 is 4.
(1/4) × (1/1) = 1/4 and (1/2) × (2/2) = 2/4  
Again (1/4) ×(8/8) = 8/32 and (2/4) × (8/8) = 16/32
Five rational numbers between 1/4 and 1/2 are 9/32, 10/32, 11/32, 12/32, 12/32.
Exercise 1.2 Page Number 20

6

 Write 5 rational numbers greater than -2. 

Answer

Five rational numbers greater than  -2 are:
-3/2, -1, -1/2, 0, 1/2 
[Other rational numbers may also be possible]
Exercise 1.2 Page Number 20

7

Find ten rational numbers between 3/5 and 3/4. 

Answer

3/5  and   3/4
L.C.M. of 5 and 4 is 20.
(3/5) × (4/4) = 12/20 and (3/4) × (5/5) = 15/20  
Again (12/20) × (8/8) = 96/160 and (15/20) × (8/8) = 120/160  
  Ten rational numbers between 3/5 and 3/4  are:
97/160, 98/160, 99/160, 100/160, 101/160, 102/160, 103/160, 104/160, 105/160, 106/160 
Exercise 1.2 Page Number 20

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