CBSE NCERT Solutions for Class 8 Mathematics Chapter 1

Class- VIII-CBSE-Mathematics

Rational Numbers

CBSE NCERT Solutions for Class 8 Mathematics Chapter 1

Back of Chapter Questions

Exercise 1.1

1. Using appropriate properties find.

(i)

-2 ? 3 + 5 - 3 ? 1

35 2 56

(ii)

2 ? (-3) - 1 ? 3 + 1 ? 2

5

7

6 2 14 5

Solution:

(i) -2 ? 3 + 5 - 3 ? 1 = -2 ? 3 - 3 ? 1 + 5 (by commutativity)

3 5256 3 5562

2 -3 -3 1 5 =3?( 5 )+( 5 )?6+2

=

(-3)

5

(2

3

+

1)

6

+

5 2

(by

distributivity)

-3 5 5 = 5 ?6+2

-1 5 = 2 +2

-1 + 5 =2

4 =2

=2

Hence,

-

2 3

?

3 5

+

5 2

-

3 5

?

1 6

=

2

(ii) 2 ? (-3) - 1 ? 3 + 1 ? 2 = 2 ? (-3) + 2 ? 2 - 1 ? 3 (by commutativity)

5

7

6 2 14 5 5

7

14 5 6 2

=

2 5

(-3

7

+

1)

14

-

1 6

?

3 2

(by

distributivity)

2 -6 + 1 1 3 = 5 ( 14 ) - 6 ? 2

2 -5 1 1 = 5 (14 ) - 2 ? 2

2 (-5) 1 = 5 ? 14 - 4

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Class- VIII-CBSE-Mathematics

Rational Numbers

-2 1 = 14 - 4

-11 = 28

Hence,

2 5

?

(-3)

7

-

1 6

?

3 2

+

1 14

?

2 5

=

-11 28

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:

We know that for any number , + (-) = 0, So, - is called the additive inverse of .

(i) Additive inverse of 2 is -2

8 8

(ii)

Additive

inverse

of

-5 9

is

5 9

(iii)

-6 = 6

-5 5

Hence,

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

3. Verify that ? (? ) = for

(i)

x = 11

15

(ii) = - 13

17

Solution:

(i) The additive inverse of 11 is - 11

15

15

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Class- VIII-CBSE-Mathematics

Rational Numbers

Since

11 15

+

(-

11)

15

=

0

11

11

15 = - (- 15)

Hence verified.

(ii)

The

additive

inverse

of

-

13 17

is

13 17

Since

-

13 17

+

(13)

17

=

0

13

13

17 = - (- 17)

Hence verified.

4. Find the multiplicative inverse of the following.

(i) -13

(ii) - 13

19

(iii)

1 5

(iv) -5 ? -3

87

(v)

-1 ? -2

5

(vi) -1

Solution:

As we know that a rational number rational number if ? = 1

is

the

multiplicative

inverse

of

another

So,

=

Or

we

can

say

that

multiplicative

inverse

of

is

(i)

Multiplicative

inverse

of

-13

is

-1 13

Since -13 ? -1 = 1

13

(ii)

Multiplicative

inverse

of

-13 19

=

-19 13

Since -13 ? -19 = 1

19 13

(ii)

Multiplicative

inverse

of

1 5

is

5

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Class- VIII-CBSE-Mathematics

Rational Numbers

Since

1 5

?

5

=

1

(iv)

Multiplicative

inverse

of

-5 8

?

-3 7

=

15 56

is

56 15

Since

15 56

? 56 =

15

1

(v)

Multiplicative

inverse

of

-1

?

-2 5

=

2 5

is

5 2

Since 2 ? 5 = 1

5 2

(vi) Multiplicative inverse of -1 is -1

Since -1 ? -1 = 1

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

(i)

-4 ? 1 = 1 ? -4 = -4

5

55

(ii)

-13 ? -2 = -2 ? -13

17 7 7 17

(iii) -19 ? 29 = 1

29 -19

Solution:

(i) Multiplicative identity

(ii) Commutative property

(iii) Multiplicative inverse property

6. Multiply 6 by the reciprocal of -7.

13

16

Solution:

The

reciprocal

of

-7 16

is

-16 7

According to the question,

-16 6 -96 7 ? 13 = 91

Hence,

Multiplication

of

6 13

by

the

reciprocal

of

-7 16

7. Tell what property allows you to compute 1 ? (6 ? 4) as (1 ? 6) ? 4.

3

3

3

3

Solution:

By using associative property of multiplication,

? ( ? ) = ( ? ) ?

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Class- VIII-CBSE-Mathematics

Rational Numbers

8.

Is

8 9

the

multiplicative

inverse

of

-1

1 8

?

Why

or

why

not?

Solution:

-1

1 8

is

eqaul

to

-9 8

So,

multiplicative

inverse

of

-9 8

is

-8 9

Since, -9 ? -8 = 1

8

9

Hence,

8 9

is

not

the

multiplicative

inverse

of

-1

1 8

9.

Is

0.3

the

multiplicative

inverse

of

3

1 3

?

Why

or

why

not?

Solution:

3 0.3 = 10

As

we

know,

multiplicative

inverse

of

is

So multiplicative inverse of 3 is 10

10 3

Which is equal to 3 1

3

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) 0

(ii) 1, -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 _______.

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Class- VIII-CBSE-Mathematics

Rational Numbers

(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

,

where

0

is

(v) The product of two rational number is always a rational number.

(vi) The reciprocal of a positive rational number is positive.

Exercise 1.2

1. Represent these numbers on the number line

(i)

7 4

(ii)

-5 6

Solution:

(i)

7 = 13

4

4

(ii)

Let

M

=

-5 6

2.

Represent

-2 11

,

-5 11

,

-9 11

on

the

number

line.

Solution:

Let -2 = C,

11

-5 11

=

B

and

-9 11 = A

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Class- VIII-CBSE-Mathematics

Rational Numbers

3. Write five rational numbers which are smaller than 2.

Solution:

The rational numbers smaller than 2 are

(i)

1 3

(ii)

2 3

(iii)

5 3

(iv)

4 3

(v)

1 2

4.

Find

ten

rational

numbers

between

-2 5

and

12.

Solution:

Rational numbers are -2 and 1

5

2

Here, L.C.M of 5 and 2 is 10.

So,

-2 5

=

-2 5

?

2 2

=

-4 10

Also,

1 2

=

1 2

?

5 5

=

5 10

Again, -4 = -4 ? 2 = -8

10 10 2 20

and,

5 10

=

5 10

?

2 2

=

10 20

Hence, -2 = -8 and 1 = 10

5 20

2 20

Ten rational numbers between -2 and 1 are -7 , -6 , -5 , -4 , -3 , -2 , -1 , 0, 1 , 2

5

2 20 20 20 20 20 20 20 20 20

5. Find five rational numbers between

(i)

2 3

and

4 5

(ii)

-3 2

and

53.

(iii) 1 and 1

4

2

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Class- VIII-CBSE-Mathematics

Rational Numbers

Solution:

(i) 2 and 4

3

5

L.C.M. of 3 and 5 is 15

So,

2 3

=

2 3

?

5 5

=

10 15

and 4 = 4 ? 3 = 12

5 5 3 15

Again,

10 15

=

10 15

?

4 4

=

40 60

and

12 15

=

12 15

?

4 4

=

48 60

Hence,

2 3

=

40 60

and

4 5

=

48 60

Five rational numbers bewteen 2 and 4 are 41 , 42 , 43 , 44 , 45

3

5 60 60 60 60 60

(ii)

-3 2

and

5 3

LCM of 2 and 3 is 6

So, -3 = -3 ? 3 = -9

2 23 6

and

5 3

=

5 3

?

2 2

=

10 6

Hence,

-3 2

=

-

9 6

and

5 3

=

10 6

Five

rational

numbers

bewteen

-3 2

and

5 3

are

-2 6

,

-1 6

,

0,

1 6

,

2 6

(iii)

1 4

and

1 2

LCM of 4 and 2 is 4

So,

1 4

=

1 4

?

1 1

=

1 4

&

1 2

=

1 2

?

2 2

=

2 4

Again,

1 4

=

1 4

?

8 8

=

8 32

and

2 4

=

2 4

?

8 8

=

16 32

Hence,

1 4

=

8 32

and

1 2

=

16 32

Five

rational

numbers

bewteen

1 4

and

1 2

are

9 32

,

10 32

,

11 32

,

12 32

,

13.

32

6. Write five rational numbers greater than ? 2

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