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Chapter 4: Rational Numbers Exercise – 4.2 Question: 1 Express each of the following as rational number with positive denominator: (i) (-15)/(-28) (ii) 6/(-9) (iii) (-28)/(-11) (iv) 19/(- 7) Solution: Rational number with positive denominators: (i) Multiplying the number by -1, we get: -15 – 28 = -15 x -1 - 28 x – 1 = 1528 (ii) Multiplying the number by -1, we get: 6 – 9 = 6x – 1 - 9x – 1 = - 69 (iii) Multiplying the number by -1, we get: – 28 – 11= – 28 x – 1 – 11 x – 1 = 2811 (iv) Multiplying the number by -1, we get: 19 – 7 = 19 x – 1 - 7 x - 1= -197 Question: 2 Express 3/5 as a rational number with numerator: (i) 6 (ii) -15 (iii) 21 (iv) – 27 Solution: Rational number with numerator: (i) 6 is: (Multiplying numerator and denominator by 2) (ii) -15 is: (Multiplying numerator and denominator by - 5) (iii) 21 is: (Multiplying numerator and denominator by 7) (iv) - 27 is: (Multiplying numerator and denominator by - 9) Question: 3 Express 5/7 as a rational number with denominator: (i) -14 (ii) 70 (iii) -28 (iv) - 84 Solution: 5/7 as a rational number with denominator: (1) -14 is: (Multiplying numerator and denominator by -2) (ii) 70 is: (Multiplying numerator and denominator by 10) (iii) - 28 is: (Multiplying numerator and denominator by - 4) (iv) - 84 is: (Multiplying numerator and denominator by -12) Question: 4 Express 3/4 as a rational number with denominator: (i) 20 (ii) 36 (iii) 44 (iv) - 80 Solution: 3/4 as rational number with denominator: (i) 20 is: (Multiplying numerator and denominator by 5) (ii) 36 is: (Multiplying numerator and denominator by 9) (iii) 44 is: (Multiplying numerator and denominator by 11) (iv) - 80 is: (Multiplying numerator and denominator by - 20) Question: 5 Express 2/5 as a rational number with numerator: (i) -56 (ii) 154 (iii) -750 (iv) - 80 Solution: 2/5 as a rational number with numerator: (i) - 56 is: (Multiplying numerator and denominator by -28) (ii) 154 is: (Multiplying numerator and denominator by 77) (iii) – 750 is: (Multiplying numerator and denominator by -375) (iv) 500 is: (Multiplying numerator and denominator by 250) Question: 6 Express (-192)/108 as a rational number with numerator: (i) 64 (ii) -16 (iii) 32 (iv) - 48 Solution: Rational number with numerator: (i) 64 as numerator: (Dividing the numerator and denomintor by -3) (ii) -16 as numerator: (Dividing the numerator and denomintor by 12) (iii) 32 as numerator: (Dividing the numerator and denomintor by - 6) (iv) - 48 as numerator: (Dividing the numerator and denomintor by 4) Question: 7 Express 168/(-294) as a rational number with denominator: (i) 14 (ii) -7 (iii) - 49 (iv) 1470 Solution: Rational number with denominator: (i) 14 as denominator: (Dividing the numerator and denomintor by -21) (ii) -7 as denominator: (Dividing the numerator and denomintor by 42) (iii) - 49 as denominator: (Dividing the numerator and denomintor by 6) (iv) 1470 as denominator: (Multiplying the numerator and denomintor by – 5) Question: 8 Write (-14)/42 form so that numerator is equal to: (i) -2 (ii) 7 (iii) 42 (iv) -70 Solution: Rational number with numerator: (i) -2 is: (Dividing numerator and denominator by 7) (ii) 7 is: (Dividing numerator and denominator by -2) (iii) 42 is: (Multiplying numerator and denominator by -3) (iv) -70 is: (Multiplying numerator and denominator by 5) Question: 9 Select those rational numbers which can be written as a rational number with numerator 6: Solution: Given rational numbers that can be written as a rational number with numerator 6 are: 1/22 (On multiplying by 6) = 6/132, 2/3 (On multiplying by 3) = 6/9, 3/4 (On multiplying by 2) = 6/8, (-6)/7 (On multiplying by -1) = 6/(-7) Question: 10 Select those rational numbers which can be written as a rational number with denominator 4: Solution: Given rational numbers that can be written as a rational number with denominator 4 are: 7/8 (On dividing by 2) = 3.5/4, 64/16 (On dividing by 4) = 16/4, 36/-12(On dividing by 3) = 12/-4 = -12/4, 16/17 can’t be expressed with a denominator 4. 5/- 4(On multiplying by -1) = - 5/4 140/28(On dividing by 7) = 20/4 Question: 11 In each of the following, find an equivalent form of the rational number having common denominator: (i) 3/4 and 5/12 (ii) 2/3, 7/6 and 11/12 (iii) 5/7, 3/8, 9/14 and 20/21< Solution: Equivalent forms of the rational number having common denominator are: (i) 3/4 = (3 × 3)/(4 × 3) = 9/12 and 512. (ii) 2/3 = (2 × 4)/(3 × 4) = 8/12 and 7/6 = (7×2)/(6×2) = 14/12 and 11/12 Forms are 8/12, 14/12 and 11/12 (iii) 5/7 = (5 × 24)/(7 × 24) = 120/168, 3/8 = (3 × 21)/(8 × 21) = 63/168, 9/14 = (9 × 12)/(14 × 12) = 108/168 and 20/21 = (20 × 8)/(21×8) =160/168 Forms are 120/168, 63/168, 108/168 and 160/168.
Express each of the following as rational number with positive denominator:
(i) (-15)/(-28)
(ii) 6/(-9)
(iii) (-28)/(-11)
(iv) 19/(- 7)
Rational number with positive denominators:
(i) Multiplying the number by -1, we get: -15 – 28 = -15 x -1 - 28 x – 1 = 1528
(ii) Multiplying the number by -1, we get: 6 – 9 = 6x – 1 - 9x – 1 = - 69
(iii) Multiplying the number by -1, we get: – 28 – 11= – 28 x – 1 – 11 x – 1 = 2811
(iv) Multiplying the number by -1, we get: 19 – 7 = 19 x – 1 - 7 x - 1= -197
Express 3/5 as a rational number with numerator:
(i) 6
(ii) -15
(iii) 21
(iv) – 27
Rational number with numerator:
(i) 6 is:
(Multiplying numerator and denominator by 2)
(ii) -15 is:
(Multiplying numerator and denominator by - 5)
(iii) 21 is:
(Multiplying numerator and denominator by 7)
(iv) - 27 is:
(Multiplying numerator and denominator by - 9)
Express 5/7 as a rational number with denominator:
(i) -14
(ii) 70
(iii) -28
(iv) - 84
5/7 as a rational number with denominator:
(1) -14 is:
(Multiplying numerator and denominator by -2)
(ii) 70 is:
(Multiplying numerator and denominator by 10)
(iii) - 28 is:
(Multiplying numerator and denominator by - 4)
(iv) - 84 is:
(Multiplying numerator and denominator by -12)
Express 3/4 as a rational number with denominator:
(i) 20
(ii) 36
(iii) 44
(iv) - 80
3/4 as rational number with denominator:
(i) 20 is:
(Multiplying numerator and denominator by 5)
(ii) 36 is:
(Multiplying numerator and denominator by 9)
(iii) 44 is:
(Multiplying numerator and denominator by 11)
(iv) - 80 is:
(Multiplying numerator and denominator by - 20)
Express 2/5 as a rational number with numerator:
(i) -56
(ii) 154
(iii) -750
2/5 as a rational number with numerator:
(i) - 56 is:
(Multiplying numerator and denominator by -28)
(ii) 154 is:
(Multiplying numerator and denominator by 77)
(iii) – 750 is:
(Multiplying numerator and denominator by -375)
(iv) 500 is:
(Multiplying numerator and denominator by 250)
Express (-192)/108 as a rational number with numerator:
(i) 64
(ii) -16
(iii) 32
(iv) - 48
(i) 64 as numerator:
(Dividing the numerator and denomintor by -3)
(ii) -16 as numerator:
(Dividing the numerator and denomintor by 12)
(iii) 32 as numerator:
(Dividing the numerator and denomintor by - 6)
(iv) - 48 as numerator:
(Dividing the numerator and denomintor by 4)
Express 168/(-294) as a rational number with denominator:
(i) 14
(ii) -7
(iii) - 49
(iv) 1470
Rational number with denominator:
(i) 14 as denominator:
(Dividing the numerator and denomintor by -21)
(ii) -7 as denominator:
(Dividing the numerator and denomintor by 42)
(iii) - 49 as denominator:
(Dividing the numerator and denomintor by 6)
(iv) 1470 as denominator:
(Multiplying the numerator and denomintor by – 5)
Write (-14)/42 form so that numerator is equal to:
(i) -2
(ii) 7
(iii) 42
(iv) -70
(i) -2 is:
(Dividing numerator and denominator by 7)
(ii) 7 is:
(Dividing numerator and denominator by -2)
(iii) 42 is:
(Multiplying numerator and denominator by -3)
(iv) -70 is:
Select those rational numbers which can be written as a rational number with numerator 6:
Given rational numbers that can be written as a rational number with numerator 6 are:
1/22 (On multiplying by 6) = 6/132, 2/3 (On multiplying by 3)
= 6/9, 3/4 (On multiplying by 2)
= 6/8, (-6)/7 (On multiplying by -1)
= 6/(-7)
Select those rational numbers which can be written as a rational number with denominator 4:
Given rational numbers that can be written as a rational number with denominator 4 are:
7/8 (On dividing by 2) = 3.5/4,
64/16 (On dividing by 4) = 16/4,
36/-12(On dividing by 3) = 12/-4 = -12/4,
16/17 can’t be expressed with a denominator 4.
5/- 4(On multiplying by -1) = - 5/4
140/28(On dividing by 7) = 20/4
In each of the following, find an equivalent form of the rational number having
common denominator:
(i) 3/4 and 5/12
(ii) 2/3, 7/6 and 11/12
(iii) 5/7, 3/8, 9/14 and 20/21<
Equivalent forms of the rational number having common denominator are:
(i) 3/4 = (3 × 3)/(4 × 3) = 9/12 and 512.
(ii) 2/3 = (2 × 4)/(3 × 4) = 8/12 and 7/6 = (7×2)/(6×2) = 14/12 and 11/12
Forms are 8/12, 14/12 and 11/12
(iii) 5/7 = (5 × 24)/(7 × 24) = 120/168, 3/8 = (3 × 21)/(8 × 21) = 63/168, 9/14
= (9 × 12)/(14 × 12)
= 108/168 and 20/21
= (20 × 8)/(21×8)
=160/168
Forms are 120/168, 63/168, 108/168 and 160/168.
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