Exercise 4.2
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
2. Express 3/5 as a rational number with numerator:
(i) 6
(ii) -15
(iii) 21
(iv) – 27
Solution:
(i) Given (3/7)
To get numerators 6 we have to multiply both and denominators by 2
Then we get, (3/5) × (2/2) = (6/10)
Therefore (3/5) as a rational numbers with numerator 6 is (6/10)
(ii) Given (3/5)
To get numerators – 15 we have to multiply both numerators and denominator by -5
Then we get, (3/5) × (-5/-5)
= (-15/-25)
Therefore (3/5) as rational numbers with numerators -15 is (-15/-25)
(iii) Given (3/5)
To get numerator -21 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 (21/-45)
3. Express 5/7 as a rational number with denominator:
(i) -14
(ii) 70
(iii) -28
(iv) – 84
Solution:
5/7 as rational number with denominator:
(i) -14 is:
(Multiply numerator and denominator by 10)
(ii) 70 is:
(Multiply numerator and denominator by 10)
(iii) -28 is:
(Multiply numerator and denominator by -4)
(iv) -28 is:
(Multiply numerator number denominator by -12)
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 rational numbers 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 denominators 36 is (27/36)
(iii) Given (*3/4)
To get denominator 44 we have to multiply both numerators and denominators by 11
Then we get, (3/4) ×(11/11)
= (33/44)
Therefore (3/4) as rational numbers with denominator 44 is (33/44)
(iv) Given (3/4)
To get denominators – 80 we have to multiply both numerators and denominators by -20
Then we get, (3/4) ×(-20/-20)
= (-60/-80)
Therefore (3/4) as rational number with denominators -80 is (-60/-80)
5. Express 2/5 as a rational number with numerator:
(i) -56
(ii) 154
(iii) -750
(iv) – 80
Solution:
(i) -56
Clearly, – 56÷2
= -28
Multiplying the numerators and denominators of 2/5 by – 28
We have,
(ii) 154
Clearly, 154÷2 = – 28
Multiply the numerators and denominators of 2/5 by 72
We have,
(iii) -750
Clearly, -750÷2= -375
Multiplying the numerators and denominators of 2/5 by 375.
(iv) – 80
Clearly, 500÷2=250
Multiplying the numerators and denominators of 2/5 by 250.
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 numerators 64 we have to divide both numerators and denominators by – 3
Then we get, (-192/108) ÷(3/-3)
= (64/-36)
Therefore (-192/108) As rational number with numerators 64 is ( 64/-36)
(ii) Given (-192/108)
To get numerators -16 we have to divide both numerators and denominators by -6
Then we get, (-192/108) ÷ (12/12)
= (-16/9)
Therefore (-192/108) as rational number with numerator -16 is (-16/9)
(iii) Given (-192/108)
To get numerators 32 we have to divide booth numerators and denominators by -6
Then we get, (192/108) ÷ (-6/-6)
= (32/-18)
Therefore (-192/108) as rational number with numerator 32 is (32/-18)
(iv) Given (-192/108)
To get numerators – 48 we have to divide both numerators and denominators by 4
Then we get, (-192/108)÷ (4/4)
= (-48/27)
Therefore (-192/108) as a rational number with numerators -48 (-48/27)
7. Express 168/(-294) as a rational number with denominator:
(i) 14
(ii) -7
(iii) – 49
(iv) 1470
Solution:
Rational number with denominators:
(i) 14 as denominator:
(Dividing the numerator and denominator by -21)
(ii) -7 as denominator:
(Dividing the numerator and denominator by 42)
(iii) – 49 as denominator:
(Dividing the numerator and denominator by 6)
(iv) 1470 as denominator:
(Multiplying the numerator and denominator by – 5)
8. Write (-14)/42 form so that numerator is equal to:
(i) -2
(ii) 7
(iii) 42
(iv) -70
Solution:
(i) Clearly, – 2 ÷ 14 = 1/7
(ii) Clearly, 7÷-14=1/-2
(iii) Clearly, 42÷ -14 = -3
(iv) Clearly, – 70÷-14 = 5
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)
10. Select those rational numbers which can be written as a rational number with denominator 4:
(7/8), (64/16), (36/-12), (-16/17), (5/-4),(140/28)
Solution:
Given rational numbers that can be written as rational numbers with denominators 4
Are:
(7/8) = (3.5/4) (On dividing both denominators and denominator by 2)
(64/16) = (16/4) (On dividing both denominators and denominator by 4)
(36/-12) = (16/4) (On dividing both denominators and denominator by -3)
(5/-4) = (-5/4) (On dividing both denominators and denominator by -1)
(140/28) = (20/4) (On dividing both denominators and denominator by 7)
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.