Class 8 NCERT Solutions- Chapter 1 Rational Numbers – 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.