## Rational Numbers – Solved Examples

Example 1: Write the numerator of the following rational numbers.

(i) (2/7) (ii) (-4/5) (iii) (-11/3) (iv) (-10/-17) (v) (13/-14) (vi) (9/8) (vii) (8/9) (viii) (29/4) (ix) (12/5) (x) (22/7)

Solution:

(i) Numerator of (2/7) is 2

(ii) Numerator of (-4/5) is -4

(iii) Numerator of (-11/3) is -11

(iv) Numerator of (-10/-17) is -10

(v) Numerator of (13/-14) is 13

(vi) Numerator of (9/8) is 9

(vii) Numerator of (8/9) is 8

(viii) Numerator of 29/4 is 29

(ix) Numerator of (12/5) is 12

(x) Numerator of (22/7) is 22

Example 2: Write the denominator of the following rational numbers.

(i) (2/5) (ii) (-14/7) (iii) (11/13) (iv) (-1/3) (v) (13/-14) (vi) (9/-4) (vii) (7/4) (viii) (19/6) (ix) (12/20) (x) (22/17)

Solution:

(i) Denominator of (2/5) is 5

(ii) Denominator of (-14/7) is 7

(iii) Denominator of (11/13) is 13

(iv) Denominator of (-1/3) is 3

(v) Denominator of (13/-14) is -14

(vi) Denominator of (9/-4) is -4

(vii) Denominator of (7/4) is 4

(viii) Denominator of (19/6) is 6

(ix) Denominator of (12/20) is 20

(x) Denominator of (22/17) is 17

Example 3: Write down the rational numbers as integers?

Solution:

Given rational numbers are (i) (2/1), (ii) (-4/1), (iii) (-11/1), (iv) (-10/1), (v) (13/1)

The integers of the given rational numbers are

(i) Integer of (2/1) is 2

(ii) Integer of (-4/1) is -4

(iii) Integer of (-11/1) is -11

(iv) Integer of (-10/1) is -10

(v) Integer of (13/1) is 13

Example 4: Write down the integers as rational numbers?

(i) (5) (ii) (-7) (iii) (-12) (iv) (-19) (v) (18)

Solution:

The rational numbers of given integers are

(i) (5/1)

(ii) (-7/1)

(iii) (-12/1)

(iv) (-19/1)

(v) (18/1)

Example 5: Which of the following rational numbers are negative?

(i) (2/9) (ii) (4/-5) (iii) (-11/8) (iv) (-19/4) (v) (-18/-7)

Solution:

Given (i) (2/9) (ii) (4/-5) (iii) (-11/8) (iv) (-19/4) (v) (-18/-7)

A rational number is said to be negative rational number, if any of its numerator or denominator is negative.

Therefore, negative rational number are (4/-5), (-11/8), and (-19/4)

Example 6: Which of the following rational numbers are positive?

(i) (-4/-9) (ii) (8/-5) (iii) (13/8) (iv) (-9/4) (v) (-15/-7)

Solution:

Given (i) (-4/-9) (ii) (8/-5) (iii) (13/8) (iv) (-9/4) (v) (-15/-7)

A rational number is said to be positive rational number, if its numerator and denominator are either positive integers or both negative integers.

Therefore, positive rational number are (-4/-9), (13/8), and (-15/-7)

Example 7: Write down the rational number whose numerator is 5 x (-4), and denominator is (5 + 2) x (8 – 2)

Solution:

Given numerator = 5 x (-4) = -20

Denominator = (5 + 2) x (8 – 2) = 7 x 6 = 42

Therefore, the rational number = -20/42

Example 8: Write down the rational number whose numerator is (-3) x (-5), and denominator is (25 – 20) x (3 + 2)

Solution:

Given numerator = (-3) x (-5) = 15

Denominator = (25 – 20) x (3 + 2) = 5 x 5 = 25

Therefore, the rational number = 15/25 = 3/5

Example 9: Write down the rational number whose numerator is smallest two digit number, and denominator is largest three digit number.

Solution:

Smallest two digit number = 10

Largest three digit number = 999

Therefore, the rational number = 10/999

Example 10: Express each of the following as a rational number with positive denominator?

(i) (-11/-10), (ii) (8/-7), (iii) (13/-2), (iv) (12/-5), (v) (-2/-7), (vi) (-4/-5), (vii) (11/-3), (viii) (-15/-17) (ix) (13/-14) (x) (-9/-8), (xi) (3/-4).

Solution:

(i) Given (-11/-10)

Multiplying both numerator and denominator by (-1/-1),

we get rational number with positive denominator.

(-11/-10) x (-1/-1) = (11/10)

(ii) Given (8/-7)

Multiplying both numerator and denominator by (-1/-1),

we get rational number with positive denominator.

(8/-7) x (-1/-1) = (-8/7)

(iii) Given (13/-2)

Multiplying both numerator and denominator by (-1/-1),

we get rational number with positive denominator.

(13/-2) x (-1/-1) = (-13/2)

(iv) Given (12/-5)

Multiplying both numerator and denominator by (-1/-1),

we get rational number with positive denominator.

(12/-5) x (-1/-1) = (-12/5)

(v) Given (-2/-7)

Multiplying both numerator and denominator by (-1/-1),

we get rational number with positive denominator.

(-2/-7) x (-1/-1) = (2/7)

(vi) Given (-4/-5)

Multiplying both numerator and denominator by (-1/-1),

we get rational number with positive denominator.

(-4/-5) x (-1/-1) = (4/5)

(vii) Given (-11/-3)

Multiplying both numerator and denominator by (-1/-1),

we get rational number with positive denominator.

(-11/-3) x (-1/-1) = (11/3)

(viii) Given (-15/-17)

Multiplying both numerator and denominator by (-1/-1),

we get rational number with positive denominator.

(-15/-17) x (-1/-1) = (15/17)

(ix) Given (-13/-14)

Multiplying both numerator and denominator by (-1/-1), we get rational number with positive denominator.

(-13/-14) x (-1/-1) = (13/14)

(x) Given (-9/-8)

Multiplying both numerator and denominator by (-1/-1),

we get rational number with positive denominator.

(-9/-8) x (-1/-1) = (9/8)

(xi) Given (3/-4)

Multiplying both numerator and denominator by (-1/-1),

we get rational number with positive denominator.

(3/-4) x (-1/-1) = (-3/4)

Example 11: Separate positive and negative rational numbers from the following rational numbers?

(-3/-4), (-20/14), (13/-8), (-5/-2), (-3/4), (15/-3), (-7/-3)

Solution:

Given (-3/-4), (-20/14), (13/-8), (-5/-2), (-3/4), (15/-3), (-7/-3).

A rational number is said to be positive rational number, if its numerator and denominator are either positive integers or both negative integers.

Therefore, positive rational numbers are (-3/-4), (-5/-2) and (-7/-3)

A rational number is said to be negative rational number if any of its numerator or denominator is negative.

Therefore, negative rational numbers are (-20/14), (13/-8), (-3/4) and (15/-3)

Example 12: Express (2/5) as a rational number with numerator:

(i) (8) (ii) 12 (iii) -16 (iv) -22

Solution:

Given (2/5)

To get numerator 8, we have to multiply both numerator and denominator by 4,

Then we get,

(2/5) x (4/4) = 8/20

Therefore,(2/5) as a rational number with numerator 8 is (8/20).

Given (2/5)

To get numerator 12, we have to multiply both numerator and denominator by 6

Then we get,

(2/5) x (6/6) = 12/30

Therefore,(2/5) as a rational number with numerator 12 is (12/30)

Given (2/5)

To get numerator -16, we have to multiply both numerator and denominator by -8

Then we get,

(2/5) x (-8/-8) = -16/-40

Therefore,(2/5) as a rational number with numerator -16 is (-16/-40)

Given (2/5)

To get numerator -22, we have to multiply both numerator and denominator by -11

Then we get,

(2/5) x (-11/-11) = -22/-55

Therefore,(2/5) as a rational number with numerator -22 is (-22/-55)

Example 13: Express (4/3) as a rational number with denominator:

(i) (-9) (ii) 15 (iii) -27 (iv) -30

Given (4/3)

To get denominator (-9), we have to multiply both numerator and denominator by (-3)

Then we get,

(4/3) x (-3/-3) = -12/-9

Therefore, (4/3) as a rational number with denominator -9 is (-12/-9)

Given (4/3)

To get denominator (15), we have to multiply both numerator and denominator by (5)

Then we get,

(4/3) x (5/5) = 20/15

Therefore, (4/3) as a rational number with denominator 15 is (20/15)

Given (4/3)

To get denominator (-27), we have to multiply both numerator and denominator by (-9)

Then we get,

(4/3) x (-9/-9) = -36/-27

Therefore, (4/3) as a rational number with denominator -27 is (-36/-27)

Given (4/3)

To get denominator (-30), we have to multiply both numerator and denominator by (-10)

Then we get,

(4/3) x (-10/-10) = -40/-30

Therefore, (4/3) as a rational number with denominator -30 is (-40/-30)

Example 14: Express (3/5) as a rational number with denominator,

(i) (15) (ii) -25 (iii) 60 (iv) 65

Given (3/5)

To get denominator (15), we have to multiply both numerator and denominator by (3)

Then we get,

(3/5) x (3/3) = (9/15)

Therefore, (3/5) as a rational number with denominator 15 is (9/15)

Given (3/5)

To get denominator (-25), we have to multiply both numerator and denominator by (-5)

Then we get,

(3/5) x (-5/-5) = (-15/-25)

Therefore, (3/5) as a rational number with denominator -25 is (-15/-25)

Given (3/5)

To get denominator (60), we have to multiply both numerator and denominator by (12)

Then we get,

(3/5) x (12/12) = (36/60)

Therefore, (3/5) as a rational number with denominator 60 is (36/60)

Given (3/5)

To get denominator (60), we have to multiply both numerator and denominator by (12)

Then we get,

(3/5) x (12/12) = (36/60)

Therefore,(3/5) as a rational number with denominator 60 is (36/60)

Given (3/5)

To get denominator (65), we have to multiply both numerator and denominator by (13)

Then we get,

(3/5) x (13/13) = (39/65)

Therefore, (3/5) as a rational number with denominator 65 is (39/65)

Example 15: Express (14/5) as a rational number with numerator

(i) (84) (ii) -140 (iii) 280 (iv) 196

Given (14/5)

To get numerator 84, we have to multiply both numerator and denominator by 6

Then we get,

(14/5) x (6/6) = 84/30

Therefore, (14/5) as a rational number with numerator 84 is (84/30)

Given (14/5)

To get numerator -140, we have to multiply both numerator and denominator by -10

Then we get,

(14/5) x (-10/-10) = -140/-50

Therefore, (14/5) as a rational number with numerator -140 is (-140/50)

Given (14/5)

To get numerator 280, we have to multiply both numerator and denominator by 20

Then we get,

(14/5) x (20/20) = 280/100

Therefore, (14/5) as a rational number with numerator 280 is (280/100)

Given (14/5)

To get numerator 196, we have to multiply both numerator and denominator by 14

Then we get,

(14/5) x (14/14) = 196/70

Therefore, (14/5) as a rational number with numerator 196 is (196/70)

Example 16: Express (144/168) as a rational number with numerator

(i) (12) (ii) -24 (iii) 48 (iv) -72

Solution:

Given (144/168)

To get numerator 12, we have to divide both numerator and denominator by 12

Then we get,

(144/168) ÷ (12/12) = 12/14

Therefore, (144/168) as a rational number with numerator 12 is (12/14)

Given (144/168)

To get numerator -24, we have to divide both numerator and denominator by -6

Then we get,

(144/168) ÷ (-6/-6) = -24/-28

Therefore, (144/168) as a rational number with numerator -24 is (-24/-28)

Given (144/168)

To get numerator 48, we have to divide both numerator and denominator by 3

Then we get,

(144/168) ÷ (3/3) = 48/56

Therefore, (144/168) as a rational number with numerator 48 is (48/56)

Given (144/168)

To get numerator -72, we have to divide both numerator and denominator by -2

Then we get,(144/168) ÷ (-2/-2) = -72/-84

Therefore, (144/168) as a rational number with numerator -72 is (-72/-84)

Example 17: Express (196/256) as a rational number with denominator: (i) (128) (ii) -64

Solution:

Given (196/256)

To get denominator 128, we have to divide both numerator and denominator by 2

Then we get,

(196/256) ÷ (2/2)

= 98/128

Therefore, (196/256) as a rational number with denominator 128 is (98/128)

Given (196/256)

To get denominator -64, we have to divide both numerator and denominator by -4

Then we get,

(196/256) ÷ (-4/-4)

= -49/-64

Therefore, (196/256) as a rational number with denominator -64 is (-49/-64)

Given (196/256)

Example 18: Express (-14/42) as a rational number with numerator:

Solution:

(i) (-2) (ii) 7 (iii) 42 (iv) -70

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)

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 7 is (7/21)

Given (-14/42)

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

Then we get,

(-14/42) x (-3/-3)

= 42/-126

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

Given (-14/42)

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

Then we get,

(-14/42) x (5/5)

= -70/210

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

Example 19: Select from given rational numbers which can be written as a rational number with numerator 4.

(i) (1/14) (ii) 3/7 (iii) 2/5 (iv) -7/8, (vi) -4/3

Solution:

Given rational numbers that can be written as a rational number with numerator 4 are

(i) (1/14)

multiplying by (1/14) by 4, we can written as

(1/14) x 4/4 = 4/56

(ii) (2/5)

multiplying by (2/5) by 2,

we can written as

(2/5) x 2/2 = 4/10

(iii) (-4/3)

multiplying by (-4/3) by -1,

we can written as

(-4/3) x (-1/-1) = 4/-3

Therefore, rational numbers that can be written as a rational number with numerator 4 are (1/14), (2/5), (-4/3)

Example 20: Select from given rational numbers which can be written as a rational number with denominator 6.

(i) (6/12) (ii) 16/24 (iii) 13/48 (iv) -5/8, (vi) -4/7

Solution:

Given rational numbers that can be written as a rational number with denominator 6

Dividing both numerator and denominator by 2 we get,

(i) (6/12)

= (6/12 ÷ 2/2)

= (2/6)

(ii) 16/24

Given rational numbers that can be written as a rational number with denominator 4

Dividing both numerator and denominator by 4

we get, (i) (16/24)

= (16/24 ÷ 4/4)

= (4/6)

(iii) 13/48

Given rational numbers that can be written as a rational number with denominator 8

Dividing both numerator and denominator by 8 we get,

(13/48)

= (13/48 ÷ 8/8)

= (1.6/6)

Therefore, rational numbers that can be written as a rational number with denominator 6 are

(6/12), (16/24), (-4/3), and (13/48).

Example 21: In each of the following, find an equivalent form if the rational number having a common denominator.

(i) (2/3) and (7/12),

Solution: Given (2/3) and (7/12),

on multiplying both numerator and denominator by 4

(2/3) = (2 x 4)/(7 x 4)

= (8/12)

Equivalent forms with same denominators are 8/12 and 7/12

Example 22: In each of the following, find an equivalent form, if the rational number having a common denominator.

(i) (3/4), (ii) (5/6) and (13/12),

Given (i) (3/4), (ii) (5/6) and (13/12),

(i) (3/4)

on multiplying both numerator and denominator by 3

(3/4) = (3 x 3)/(4 x 3) = (9/12) and

(ii) (5/6)

on multiplying both numerator and denominator by 2

(5/6) = (5 x 2)/(6 x2)

= (10/12)

(iii) (13/12) = 13/12

Equivalent forms with same denominators are 9/12, 10/12 and 13/12

Example 23: In each of the following, find an equivalent form, if the rational number having a common denominator.

Given (i) (6/4), (ii) (7/8), (iii) (13/6), and 9/12.

(i) (6/4)

on multiplying both numerator and denominator by 6

(6/4) = (6 x 6)/(4 x 6) = (36/24)

(ii) (7/8),

on multiplying both numerator and denominator by 3

(7/8) = (7 x 3)/(8 x 3) = (21/24)

(iii) (13/6)

on multiplying both numerator and denominator by 2

(13/6) = (13 x 4)/(6 x 4) = (52/24)

(iv) 9/12

on multiplying both numerator and denominator by 2

(9/12) = (9 x 2)/(12 x 2) = (18/24)

Equivalent forms with same denominators are 36/24, 21/24, 52/24 and 18/24