Rational Numbers – Solved Examples
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