Rational Numbers: Exercise 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.

NCERT Solutions for Class 8 Maths Chapter 1- Rational Numbers is a resourceful article which was developed by GFG experts to aid students in answering questions they may have as they go through problems from the NCERT textbook.

This chapter contains the following topics:

• Rational Number
• Operations on Rational Numbers
  • Addition and subtraction of Rational Numbers
  • Multiplication and division of Rational Numbers
• Properties of Rational Numbers
  • Closure Property
  • Commutative Property
  • Associative Property
  • Distributive Property
  • Identity Property
  • Inverse Property
• Properties of Whole Numbers
• Properties of Integers
• Negative of a Number
• Reciprocal
• Representation of Rational Numbers on the Number Line
• Rational Numbers between Two Rational Numbers