Page 1
Exercise 4.2 page no: 4.8
1. Express each of the following as a rational number with positive denominator.
(i) (-15/-28)
(ii) (6/-9)
(iii) (-28/-11)
(iv) (19/-7)
Solution:
(i) Given (-15/-28)
Multiplying both numerator and denominator we can rational number with positive
denominator.
(-15/-28) = (-15/-28) × (-1/-1)
= (15/28)
(ii) Given (6/-9)
Multiplying both numerator and denominator we can rational number with positive
denominator.
(6/-9) = (6/-9) × (-1/-1)
= (-6/9)
(iii) Given (-28/-11)
Multiplying both numerator and denominator we can rational number with positive
denominator.
(-28/-11) = (-28/-11) × (-1/-1)
= (28/11)
(iv) Given (19/-7)
Multiplying both numerator and denominator we can rational number with positive
denominator.
(19/-7) = (19/-7) × (-1/-1)
= (-19/7)
2. Express (3/5) as a rational number with numerator:
(i) 6
(ii) -15
Page 2
Exercise 4.2 page no: 4.8
1. Express each of the following as a rational number with positive denominator.
(i) (-15/-28)
(ii) (6/-9)
(iii) (-28/-11)
(iv) (19/-7)
Solution:
(i) Given (-15/-28)
Multiplying both numerator and denominator we can rational number with positive
denominator.
(-15/-28) = (-15/-28) × (-1/-1)
= (15/28)
(ii) Given (6/-9)
Multiplying both numerator and denominator we can rational number with positive
denominator.
(6/-9) = (6/-9) × (-1/-1)
= (-6/9)
(iii) Given (-28/-11)
Multiplying both numerator and denominator we can rational number with positive
denominator.
(-28/-11) = (-28/-11) × (-1/-1)
= (28/11)
(iv) Given (19/-7)
Multiplying both numerator and denominator we can rational number with positive
denominator.
(19/-7) = (19/-7) × (-1/-1)
= (-19/7)
2. Express (3/5) as a rational number with numerator:
(i) 6
(ii) -15
(iii) 21
(iv) -27
Solution:
(i) Given (3/5)
To get numerator 6 we have to multiply both numerator and denominator by 2
Then we get, (3/5) × (2/2) = (6/10)
Therefore (3/5) as a rational number with numerator 6 is (6/10)
(ii) Given (3/5)
To get numerator -15 we have to multiply both numerator and denominator by -5
Then we get, (3/5) × (-5/-5)
= (-15/-25)
Therefore (3/5) as a rational number with numerator -15 is (-15/-25)
(iii) Given (3/5)
To get numerator 21 we have to multiply both numerator and denominator by 7
Then we get, (3/5) × (7/7)
= (21/35)
Therefore (3/5) as a rational number with numerator 21 is (21/35)
(iv) Given (3/5)
To get numerator -27 we have to multiply both numerator and denominator by -9
Then we get, (3/5) × (-9/-9)
= (-27/-45)
Therefore (3/5) as a rational number with numerator -27 is (-27/-45)
3. Express (5/7) as a rational number with denominator:
(i) -14
(ii) 70
(iii) -28
(iv) -84
Solution:
(i) Given (5/7)
To get denominator -14 we have to multiply both numerator and denominator by -2
Then we get, (5/7) × (-2/-2)
Page 3
Exercise 4.2 page no: 4.8
1. Express each of the following as a rational number with positive denominator.
(i) (-15/-28)
(ii) (6/-9)
(iii) (-28/-11)
(iv) (19/-7)
Solution:
(i) Given (-15/-28)
Multiplying both numerator and denominator we can rational number with positive
denominator.
(-15/-28) = (-15/-28) × (-1/-1)
= (15/28)
(ii) Given (6/-9)
Multiplying both numerator and denominator we can rational number with positive
denominator.
(6/-9) = (6/-9) × (-1/-1)
= (-6/9)
(iii) Given (-28/-11)
Multiplying both numerator and denominator we can rational number with positive
denominator.
(-28/-11) = (-28/-11) × (-1/-1)
= (28/11)
(iv) Given (19/-7)
Multiplying both numerator and denominator we can rational number with positive
denominator.
(19/-7) = (19/-7) × (-1/-1)
= (-19/7)
2. Express (3/5) as a rational number with numerator:
(i) 6
(ii) -15
(iii) 21
(iv) -27
Solution:
(i) Given (3/5)
To get numerator 6 we have to multiply both numerator and denominator by 2
Then we get, (3/5) × (2/2) = (6/10)
Therefore (3/5) as a rational number with numerator 6 is (6/10)
(ii) Given (3/5)
To get numerator -15 we have to multiply both numerator and denominator by -5
Then we get, (3/5) × (-5/-5)
= (-15/-25)
Therefore (3/5) as a rational number with numerator -15 is (-15/-25)
(iii) Given (3/5)
To get numerator 21 we have to multiply both numerator and denominator by 7
Then we get, (3/5) × (7/7)
= (21/35)
Therefore (3/5) as a rational number with numerator 21 is (21/35)
(iv) Given (3/5)
To get numerator -27 we have to multiply both numerator and denominator by -9
Then we get, (3/5) × (-9/-9)
= (-27/-45)
Therefore (3/5) as a rational number with numerator -27 is (-27/-45)
3. Express (5/7) as a rational number with denominator:
(i) -14
(ii) 70
(iii) -28
(iv) -84
Solution:
(i) Given (5/7)
To get denominator -14 we have to multiply both numerator and denominator by -2
Then we get, (5/7) × (-2/-2)
= (-10/-14)
Therefore (5/7) as a rational number with denominator -14 is (-10/-14)
(ii) Given (5/7)
To get denominator 70 we have to multiply both numerator and denominator by -2
Then we get, (5/7) × (10/10)
= (50/70)
Therefore (5/7) as a rational number with denominator 70 is (50/70)
(iii) Given (5/7)
To get denominator -28 we have to multiply both numerator and denominator by -4
Then we get, (5/7) × (-4/-4)
= (-20/-28)
Therefore (5/7) as a rational number with denominator -28 is (-20/-28)
(iv) Given (5/7)
To get denominator -84 we have to multiply both numerator and denominator by -12
Then we get, (5/7) × (-12/-12)
= (-60/-84)
Therefore (5/7) as a rational number with denominator -84 is (-60/-84)
4. Express (3/4) as a rational number with denominator:
(i) 20
(ii) 36
(iii) 44
(iv) -80
Solution:
(i) Given (3/4)
To get denominator 20 we have to multiply both numerator and denominator by 5
Then we get, (3/4) × (5/5)
= (15/20)
Therefore (3/4) as a rational number with denominator 20 is (15/20)
(ii) Given (3/4)
To get denominator 36 we have to multiply both numerator and denominator by 9
Then we get, (3/4) × (9/9)
Page 4
Exercise 4.2 page no: 4.8
1. Express each of the following as a rational number with positive denominator.
(i) (-15/-28)
(ii) (6/-9)
(iii) (-28/-11)
(iv) (19/-7)
Solution:
(i) Given (-15/-28)
Multiplying both numerator and denominator we can rational number with positive
denominator.
(-15/-28) = (-15/-28) × (-1/-1)
= (15/28)
(ii) Given (6/-9)
Multiplying both numerator and denominator we can rational number with positive
denominator.
(6/-9) = (6/-9) × (-1/-1)
= (-6/9)
(iii) Given (-28/-11)
Multiplying both numerator and denominator we can rational number with positive
denominator.
(-28/-11) = (-28/-11) × (-1/-1)
= (28/11)
(iv) Given (19/-7)
Multiplying both numerator and denominator we can rational number with positive
denominator.
(19/-7) = (19/-7) × (-1/-1)
= (-19/7)
2. Express (3/5) as a rational number with numerator:
(i) 6
(ii) -15
(iii) 21
(iv) -27
Solution:
(i) Given (3/5)
To get numerator 6 we have to multiply both numerator and denominator by 2
Then we get, (3/5) × (2/2) = (6/10)
Therefore (3/5) as a rational number with numerator 6 is (6/10)
(ii) Given (3/5)
To get numerator -15 we have to multiply both numerator and denominator by -5
Then we get, (3/5) × (-5/-5)
= (-15/-25)
Therefore (3/5) as a rational number with numerator -15 is (-15/-25)
(iii) Given (3/5)
To get numerator 21 we have to multiply both numerator and denominator by 7
Then we get, (3/5) × (7/7)
= (21/35)
Therefore (3/5) as a rational number with numerator 21 is (21/35)
(iv) Given (3/5)
To get numerator -27 we have to multiply both numerator and denominator by -9
Then we get, (3/5) × (-9/-9)
= (-27/-45)
Therefore (3/5) as a rational number with numerator -27 is (-27/-45)
3. Express (5/7) as a rational number with denominator:
(i) -14
(ii) 70
(iii) -28
(iv) -84
Solution:
(i) Given (5/7)
To get denominator -14 we have to multiply both numerator and denominator by -2
Then we get, (5/7) × (-2/-2)
= (-10/-14)
Therefore (5/7) as a rational number with denominator -14 is (-10/-14)
(ii) Given (5/7)
To get denominator 70 we have to multiply both numerator and denominator by -2
Then we get, (5/7) × (10/10)
= (50/70)
Therefore (5/7) as a rational number with denominator 70 is (50/70)
(iii) Given (5/7)
To get denominator -28 we have to multiply both numerator and denominator by -4
Then we get, (5/7) × (-4/-4)
= (-20/-28)
Therefore (5/7) as a rational number with denominator -28 is (-20/-28)
(iv) Given (5/7)
To get denominator -84 we have to multiply both numerator and denominator by -12
Then we get, (5/7) × (-12/-12)
= (-60/-84)
Therefore (5/7) as a rational number with denominator -84 is (-60/-84)
4. Express (3/4) as a rational number with denominator:
(i) 20
(ii) 36
(iii) 44
(iv) -80
Solution:
(i) Given (3/4)
To get denominator 20 we have to multiply both numerator and denominator by 5
Then we get, (3/4) × (5/5)
= (15/20)
Therefore (3/4) as a rational number with denominator 20 is (15/20)
(ii) Given (3/4)
To get denominator 36 we have to multiply both numerator and denominator by 9
Then we get, (3/4) × (9/9)
= (27/36)
Therefore (3/4) as a rational number with denominator 36 is (27/36)
(iii) Given (3/4)
To get denominator 44 we have to multiply both numerator and denominator by 11
Then we get, (3/4) × (11/11)
= (33/44)
Therefore (3/4) as a rational number with denominator 44 is (33/44)
(iv) Given (3/4)
To get denominator -80 we have to multiply both numerator and denominator by -20
Then we get, (3/4) × (-20/-20)
= (-60/-80)
Therefore (3/4) as a rational number with denominator -80 is (-60/-80)
5. Express (2/5) as a rational number with numerator:
(i) -56
(ii) 154
(iii) -750
(iv) 500
Solution:
(i) Given (2/5)
To get numerator -56 we have to multiply both numerator and denominator by -28
Then we get, (2/5) × (-28/-28)
= (-56/-140)
Therefore (2/5) as a rational number with numerator -56 is (-56/-140)
(ii) Given (2/5)
To get numerator 154 we have to multiply both numerator and denominator by 77
Then we get, (2/5) × (77/77)
= (154/385)
Therefore (2/5) as a rational number with numerator 154 is (154/385)
(iii) Given (2/5)
To get numerator -750 we have to multiply both numerator and denominator by -375
Then we get, (2/5) × (-375/-375)
Page 5
Exercise 4.2 page no: 4.8
1. Express each of the following as a rational number with positive denominator.
(i) (-15/-28)
(ii) (6/-9)
(iii) (-28/-11)
(iv) (19/-7)
Solution:
(i) Given (-15/-28)
Multiplying both numerator and denominator we can rational number with positive
denominator.
(-15/-28) = (-15/-28) × (-1/-1)
= (15/28)
(ii) Given (6/-9)
Multiplying both numerator and denominator we can rational number with positive
denominator.
(6/-9) = (6/-9) × (-1/-1)
= (-6/9)
(iii) Given (-28/-11)
Multiplying both numerator and denominator we can rational number with positive
denominator.
(-28/-11) = (-28/-11) × (-1/-1)
= (28/11)
(iv) Given (19/-7)
Multiplying both numerator and denominator we can rational number with positive
denominator.
(19/-7) = (19/-7) × (-1/-1)
= (-19/7)
2. Express (3/5) as a rational number with numerator:
(i) 6
(ii) -15
(iii) 21
(iv) -27
Solution:
(i) Given (3/5)
To get numerator 6 we have to multiply both numerator and denominator by 2
Then we get, (3/5) × (2/2) = (6/10)
Therefore (3/5) as a rational number with numerator 6 is (6/10)
(ii) Given (3/5)
To get numerator -15 we have to multiply both numerator and denominator by -5
Then we get, (3/5) × (-5/-5)
= (-15/-25)
Therefore (3/5) as a rational number with numerator -15 is (-15/-25)
(iii) Given (3/5)
To get numerator 21 we have to multiply both numerator and denominator by 7
Then we get, (3/5) × (7/7)
= (21/35)
Therefore (3/5) as a rational number with numerator 21 is (21/35)
(iv) Given (3/5)
To get numerator -27 we have to multiply both numerator and denominator by -9
Then we get, (3/5) × (-9/-9)
= (-27/-45)
Therefore (3/5) as a rational number with numerator -27 is (-27/-45)
3. Express (5/7) as a rational number with denominator:
(i) -14
(ii) 70
(iii) -28
(iv) -84
Solution:
(i) Given (5/7)
To get denominator -14 we have to multiply both numerator and denominator by -2
Then we get, (5/7) × (-2/-2)
= (-10/-14)
Therefore (5/7) as a rational number with denominator -14 is (-10/-14)
(ii) Given (5/7)
To get denominator 70 we have to multiply both numerator and denominator by -2
Then we get, (5/7) × (10/10)
= (50/70)
Therefore (5/7) as a rational number with denominator 70 is (50/70)
(iii) Given (5/7)
To get denominator -28 we have to multiply both numerator and denominator by -4
Then we get, (5/7) × (-4/-4)
= (-20/-28)
Therefore (5/7) as a rational number with denominator -28 is (-20/-28)
(iv) Given (5/7)
To get denominator -84 we have to multiply both numerator and denominator by -12
Then we get, (5/7) × (-12/-12)
= (-60/-84)
Therefore (5/7) as a rational number with denominator -84 is (-60/-84)
4. Express (3/4) as a rational number with denominator:
(i) 20
(ii) 36
(iii) 44
(iv) -80
Solution:
(i) Given (3/4)
To get denominator 20 we have to multiply both numerator and denominator by 5
Then we get, (3/4) × (5/5)
= (15/20)
Therefore (3/4) as a rational number with denominator 20 is (15/20)
(ii) Given (3/4)
To get denominator 36 we have to multiply both numerator and denominator by 9
Then we get, (3/4) × (9/9)
= (27/36)
Therefore (3/4) as a rational number with denominator 36 is (27/36)
(iii) Given (3/4)
To get denominator 44 we have to multiply both numerator and denominator by 11
Then we get, (3/4) × (11/11)
= (33/44)
Therefore (3/4) as a rational number with denominator 44 is (33/44)
(iv) Given (3/4)
To get denominator -80 we have to multiply both numerator and denominator by -20
Then we get, (3/4) × (-20/-20)
= (-60/-80)
Therefore (3/4) as a rational number with denominator -80 is (-60/-80)
5. Express (2/5) as a rational number with numerator:
(i) -56
(ii) 154
(iii) -750
(iv) 500
Solution:
(i) Given (2/5)
To get numerator -56 we have to multiply both numerator and denominator by -28
Then we get, (2/5) × (-28/-28)
= (-56/-140)
Therefore (2/5) as a rational number with numerator -56 is (-56/-140)
(ii) Given (2/5)
To get numerator 154 we have to multiply both numerator and denominator by 77
Then we get, (2/5) × (77/77)
= (154/385)
Therefore (2/5) as a rational number with numerator 154 is (154/385)
(iii) Given (2/5)
To get numerator -750 we have to multiply both numerator and denominator by -375
Then we get, (2/5) × (-375/-375)
= (-750/-1875)
Therefore (2/5) as a rational number with numerator -750 is (-750/-1875)
(iv) Given (2/5)
To get numerator 500 we have to multiply both numerator and denominator by 250
Then we get, (2/5) × (250/250)
= (500/1250)
Therefore (2/5) as a rational number with numerator 500 is (500/1250)
6. Express (-192/108) as a rational number with numerator:
(i) 64
(ii) -16
(iii) 32
(iv) -48
Solution:
(i) Given (-192/108)
To get numerator 64 we have to divide both numerator and denominator by -3
Then we get, (-192/108) ÷ (-3/-3)
= (64/-36)
Therefore (-192/108) as a rational number with numerator 64 is (64/-36)
(ii) Given (-192/108)
To get numerator -16 we have to divide both numerator and denominator by 12
Then we get, (-192/108) ÷ (12/12)
= (-16/9)
Therefore (-192/108) as a rational number with numerator -16 is (-16/9)
(iii) ) Given (-192/108)
To get numerator 32 we have to divide both numerator and denominator by -6
Then we get, (-192/108) ÷ (-6/-6)
= (32/-18)
Therefore (-192/108) as a rational number with numerator 32 is (32/-18)
(iv) Given (-192/108)
To get numerator -48 we have to divide both numerator and denominator by 4
Then we get, (-192/108) ÷ (4/4)
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