NCERT Solutions for Class 8th: Ch 1 Rational Numbers Maths

NCERT Solutions for Class 8th: Ch 1 Rational Numbers Maths

Page No: 14

Exercise 1.1

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

Answer

(i) -2/3 × 3/5 + 5/2 - 3/5 × 1/6
= -2/3 × 3/5 - 3/5 × 1/6 + 5/2    (by commutativity)
= 3/5(-2/3 - 1/6) + 5/2
= 3/5{(-4 - 1)/6} + 5/2
= 3/5(-5/6) + 5/2    (by distributivity)
= -15/30 + 5/2
= -1/2 + 5/2
= 4/2 = 2

(ii) 2/5 × (-3/7) - 1/6 × 3/2 + 1/14 × 2/5
= 2/5 × (-3/7) + 1/14 × 2/5 - (1/6 × 3/2)    (by commutativity)
= 2/5(-3/7 + 1/14) - 1/4
= 2/5{(-6 + 1)/14} - 1/4    (by distributivity)
= 2/5(-5/14) - 1/4
= -1/7 - 1/4
= (-4-7)/28
= -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

Answer

(i) 2/8
Additive inverse = -2/8
(ii) -5/9
Additive inverse = 5/9
(iii) -6/-5 = 6/5
Additive inverse = -6/5
(iv) 2/-9 = -2/9
Additive inverse = 2/9
(v) 19/-6 = -19/6
Additive inverse = 19/6

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

Answer

(i) x = 11/15
The additive inverse of x = 11/15 is -x = -11/15 as 11/15 + (-11/15) = 0
The same equality 11/15 + (-11/15) = 0 , shows that the additive inverse of -11/15 is 11/15 or
-(-11/15) = 11/15 i.e. -(-x) = x

(ii) x = -13/17
The additive inverse of x = -13/17 is -x = 13/17 as (-13/17) + 13/17 = 0
The same equality 13/17 + (-13/17) = 0 , shows that the additive inverse of 13/17 is -13/17 or
-(13/17) = -13/17 i.e. -(-x) = x

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

The multiplicative inverse of a number is the reciprocal of that number.

(i) -13
Multiplicative inverse = -1/13
(ii) -13/19
Multiplicative inverse = -19/13
(iii) 1/5
Multiplicative inverse = 5
(iv) -5/8 × -3/7 = 15/56
Multiplicative inverse = 56/15
(v) -1 × -2/5 = 2/5
Multiplicative inverse = 5/2
(vi) -1
Multiplicative inverse = -1

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) -4/5 × 1 = 1 × -4/5 = -4/5
Here 1 is the multiplicative identity.
(ii) -13/17 × -2/7 = -2/7 × -13/17
Commutavity
(iii) -19/29 × 29/-19 = 1
Multiplicative inverse

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

Answer

Reciprocal of -7/16 = 16/-7
A/q,
6/13 × (Reciprocal of -7/16)
= 6/13 × 16/-7 = 96/-91 = -96/91

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

Answer

By the property of associativity.
8. Is 8/9 the multiplicative inverse of ? Why or why not?

Answer

If it will be the multiplicative inverse then their product will be 1.
= -7/8
A/q,
8/9 × -7/8 = -7/9 ≠ 1
Hence, 8/9 is not the multiplicative inverse.

9. Is 0.3 the multiplicative inverse of ? Why or why not?

Answer

If it will be the multiplicative inverse then their product will be 1.
= 10/3
also, 0.3 = 3/10
A/q,
3/10 × 10/3 = 1
Hence, 0.3 is the multiplicative inverse.

Page No: 15

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 is the rational number that does not have a reciprocal.

(ii) 1 and -1 are the rational numbers that are equal to their reciprocals.

(iii) 0 is the rational number that is equal to its negative.

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

Page No: 20

Exercise 1.2

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

Answer

(i) 7/4 on the number line.
Divide line between two natural number in 4 parts. Thus, the rational number 7/4 lies at a distance of 7 points from 0 towards positive number line.

(ii) -5/6 on the number line.
Divide line between two natural number in 6 parts. Thus, the rational number -5/6 lies at a distance of 5 points from 0 towards negative number line.
 

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

Answer

-2/11, -5/11, -9/11 on the number line.
Divide line between two natural number in 11 parts. Thus, the rational number -2/11, -5/11, -9/11 lie at a distance of 2, 5, 9 points from 0 towards negative number line respectively.


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

Answer

2 can be written as 10/5.
Thus, 5 natural numbers smaller than 2 are:
9/5, 8/5, 7/5, 6/5 and 5/5

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

Answer

The numbers -2/5 and 1/2 can be written as -8/20 and 10/20
Thus, ten rational numbers between -2/5 and 1/2 are:
-7/20, -6/20, -5/20, -4/20, -3/20, -2/20, -1/20, 0, 1/20 and 2/20

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) Five rational numbers between 2/3 and 4/5
The numbers 2/3 and 4/5 can be written as 30/45 and 36/45
Thus, five rational numbers are:
31/45, 32/45, 33/45, 34/45 and 35/45

(ii) Five rational numbers between -3/2 and 5/3
The numbers -3/2 and 5/3 can be written as -9/6 and 10/6
Thus, five rational numbers are:
-8/6, -5/6, -2/6, 0 and 2/6

(iii) Five rational numbers between 1/4 and 1/2
The numbers 1/4 and 1/2 can be written as 7/28 and 14/28
Thus, five rational numbers are:
8/28, 9/28, 10/28, 11/28 and 12/28

6. Write five rational numbers greater than -2.

Answer

-2 can be written as -16/8.
Five rational numbers greater than -2 are:
-15/8, -14/8, -13/8, -12/8 and -11/8

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

Answer

The numbers 3/5 and 3/4 can be written as 48/80 and 60/80
Thus, ten rational numbers between 3/5 and 3/4 are:
49/80, 50/80, 51/80, 52/80, 53/80, 54/80, 55/80, 56/80, 57/80 and 58/80.

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