 # RD Sharma Solutions For Class 7 Maths Exercise 4.2 Chapter 4 Rational Numbers

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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

= (15/28)

(ii) Given (6/-9)

Multiplying both numerator and denominator we can rational number with positive denominator.

(6/-9) = (6/-9) × -1

= (-6/9)

(iii) Given (-28/-11)

Multiplying both numerator and denominator we can rational number with positive denominator.

(-28/-11) = (-28/-11) × -1

= (28/11)

(iv) Given (19/-7)

Multiplying both numerator and denominator we can rational number with positive denominator.

(19/-7) = (19/-7) × -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/-150)

(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)

= (-48/27)

Therefore (-192/108) as a rational number with numerator -48 is (-48/27)

7. Express (169/-294) as a rational number with denominator:

(i) 14

(ii) -7

(iii) -49

(iv) 1470

Solution:

(i) Given (169/-294)

To get denominator 14 we have to divide both numerator and denominator by -21

Then we get, (169/-294) ÷ (-21/-21)

= (-8/14)

Therefore (169/-294) as a rational number with denominator 14 is (-8/14)

(ii) Given (169/-294)

To get denominator -7 we have to divide both numerator and denominator by 42

Then we get, (169/-294) ÷ (42/42)

= (4/-7)

Therefore (169/-294) as a rational number with denominator -7 is (4/-7)

(iii) Given (169/-294)

To get denominator -49 we have to divide both numerator and denominator by 6

Then we get, (169/-294) ÷ (6/6)

= (28/-49)

Therefore (169/-294) as a rational number with denominator -49 is (28/-49)

(iv) Given (169/-294)

To get denominator 1470 we have to multiply both numerator and denominator by -5

Then we get, (169/-294) × (-5/-5)

= (-840/1470)

Therefore (169/-294) as a rational number with denominator 1470 is (-840/1470)

8. Write (-14/42) in a form so that the numerator is equal to:

(i) -2

(ii) 7

(iii) 42

(iv) -70

Solution:

(i) Given (-14/42)

To get numerator -2 we have to divide both numerator and denominator by 7

Then we get, (-14/42) ÷ (7/7)

= (-2/6)

Therefore (-14/42) as a rational number with numerator -2 is (-2/6)

(ii) Given (-14/42)

To get numerator 7 we have to divide both numerator and denominator by -2

Then we get, (-14/42) ÷ (-2/-2)

= (7/-21)

Therefore (-14/42) as a rational number with numerator -14 is (-14/21)

(iii) Given (-14/42)

To get numerator 42 we have to multiply both numerator and denominator by -3

Then we get, (-14/42) × (-3/-3)

= (42/-126)

Therefore (-14/42) as a rational number with numerator 42 is (42/-126)

(iv) Given (-14/42)

To get numerator -70 we have to multiply both numerator and denominator by 5

Then we get, (-14/42) × (5/5)

= (-70/210)

Therefore (-14/42) as a rational number with numerator -70 is (-70/210)

9. Select those rational numbers which can be written as a rational number with numerator 6:

(1/22), (2/3), (3/4), (4/-5), (5/6), (-6/7), (-7/8)

Solution:

Given rational numbers that can be written as a rational number with numerator 6 are:

Consider (1/22)

On multiplying by 6, (1/22) can be written as

(1/22) = (6/132)

Consider (2/3)

On multiplying by 3, (2/3) can be written as

(2/3) = (6/9)

Consider (3/4)

On multiplying by 2, (3/4) can be written as

(3/4) = (6/8)

Consider (-6/7)

On multiplying by -1, (-6/7) can be written as

(-6/7) = (6/-7)

Therefore rational numbers that can be written as a rational number with numerator 6 are (1/22), (2/3), (3/4) and (-6/7)

10. Select those rational numbers which can be written as 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 a rational number with denominator 4 are:

(7/8) = (3.5/4) (On dividing both denominator and denominator by 2)

(64/16) = (16/4) (On dividing both denominator and numerator by 4)

(36/-12) = (-12/4) (On dividing both denominator and numerator by -3)

(5/- 4) = (- 5/4) (On multiplying both denominator and numerator by -1)

(140/28) = (20/4) (On dividing both numerator and denominator by 7)

11. In each of the following, find an equivalent form of the rational number having a 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:

(i) Given (3/4) and (5/12)

On multiplying both numerator and denominator by 3

(3/4) = (3 × 3)/ (4 × 3) = (9/12)

Equivalent forms with same denominators are (9/12) and (5/12)

(ii) Given (2/3), (7/6) and (11/12)

On multiplying both numerator and denominator by 4

(2/3) = (2 × 4)/ (3 × 4) = (8/12)

And (7/6) = (7 × 2)/ (6 × 2) = (14/12)

Equivalent forms are (8/12), (14/12) and (11/12) having same denominators

(iii) Given (5/7), (3/8), (9/14) and (20/21)

(5/7) = (5 × 24)/ (7 × 24) = (120/168) [on multiplying both numerator and denominator by 24]

(3/8) = (3 × 21)/ (8 × 21) = (63/168) [on multiplying both numerator and denominator by 21]

(9/14) = (9 × 12)/ (14 × 12) = (108/168) [on multiplying both numerator and denominator by 12]

(20/21) = (20 × 8)/ (21×8) = (160/168) [on multiplying both numerator and denominator by 8]

Forms are (120/168), (63/168), (108/168) and (160/168) having same denominators.