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**Joy of Mathematics Class 8 Solutions Chapter 2 Rational Numbers**

Welcome to NCTB Solutions. Here with this post we are going to help 8th class students for the Solutions of Joy of Mathematics Class 8 Book, Chapter 2 Rational Numbers. Here students can easily find step by step solutions of all the problems for Rational Numbers. Exercise wise proper solutions for every problems. All the problem are solved with easily understandable methods so that all the students can understand easily. Here students will find solutions for Exercise 2.1 and 2.2

**Rational Numbers Exercise 2.1 Solution**

**Question no – (1) **

**Solution :**

**(a)** Given, – 35/25

= – 5 × 7/5 × 5

= – 7/5

**(b)** Given, 32/- 40

= 2 × 2 × 2 × 2 × 2/- 2 × 5 × 2 × 2

= – 4/5

**(c)** Given, – 225/465

= – 5 × 5 × 3 × 3/5 × 3 × 31

= – 15/31

**(d)** Given, – 484/- 572

= 2 × 2 × 11 × 11 /2 × 2 × 13 × 11

= 11/13

**Question no – (2) **

**Solution :**

**(a)** numerator 12

= 3 × 4/7 × 4

= 12/28

**(b)** Numerator – 27

= 3 × (- 9)/7 × (- 9)

= – 27/- 63

**(c)** Denominator 91

= 3 × 13/7 × 13

= 39/91

**(d)** Denominator – 63

= 3 × (- 9)/7 × – (9)

= – 27/- 63

**Question no – (3) **

**Solution :**

**(a)** **0 and 1**

= A rational number lying between

0 + 1/2 = 1/2

= 1/2 + 1/2 = 1 + 2/2/2 = 3/4

= 1/2 (1/2 + 3/4) = 1/2 (2 + 3/4) = 1/2 × 5/4 = 8/5

= 1/2 (3/4 + 5/8) = 1/2 (6 + 5/8) = 1/2 × 11/8 = 11/16

= 1/2 (5/8 + 11/16) = 1/2 (10 + 11/16) = 1/2 × 21/16

= 21/32

**∴** five rational numbers between 0 and 1

1/2, 3/4, 5/8, 11/16, 21/32

**(b) 3 and 4**

= A rational number lying between 3 and 4 = 3 + 4/2 = 7/2

= 1/2 (4 + 7/2) = 1/2

= (8 + 7/2) = 1/2 × 15/2

= 15/4

= 1/2 (7/5 + 15/4) = 1/2 (14 + 15/4) = 1/2 × 29/4

= 29/8

= 1/2 (15/4 + 29/8) = 1/2 (30 +39/8)

= 1/2 × 69/8

= 69/16

= 1/2 (29/8 + 69/16) = 1/2 (58 + 69/16)

= 1/2 × 127/16 = 127/32

**∴** Five rational number between 3 and 4 are 7/2, 15/4, 29/8, 69/16, 127/32

**(c) – 3 and – 3.5**

= A rational number lying between – 3 and – 3.5

= – 3 + (- 3.5)/2

= – 3 – 3.5/2

= – 6.5/2

= 1/2 (- 3.5 – 6.5/2) = 1/2 (-7.0 – 6.5/2) = 1/2 (- 13.5/2)

= – 13.5/4

= 1/2 (- 6.5/2 – 13.5/4) = 1/2 (- 13 – 13.5/4) = – 26.5/8

= 1/2 (- 13.5/4 – 26.5/8) = 1/2 (- 27 – 26.5/8) = – 53.5/16

= 1/2 (- 26.5/8 – 53.5/16) = 1/2 (- 53 – 53.5/16) = – 106.5/32

**∴** Five rational number lying between – 3 and – 3.5 are – 6.5/2, – 13.5/4, – 26.5/8, – 53.5/16, – 106.5/32

**(d) 1/2 and 3/4**

= A rational number lying between 1/2 and 3/4

= 1/2 (1/2 + 3/4)

= 1/2 (2 + 3/4)

= 1/2 × 5/4 = 5/8

= 1/2 (3/4 + 5/8) = 1/2 (6 + 5/8) = 1/2 × 11/8 = 11/16

= 1/2 (5/8 + 11/16) = 1/2 (10 + 11/16) = 1/2 × 21/16 = 21/32

= 1/2 (11/16 + 21/32) = 1/2 (22 + 21/32) = 1/2 × 43/32 = 43/64

∴ Five rational number between 1/2 and 3/4 are, 5/8, 11/16, 21/32, 43/64

**(e) – 1/2 and 1/2**

= A rational number lying between – 1/2 and 1/2 = 1/2 (- 1/2 + 1/2)

= 0

**∴** 1/2 (0 + 1/2) = 1/4

= 1/2 (1/2 + 1/4) = 1/2 (2 + 1/4) = 3/8

= 1/2 (1/4 + 3/8) = 1/2 (2 + 3/8) = 1/2 × 5/8 = 5/16

= 1/2 (3/8 + 5/16) = 1/2 (6 + 5/16) = 1/2 × 11/16 = 11/32

= 1/2 (5/16 + 11/32) = 1/2 (10 + 11/32) = 21/64

**∴** Five rational number between – 1/2 and 1/2 are, 1/4, 3/8, 5/16, 11/32, 21/64

**(f) – 3/5 and – 1/3**

A rational number between – 3/5 and – 1/3

= 1/2 (- 3/5 – 1/3) = 1/2 (- 9 – 5/15) = 1/2 × (- 14/15) = – 14/30

**∴** 1/2 (- 1/3 – 14/30) = 1/2 (- 10 – 14/30) = 1/2 × (- 24/30) = – 24/60

**∴** 1/2 (- 14/30 – 24/60) = 1/2 (- 28 – 24/60) = 1/2 (- 52/60) – 52/120

**∴** 1/2 (- 3/5 – 14/30) 1/2 (- 18 – 14/30) = 1/2 (- 32/30) = – 32/60

**∴** 1/2 (- 14/30 – 32/60) = 1/2 (- 28 – 32/60) = 1/2 (- 60/60) = – 1/2

**∴** Five rational number between – 3/5 and – 1/3

are, – 14/30, – 24/60, – 52/120, – 32/60, – 1/2

**Question no – (4)**

**Solution :**

**(a) 3/8**

= Three rational numbers equivalent to 3/8 are 3 × 2/8 × 2 = 6/16, 3 × 4/8 × 4 = 12/32 3 × 3/8 × 3 = 9/24

**(b) – 4/15**

= Three rational numbers equivalent to – 4/15 are – 4 × 2/15 × 2 = – 8/30, – 4 × 3/15 × 3 = – 12/45 – 4 ×4/15 × 4 = – 16/60

**(c) – 3/- 5**

= Three rational numbers equivalent to – 3/- 5 are – 3 × 2/- 5 × 2 = – 6/- 10, – 3 × 3/- 5 × 3 = – 9/- 15 – 3 × 4/- 5 × 4 = – 12/- 20

**(d) – 2/9**

– Three rational numbers equivalent to – 2/9 are

= – 2 × 2/9 × 2 = – 4/18,

= – 2 × 4 /9 × 4 = – 8/36

= – 2 × 3/9 × 3 = – 6/27

**(e) 3/- 7**

= Three rational numbers equivalent to 3/-7 are,

3 × 2/- 7 × 2 = 6/- 14, 3 × 3/- 7 × 3 = 9/-21, 3 × 4/-7 × 4 = 12/-28

**(f) – 6/5**

=> Three rational numbers equivalent to – 6/5 are

– 6 × 2/5 × 2 = – 12/10, – 6 × 3/5 × 3 = – 18/15, – 6 × 4/5 × 4 = – 24/20

**Question no – (5) **

**Solution :**

**(a) 2/3, 3/2**

= This pairs are not equivalent rational number

**(b) – 3/8, 6/- 16**

= – 3 × 2/8 × 2

= – 6/16

= This pairs are equivalent rational number

**(c) 10/-14, – 40/56**

= 10 × 4/- 14 × 4

= 40/- 56

= This pairs are equivalent rational number

**(d) – 1/4, 4/16**

= – 1 × 4/4 × 4

= – 4/16

= This pairs are equivalent rational number

**(e) – 45/54, 15/- 18**

= – 45 × 1/3/54 × 1/3 = 15/- 18

= This pairs are equivalent rational number

**(f) 12/48, – 4/16**

= This is not equivalent rational number

**Question no – (6)**

**Solution :**

**(a) 4/5, 3/10, 7/15**

= L.C.M. of 5, 10, 15 = 30

= 4 × 6/5 × 6, 3 × 3/10 × 3, 7 × 2/15 × 2

= 24/30, 9/30, 14/30

**∴** Ascending order = 3/10 < 7/15 < 4/5

**(b) – 5/2, 3/8, 7/4**

= L.C.M. of 2, 8, 4 = 8

= – 5 × 4/2 × 4, = 3 × 1/8 × 1, 7 × 2/4 × 2

= – 20/8, 3/8, 14/8

**∴** Ascending order = – 5/2 < 3/8 < 7/4

**(c) 7/8, 2/3, 11/12**

= LCM of 8, 3, 12, = 24

= 7 × 3/8 × 3 = 2 × 8/3 × 8, 11 × 2/12 × 2

= 21/24, 16/24, 22/24

**∴** Ascending order = 2/3 < 7/8 < 11/12

**(d) – 1/3, – 3/8, 23/24**

= L.C.M. of 3, 8, 24 = 24

**∴ **1 × 8/3 × 8, – 3 × 3/8 × 3, 23 × 1/24 × 1

= – 8/24, – 9/24, 23/24

**∴ **Ascending order = – 3/8 < – 1/3 < 23/24

**(e) – 5/6, 11/12, 2/3**

= L.C.M. of 6, 12, 3, = 12

= – 5 × 2/6 × 2 = 11 × 1/12 × 1, 2 × 4/3 × 4

= – 10/12, 11/12, 8/12

**∴** Ascending order = 5/6 < 2/3 < 11/12

**(f) 1/2, 3/4, 2/3**

= L.C.M. of 2, 4, 3, = 12

**∴** 1 × 6/2 × 6, 3 × 3/4 × 3, 2 × 4/3 × 4

= 6/12, 9/12, 8/12

**∴** Ascending order = 1/2 < 2/3 < 3/4

**Question no – (7) **

**Solution : **

**(a) 2/3, 3/4, 5/6**

= L.C.M. of 2, 4, 8 = 12

= 2 × 4/3 × 4, 3 × 3/4 × 3, 5 × 2/6 × 2

= 8/12, 9/12, 10/12

**∴** Descending order = 5/6 > 3/4 > 2/3

**(b) – 2/3, – 3/4, 5/- 6**

= L.C.M. of 3, 4, – 6 = 12

= – 2 × 4/3 × 4, – 3 × 3/4 × 3, 5 × 2/- 6 × 2

= – 8/1, – 9/12, 10/- 12

**∴** Descending order = – 2/3 > – 3/4 > 5/- 6

**(c) – 1/2, 4/- 5, 2/3**

= L.C.M. of 2, – 5, 3 = 30

= – 1 × 15/2 × 15, 4 × 6/- 5 × 6, 2 × 10/3 × 10

= – 15/30, 24/- 30, 20/30, 20/30

**∴** Descending order = 2/3 > – 1/2 > – 4/5

**(d) – 3/4, – 3/10, 8/- 3**

= L.C.M. of 4, 10, – 3 = 60

= – 3 × 15/4 × 15, – 3 × 6/10 × 6, 8 × 20/- 3 × 20

= – 45/60, – 18/60. 10/- 60

**∴** Descending order = – 3/10 > – 3/4 > 8/- 3

**(e) 4/7, 2/5, 5/9**

= L.C.M. of 7, 5, 9, = 315

= 4 × 45/7 × 45, 2 × 63/5 × 63, 5 × 35/9 × 35

= 180/315, 126/315, 175/315

**∴** Descending order = 4/7 > 5/9 > 2/5

**(f) – 3/10, 7/15, – 3/5**

= L.C.M. of 10, 15, 5, = 30

= – 3 × 3/10 × 3, 7 × 2/15 × 2, – 3 × 6/5 × 6

= – 9/30, 14/30, – 18/30

**∴** Descending order = 7/15 > – 3/10 > -3/5

**Question no – (8) **

**Solution : **

**(a) – 3/4**

= Since in – 3/4 denominators is 4 we divide line between 0 and – 1 into 4 equal

**(b) 3/4**

Since, in 3/4 denominator is 4

We divide line between 0 and 1 into 4 equal parts.

**(c) 1/2**

= Since in 1/2 denominator is 2

We divide line between 0 and 1 into 2 equal parts.

**(d) 7/8**

= Since in 7/8 denominator is 8

We divide line between 0 and 1 into 8 equal parts.

**(e) 1/3**

In 1/2 denominator is = 3

We divide line between a and 1 into 2 equal parts.

**(f) – 1/3**

= In – 1/3 denominator is = – 3

**(g) – 7/3**

**Rational Numbers Exercise 2.2 Solution**

**Question no – (1)**

**Solution :**

**(a)** 2/7 and 6/7

= 2/7 + 6/7

= 2 + 6/7

= 8/7

**(b)** 2/11 and 7/11

= 2/11 + 7/11

= 2 + 7/11

= 9/11

**(c)** 2/5 and – 1/5

= 2/5 – 1/5

= 2 – 1/5

= 1/5

**(d)** 5/8 and 1/- 8

= 5/8 – 1/8

= 5 – 1/8

= 4/8

= 1/2

**Question no – (2)**

**Solution :**

**(a)** 1/3 + (- 4/3)

= 1 + (- 4/3)

= 1 + (- 4)/3

= – 3 /3 [∵ 1 + (- 4) = – 3]

= – 1

**(b)** 3/5 + 4/5

= 3/5 + (- 4/5)

= 3 + (- 4)/5

= – 1/5 [∵ 3 + (- 4) = – 1]

**(c)** 2/9 + – 4/- 9

= 2/9 + 4/9

= 2 + 4/9

= 6/9

= 2/3

**(d)** – 9/13 + 5/- 13

= – 9/13 + (- 5/13)

= – 9 + (- 5)/13

= – 14/13 [∵ – 9 + (- 5) = – 14]

**Question no – (3)**

**Solution :**

**(a)** 1/2 + 1/3 + – 2/5

= 15 + 10 + (- 12)/30 [∵ L.C.M. of 2, 3, and 5 is 30]

= 25 + (- 12)/30

= 13/30

**(b)** 1/3 + 3/4 + – 4/5

= 20 + 45 + (- 24)/60 [∵ LCM of 3, 4, and 5 is 60]

= 65 + (- 24)/60

= 41/60

**(c)** 2/3 +2/5 + – 7/15

= 10 + 6 + (- 7)/15 [∵ LCM of 3, 5 and 15 is 15]

= 6 + (- 7)/15

= 9/15

= 3/5

**(5) Subtract :**

**Solution :**

**(a)** – 7/15 from – 6/15

= – 6/15 – (- 7/15)

= – 6 + 7/15

= 1/15

**(b)** 5/13 from 15/13

= 15/13 – 5/13

= 15 – 5/13

= 10/13

**(c)** – 7/8 from 3/16

= 3/16 + 7/8

= 3 + 14/16

= 17/16

**(d)** 6/11 from 29/33

= 29/33 – 6/11

= 29 – 18/33

= 11/33

= 1/3

**Question no – (6)**

**Solution :**

= ‘0’ is the rational numbers which when added 40 itself gives the same number.

**Question no – (7)**

**Solution :**

= 2/3, – 2/5, – 16/7, – 5/6 negatives are

= – 2/3, – 2/5, 16/7, 5/6

**(8) Simplify :**

**Solution :**

**(a)** 9/4 – 3/5

= 45 – 12/20

= 33/20

**(b)** 5/3 – 4/7

= 35 – 12/21

= 23/21

**(c)** 5/7 – 4/21

= 15 – 4/21

= 11/21

**(d)** 7/33 – 5/11

= 7 – 15/33

= – 8/33

**Question no – (9)**

**Solution :**

= 7/3 + required number = – 4

= required number = 7/3 – 4

= 7 – 12/3

= – 5/3

**Question no – (10)**

**Solution :**

= – 7/3 + required number = – 5

= required number – 5 – 7/12

= – 60 – 7/12

= – 67/12

**Question no – (11)**

**Solution :**

= – 1 + required number = 8/9

= required number = 1 + 8/9

= 9 + 8/9

= 17/9

**Question no – (12)**

**Solution :**

= – 3/5 – required number = 5/6

= – required number = 5/6 + 3/5

= 25 + 18/30 [∵ LCM of 6, 5 = 30]

= 43/30

= required number = – 43/30

**Question no – (13) **

**Solution :**

**(a)** 4/7 × 3/10

= 6/35

**(b)** 11/18 × – 9/65

= – 11/130

**(c)** – 5/7 × 35/36

= – 25/36

**(d)** – 4/5 × – 25/12

= 5/3

**(e)** – 9/11 × 33/(- 35)

= 27/35

**(f)** – 6/25 × 15/32

= – 9/80

**Question no – (14) **

**Solution :**

**(a)** – 4/9 × 27/25

= – 12/25

**(b)** 7/6 × – 24/35

= – 4/5

**(c)** 7/18 × – 12/35

= – 2/15

**(d)** 8/13 × 65/(- 24)

= – 5/3

**(e)** 1/- 3 × – 27/66

= – 11/3 × – 27/66

= 3/2

**(f)** 7/- 36 × 72/21

= – 2/3

**Question no – (15) **

**Solution :**

**(a)** 5/7 × 49/75 × 45/28

= 7/8

**(b)** – 9/18 × 12/- 45 × 45/- 8

= – 9/18 × – 12/45 × – 45/8

= – 3/4

**(c)** – 111/330 × 55/660 ×- 48/5

= 4/15

**(d)** – 2/3 × 3/- 4 × – 4/- 5

= – 2/3 × – 3/4 × 4/5

= 2/5

**Question no – (16)**

**Solution :**

**(a)** Reciprocal of 4/7 is

= 7/4

**(b)** Reciprocal of – 1/10 is

= – 10/1

= – 10

**(c)** Reciprocal of 1 is

= 1/1

= 1

**(d)** Reciprocal of – 1 is

= – 1/1

= – 1

**(e)** Reciprocal of 4 is

= 1/4

**(f)** Reciprocal of 0 is

= 0/0

= 0

**Question no – (17)**

**Solution :**

**(a)** 5 ÷ 1/5

= 5 × 5/1

= 25

**(b)** 6/5 ÷ 36/25

= 6/5 × 25/26

= 5/6

**(c)** – 3/4 ÷ 1/12

= – 3/4 × 12/1

= – 9

**(d)** 3/8 ÷ 21/16

= 3/8 × 16/21

= 2/7

**Question no – (18)**

**Solution :**

= Product of two rational numbers = 15

**∴** required number x one number = 15

= required number × (- 5/2) = 15

= required number = 15 ÷ (- 5/2)

= 15 ÷ (- 2/5)

= – 6

**Question no – (19)**

**Solution :**

Product of two rational number = – 24/9

**∴** required number × another number = – 24/9

= required number × (- 4/3) = – 24/9

= required number = – 24/9 ÷ (- 4/3)

= – 24/9 × – 3/- 4

= 2

**Question no – (20)**

**Solution :**

= required number x one number = 9/35

= required number × (- 3/5) = 9/35

= required number = 9/35 ÷ (- 3/5)

= 9/35 × – 5/3

= – 3/7

**Question no – (21)**

**Solution :**

= Price of 12 pens is = 45

Price of 1 pens is

= 45/12

= 15/4

**Question no – (22)**

**Solution :**

The cost of 2 3/4 metres of cloths is = 638

**∴** Cost of cloth per metre

= 683 ÷ 11/4

= 683 × 4/11

**Question no – (23)**

**Solution :**

Length of the rope = 40 metres and 50 cm

= 40 m + 50/100m

= (40 + 1/2) metres

= 80 + 1/2 metres

= 8 1/2 metres

If each piece is 2 1/4 = 9/4 meter long then pieces are

= 81/2 ÷ 9/4

= 81/2 × 4/9

= 18

**Question no – (24)**

**Solution :**

**(a)** 2/3 + 4/5

= 10 + 12/15

= 22/15

**∴** This is a rational number.

**(b)** L.H.S

3/5 + 5/7

= 21 + 25/25

= 46/35

R.H.S, 5/7 + 3/5

= 25 + 21/35

= 46/35

**∴ **L.H.S = R.H.S …..(Proved)

**(c)** L.H.S

= (- 3/7 + 5/2) + 5/6

= (6 + 35/14) + 5/6

= 29/14 + 5/6

= 87 + 35/42

= 122/42

= 61/21

R.H.S,

= – 3/7 + (5/2 + 5/6)

= – 3/7 + 915 + 5/6)

= – 3/7 + 20/6

= 3/7 + 20/6

= – 18 + 140/42

= 112/42 = 61/21

**∴** L.H.S = R.H.S …..(Proved)

**(d)** L.H.S = – 2/3 + 2/3

= – 2 + 2/3

= 0/3

= 0

L.H.S = R.H.S …..(Proved)

**(e)** L.H.S

= 3/4 + (- 3/4)

= 3 + 3/4

= 0/4

= 0 = R.H.S ……(Proved)

**(f)** L.H.S

= 6/7 + 0

= 6 + 0/7

= 6/7 = R.H.S (Proved)

**Question no – (25)**

**Solution : **

Here, a = – 3/7 and b = 2/3

L.H.S a + b

= – 3/7 + 2/3

= – 9 + 14/21

= 6/21

R.H.S,

b + a

= 2/3 + (- 3/7)

= 14 (- 9)/21

= 5/21

**Question no – (26) **

**Solution :**

**(a)** L.H.S (3/7 × 5/9) × 2/5

= 5/21 × 2/5

= 2/21

R.H.S,

= 3/7 × (5/9 × 2/5)

= 3/7 × 2/9

= 2/21

**∴** L.H.S = R.H.S (Proved)

**(b)** L.H.S (- 7/8) [(- 13/5) + (- 7/10)]

= (- 7/8) [- 26 + (- 7)/10]

= – 7/8 × – 33/10

= 231/80

R.H.S, (- 7/8) × (- 13/5) + (- 7/8) × (- 7/10)

= 91/40 + 49/80

= 182 + 49/80

= 231/80

L.H.S = R.H.S (Proved)

**(c) **L.H.S

= 7/2 × 3/5

= 21/10

R.H.S

= 3/5 × 7/2

= 21/10

L.H.S = R.H.S (Proved)

**(d)** L.H.S

= (- 4/5) × (- 2/3)

= 8/15

R.H.S

= (- 2/3) × (- 4/5)

= 8/15

∴ L.H.S = R,H,S …..(Proved)

**Question no – (27)**

**Solution :**

**(a) L.H.S.**

(a × b × c)

= 1/5 × 1/3 × (- 2/5) [Putting the value of a, b, c]

= 1/5 × – 2/15

= – 2/75

**∴** L.H.S = R.H.S (Proved)

**(b) L.H.S**

a(b + c)

Putting the value of a, b, c

= 1/5 (1/3 + – 2/5)

= 1/5 (5 + (-6)/15)

= 1/5 × (-1/15)

= -1/75

R.H.S.

a × b + a × c

Putting the value of a, b, c

1/5 × 1/3 × + 1/5 × – 2/5

= 1/15 + -2/25

= 5 + (-6)/75

= – 1/75

**∴** L.H.S. = R.H.S ….(Proved)

**(c)** L.H.S

a(b – c)

= 1/5 (1/3 + 2/5) Putting the value of a, b, c

= 1/5 (5 + 6/15)

= 1/5 × 11/15

= 11/75

R.H.S

a × b – a × c

Putting the value of a, b,

1/5 × 1/3 – 1/5 (- 2/5)

= 1/15 + 2/25

= 5 + 6/75

= 11/75

**∴** L.H.S. = R.H.S. …..(Proved)

**(d)** L.H.S.

(a – b) × c

putting the value of a, b, c

(1/5 – 1/3) × – 2/5

= (3 – 5/15) × -2/5

= -2/15 ** **× -2/5

= 4/75

R.H.S

a × c – b × c

Putting the value of a, b, c

1/5 -2/5 – 1/3 × – 2/5

= – 2/25 + 2/25

= – 6 + 10/75

= 4/75

∴ L.H.S. = R.H.S …..(Proved)

**Previous Chapter Solution : **