Class 8 RD Sharma – Chapter 1 Rational Numbers – Exercise 1.5

Question 1. Multiply:

(i) 7/11 by 5/4

Solution:

7/11 × 5/4

Multiplying numerator with numerator of other rational number and denominator with denominator

= (7 × 5)/(11 × 4)

= 35/44

(ii) 5/7 by -3/4

Solution:

(5/7) × (-3/4)  

Multiplying numerator with numerator of other rational number and denominator with denominator

= (5 × -3)/(7 × 4)

= -15/28

(iii) -2/9 by 5/11

Solution:

-2/9 × 5/11

Multiplying numerator with numerator of other rational number and denominator with denominator

= (-2 × 5)/(9 × 11)

= -10/99

(iv) -3//17 by -5/-4

Solution:

-3/17 × 5/4

Multiplying numerator with numerator of other rational number and denominator with denominator

= (-3 × 5)/(17 × 4)

= -15/68

(v) 9/-7 by 36/-11

Solution:

(9/-7) × (36/-11)

Multiplying numerator with numerator of other rational number and denominator with denominator

= (-9 × -36)/(7 × 11)

= 324/77

(vi) -11/13 by -21/7

Solution:

Multiplying numerator with numerator of other rational number and denominator with denominator

= (-11 × -21)/(13 × 7)

= 231/91

(vii) -3/5 by -4/7

Solution:

Multiplying numerator with numerator of other rational number and denominator with denominator

= (-3 × -4)/(5 × 7)

= 12/35

(viii) -15/11 by 7

Solution:

-15/11×7/1

Multiplying numerator with numerator of other rational number and denominator with denominator

=(-15×7)/(11×1)

=-105/11

Question 2. Multiply

(i) -5/17 by 51/-60

Solution:

(-5/17) × (51/-60)

Multiplying numerator with numerator of other rational number and denominator with denominator

= (-5 × -51)/(17 × 60)

Common factor of 5 and 60

= 51/17 × 12

51 and 12 have 3 as common factor

= 17/17 × 4

= 1/4

(ii) -6/11 by -55/36

Solution:

(-6/11) × (-55/36)

Multiplying numerator with numerator of other rational number and denominator with denominator

= (-6 × -55)/(11 × 36)

Common factor of 6 and 36, 55 and 11

= 5/6

(iii) -8/25 by -5/16

Solution:

(-8/25) × (-5/16)

Multiplying numerator with numerator of other rational number and denominator with denominator

= (-8 × -5)/(25 × 16)

Common factor of 8 and 16, 5 and 25

= 1/5 × 2

= 1/10

(iv) 6/7 by -49/36

Solution:

(6/7) × (-49/36) 

Multiplying numerator with numerator of other rational number and denominator with denominator

= (6 × -49)/(7 × 36)

Common factor of 6 and 36, 49 and 7

= -7/6

(v) 8/-9 by -7/-16

Solution:

-8/9 × 7/16

Multiplying numerator with numerator of other rational number and denominator with denominator

= (-8 × 7)/(9 × 16)

Common factor of 8 and 16

= -7/9 × 2

= -7/18

(vi) -8/9 by 3/64

Solution:

-8/9 × 3/64

Multiplying numerator with numerator of other rational number and denominator with denominator

= (-8 × 3)/(9 × 64)

Common factor of 8 and 64, 3 and 9

= -1/3 × 8

= -1/24

Question 3. Simplify each of the following and express the result as a rational number in standard form:

(i) (-16/21) × (14/5)

Solution:

Multiplying numerator with numerator of other rational number and denominator with denominator

= (-16 × 14)/(21 × 5)

Common factor of 21 and 14

= (-16 × 2)/(3 × 5)

= -32/15

(ii) (7/6) × (-3/28)

Solution:

Multiplying numerator with numerator of other rational number and denominator with denominator

= (7 × -3)/(6 × 28)

Common factor of 7, 28, 3 and 6

= -1/2 × 4

= -1/8

(iii) (-19/36) × 16

Solution:

Multiplying numerator with numerator of other rational number and denominator with denominator

= (-19 × 16)/(36 × 1)

Common factor of 16 and 36

= (-19×4)/(9 × 1)

=-76/9

(iv) (-13/9) × (27/-26)

Solution:

Multiplying numerator with numerator of other rational number and denominator with denominator

=(-13× -27)/(9×26)

Common factor of 27 and 9 , 13 and 26

=(-1 × -3)/(2)

=3/2

(v) (-9/16) × (-64/-27)

Solution:

Multiplying numerator with numerator of other rational number and denominator with denominator

= (-9 × 64) / (16 × 27)

Common factor of 9 and 27, 64 and 16

= (-1 × 4) / (1 × 3)

= -4/3

(vi) (-50/7) × (14/3)

Solution:

Multiplying numerator with numerator of other rational number and denominator with denominator

= (-50 × 14)/(7 × 3)

Common factor of 14 and 7

= (-50 × 2)/(3)

= -100/3

(vii) (-11/9) × (-81/-88)

Solution:

Multiplying numerator with numerator of other rational number and denominator with denominator

= (-11 × 81)/(9 × 88)

Common factor of 1, 88, 9 and 81

= (-1 × 9)/(1 × 8)

= -9/8 

(viii) (-5/9) × (72/-25)

Solution:

Multiplying numerator with numerator of other rational number and denominator with denominator

= (-5 * -72)/(9 × 25)

Common factor of 5 and 25, 9 and 72

= (-1 × -8)/(1 × 5)

= 8/5

4. Simplify:

(i) ((25/8) × (2/5)) – ((3/5) × (-10/9))

Solution:

= ((25 × 2)/(8 × 5)) – ((3 × -10)/(5 × 9))

= 50/40 – (-30)/45

= 5/4 + 2/3

LCM of 4 and 3 is 12

= (5 × 3 + 2 × 4)/12

= (15 + 8)/12

= 23/12

(ii) ((1/2) × (1/4)) + ((1/2) × 6)

Solution:

= ((1 × 1)/(2 × 4)) + ((1 × 6)/(1 × 2))

= 1/8 + 3/1

LCM of 8 and 1 is 8

= (1 × 1 + 3 × 8)/8

= (1 + 24)/8

= 25/8

(iii) (-5 × (2/15)) – (-6 × (2/9))

Solution:

= ((-5 × 2)/(15 × 1)) – ((-6 × 2)/(1 × 9))

= (-10/15) – (-12/9)

Common factor of 10 and 15, 12 and 9

= -2/5 + 4/3

LCM of 5 and 3 is 15

= (-2 × 3 + 4 × 4)/15

= (-6 + 16)/15

= 10/15

Common factor of 10 and 15

= 2/3

(iv) ((-9/4) × (5/3)) + ((13/2) × (5/6))

Solution:

= (-9 × 5)/(4 × 3) + (13 × 5)/(2 × 6)

Common factor of 9 and 3

= (-45/12) + (65/12)

As denominators are same

= (-45 + 65)/12

= (20)/12

Common factor of 20 and 12 

= 5/3

(v) ((-4/3) × (12/-5)) + ((3/7) × (21/15))

Solution:

= (-4 * -12)/(3 × 5) + ((3 × 21)/(7 × 15))

= 48/15 + 3/5 (Common factor 3 and 15, 21 and 7)

LCM of 15 and 5 is 15

= (48 + 3 × 3)/15

= (48 + 9)/15

= 57/15

Common factor of 57 and 15

= 19/5

(vi) ((13/5) × (8/3)) – ((-5/2) × (11/3))

Solution:

= (13 × 8)/(5 × 3) – ((-5 × 11)/(2 × 3))

= 104/15 – 55/6

LCM of 15 and 6 is 3 × 5 × 2 = 30

= (104 × 2 + 55 × 5)/30

= (208 + 275)/30

= 483/30

(vii) ((13/7) × (11/26)) — ((-4/3) × (5/6))

Solution:

= ((13 × 11)/(7 × 26)) – ((-4 × 5)/(3 × 6))

Common factor of 13 and 26, 4 and 6

= 11/7 × 2 – (-2 × 5/3 × 3)

= 11/14 + 10/9

LCM of 14 and 9 is 126

= (11 × 9 + 10 × 14)/126

= (99 + 140)/126

= 239/126

Question 5. Simplify:

(i) ((3/2) × (1/6)) + ((5/3) × (7/2) – (13/8) × (4/3))

Solution:

= (3 × 1)/(2 × 6) + (5 × 7)/(3 × 2) – (13 × 4)/(8 × 3)

Common factor of 3 and 6, 4 and 8

= 1/4 + 35/6 – 13/6

LCM of 4 and 6 is 12

= (1 × 3 + 35 × 2 – 13 × 2)/12

= (3 + 70 – 26)/12

= (73 – 26)/12

= 47/12

(ii) ((1/4) × (2/7)) — (5/14) × (-2/3) + (3/7) × (9/2)

Solution:

= (1 × 2)/(4 × 7) – (5 × -2)/(14 × 3) + (3 × 9)/(7 × 2)

Common factor of 2 and 4, 2 and 14

= 1/14 – (-5/21) + 27/14

LCM of 21 and 14 is 7 × 2 × 3 = 42

= 1/14 + 5/21 + 27/14

LCM of 14 and 21 is 2 × 7 × 3 = 42

= (1 × 3 + 5 × 2 + 27 × 3)/42

= (3 + 10 + 81)/42

= (94)/42

(iii) ((13/9) × (-15/2)) + ((7/3) × (8/5) + (3/5) × (1/2))

Solution:

= (13 × -15)/(9 × 2) + ((7 × 8)/(3 × 5) + (3 × 1)/(5 × 2))

Common factor of 9 and 15

= (13 × -5)/(3 × 2) + ((56/15) + 3/10)

= -65/6 + 56/15 + 3/10

6 = 2 × 3

15 = 3 × 5

10 = 2 × 5

LCM is 2 × 3 × 5 = 30

= (-65 × 5 + 56 × 2 + 3 × 3)/30

= (-325 + 112 + 9)/30

= (-325 + 121)/30

= -204/30

(iv) ((3/11) × (5/6)) – (9/12) × (4/3) + (5/13) × (6/15)

Solution:

= (3 × 5)/(11 × 6) – ((9 × 4)/(12 × 3) + (5 × 6)/(13 × 15))

Common factor of 3 and 6, 9 and 12, 5 and 15

= 5/22 – 1/1 + 2/13

= 5/22 – 1/1 + 2/13

LCM of 22,1 and 13 is 286

= (5 × 13 – 286 + 2 × 22)/286

= (65 – 286 + 44)/286

= (65 – 330)/286

= -177/286