Exercise 4.1
Question 1: Write down the numerator of each of the following rational numbers:
(i) ((-7)/5)
(ii) (15/(-4))
(iii) ((-17)/(-21))
(iv) (8/9)
(v) 5
Solution 1:
(i) ((-7)/5) = Numerator of ((-7)/5) is -7
(ii) (15/(-4)) = Numerator of (15/(-4)) is 15
(iii) ((-17)/(-21)) = Numerator of ((-17)/(-21)) is -17
(iv) (8/9) = Numerator of (8/9) is 8
(v) 5 = Numerator of 5 is 5
Question 2: Write down the denominator of each of the following rational numbers:
(i) ((-4)/5)
(ii) (11/(-34))
(iii) ((-15)/(-82))
(iv) 15
(v) 0
Solution 2:
(i) ((-4)/5) = Denominator of ((-4)/5) is 5
(ii) (11/(-34)) = Denominator of (11/(-34)) is -34
(iii) ((-15)/(-82))= Denominator of (15/(-82)) is -82
(iv) 15 = Denominator of 15 is 1
(v) 0 = Denominator of 0 is any non-zero integer
Question 3: Write down the rational number whose numerator is (-3) × 4, and whose denominator is (34 – 23) × (7 – 4).
Solution 3:
We found that Numerator is (-3) × 4 = -12
And Denominator = (34 – 23) × (7 – 4)
= 11 × 3 = 33
Hence, the rational number = ((-12)/33)
Question :4. Write down the rational numbers as integers: (7/1), ((-12)/1), (34/1), ((-73)/1), (95/1)
Solution 4:
Integers of (7/1), ((-12)/1), (34/1), ((-73)/1), (95/1) are 7, -12, 34, -73, 95
Question :5. Write the following integers as rational numbers: -15, 17, 85, -100
Solution 5:
The rational numbers of integers are ((-15)/1), (17/1), (85/1) and ((-100)/1)
Question :6. Write down the rational number whose numerator is the smallest three-digit number and denominator is the largest four-digit number.
Solution 6:
The smallest three-digit number are 100
The Largest four-digit number are 9999
Hence, the rational number of smallest three- digit number and the largest four digit number is = (100/9999)
Question :7. Separate positive and negative rational numbers from the following rational numbers:
((-5)/(-7)), (12/(-5)), (7/4), (13/(-9)), 0, ((-18)/(-7)), ((-95)/116), ((-1)/(-9))
Solution 7:
A rational number is positive if its numerator and denominator are either positive integers or negative integers. hence, positive rational numbers are: ((-5)/(-7)), , (7/4), ((-18)/(-7)), ((-1)/(-9)).
A rational number is negative integers if its numerator and denominator are such that one of them is positive integer and another one is a negative integer.
Hence, negative rational numbers are: (12/(-5)), (13/(-9)), ((-95)/116),
Question :8. Which of the following rational numbers are positive:
(i) ((-8)/7)
(ii) (9/8)
(iii) ((-19)/(-13))
(iv) ((-21)/13)
Solution 8:
A rational number is positive if its numerator and denominator are either positive integers or both negative integers.
Hence, the positive rational numbers are (9/8) and ((-19)/(-13))
Question :9. Which of the following rational numbers are negative:
(i) ((-3)/7)
(ii) ((-5)/(-8))
(iii) (9/(-83))
(iv) ((-115)/(-197))
Solution 9:
A rational number is negative integers if its numerator and denominator are such that one of them is positive integer and another one is a negative integer.
Hence, negative rational numbers are ((-3)/7) and (9/(-83))
Exercise 4.2
Question :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 1:
(i) ((-15)/(-28))
Multiplying the fraction’s numerator and denominator by -1:
= ((-15)/(-28)) × ((-1)/(-1)) = (15/28)
(ii) (6/(-9))
Multiplying the fraction’s numerator and denominator by -1:
= (6/(-9)) × ((-1)/(-1)) = ((-6)/9)
(iii) ((-28)/(-11))
Multiplying the fraction’s numerator and denominator by
= ((-28)/(-11)) × ((-1)/(-1)) = (28/11)
(iv) (19/(-7))
Multiplying the fraction’s numerator and denominator by
= (19/(-7)) × ((-1)/(-1)) = ((-19)/7)
Question 2: Express (3/5) as a rational number with numerator:
(i) 6
(ii) -15
(iii) 21
(iv) -27
Solution 2:
(i) (3/5)
To get 6 as a numerator of given value, we have to multiply both numerator and denominator by 2
Then, (3/5) × (2/2) = (6/10)
Hence, (3/5) as a rational number with numerator 6 is (6/10)
(ii) (3/5)
To get 15 as a numerator of the given value, we have to multiply both numerator and denominator by -5
Then, (3/5) × ((-5)/(-5)) = ((-15)/(-25))
Hence, (3/5) as a rational number with numerator -15 is ((-15)/(-25))
(iii) (3/5)
To get 21 as a numerator of the given value, we have to multiply both numerator and denominator by 7
Then, (3/5) × (7/7) = (21/35)
Hence, (3/5) as a rational number with numerator 21 is (21/35)
(iv) (3/5)
To get 27 as a numerator of the given value, we have to multiply both numerator and denominator by -9
Then, (3/5) × ((-9)/(-9)) = ((-27)/(-45))
Hence, (3/5) as a rational number with numerator -27 is ((-27)/(-45))
Question 3: Express (5/7) as a rational number with denominator:
(i) -14
(ii) 70
(iii) -28
(iv) -84
Solution 3:
(i) (5/7)
To get 14 as a denominator of the given value we have to multiply both numerator and denominator by -2
Then, (5/7) × ((-2)/(-2)) = ((-10)/(-14))
Hence, (5/7) as a rational number with denominator -14 is ((-10)/(-14))
(ii) (5/7)
To get 70 as a denominator of the given value we have to multiply both numerator and denominator by -2
Then, (5/7) × (10/10) = (50/70)
Hence, (5/7) as a rational number with denominator 70 is (50/70)
(iii) (5/7)
To get -28 as a denominator of the given value we have to multiply both numerator and denominator by -4
Then, (5/7) × ((-4)/(-4)) = ((-20)/(-28))
Hence, (5/7) as a rational number with denominator -28 is ((-28)/(-20))
(iv) (5/7)
To get-84 as a denominator of the given value we have to multiply both numerator and denominator by -12
Then, (5/7) × ((-12)/(-12)) = ((-60)/(-84))
Hence, (5/7) as a rational number with denominator -84 is ((-60)/(-84))
Question 4: Express (3/4) as a rational number with denominator:
(i) 20
(ii) 36
(iii) 44
(iv) -80
Solution 4:
(i) (3/4)
To get 20 as a denominator of the given value we have to multiply both by 5
Then, (3/4) × (5/5) = (15/20)
Hence (3/4) as a rational number with denominator 20 is (15/20)
(ii) (3/4)
To get 35as a denominator of the given value we have to multiply both by 9
Then, (3/4) × (9/9) = (27/36)
Hence (3/4) as a rational number with denominator 36 is (27/36)
(iii) (3/4)
To get 44 as a denominator of the given value we have to multiply both by 11
Then, (3/4) × (11/11) = (33/44)
Hence (3/4) as a rational number with denominator 44 is (33/44)
(iv) (3/4)
To get -80 as a denominator of the given value we have to multiply both by -20
Then, (3/4) × ((-20)/(-20)) = ((-60)/(-80))
Hence (3/4) as a rational number with denominator -80 is ((-60)/(-80))
Question :5. Express (2/5) as a rational number with numerator:
(i) -56
(ii) 154
(iii) -750
(iv) 500
Solution 5:
(i) (2/5)
To get-56 as a numerator of the given value we have to multiply both by -28
Then we get, (2/5) × ((-28)/(-28))
= ((-56)/(-140))
Hence (2/5) as a rational number with numerator -56 is ((-56)/(-140))
(ii) (2/5)
To get 154 as a numerator of the given value we have to multiply both by 77
Then we get, (2/5) × (77/77)
= (154/385)
Hence (2/5) as a rational number with numerator 154 is (154/385)
(iii) (2/5)
To get -750 as a numerator of the given value we have to multiply both by -375
Then we get, (2/5) × ((-375)/(-375)) = ((-750)/(-1875))
Hence (2/5) as a rational number with numerator -750 is ((-750)/(-1875))
(iv) (2/5)
To get 500 as a numerator of the given value we have to multiply both by 250
Then, (2/5) × (250/250) = (500/1250)
Hence (2/5) as a rational number with numerator 500 is (500/1250)
Question :6. Express ((-192)/108) as a rational number with numerator:
(i) 64
(ii) -16
(iii) 32
(iv) -48
Solution 6:
(i) ((-192)/108)
To get 64 as a numerator of the given value we have to divide both by -3
Then, ((-192)/108) ÷ ((-3)/(-3)) = (64/(-36))
Hence ((-192)/108) as a rational number with numerator 64 is (64/(-36))
(ii) ((-192)/108)
To get -16 as a numerator of the given value we have to divide both by 12
Then, ((-192)/108) ÷ (12/12) = ((-16)/9)
Hence ((-192)/108) as a rational number with numerator -16 is ((-16)/9)
(iii)((-192)/108)
To get 32 as a numerator of the given number we have to divide both by -6
The, ((-192)/108) ÷ ((-6)/(-6)) =(32/(-18))
Hence ((-192)/108) as a rational number with numerator 32 is (32/-18)
(iv) ((-192)/108)
To get -48 as a numerator of the given number we have to divide both by 4
Then, ((-192)/108) ÷ (4/4) = ((-48)/27)
Hence ((-192)/108) as a rational number with numerator -48 is ((-48)/27) (-48/27)
Question :7. Express (169/(-294)) as a rational number with denominator:
(i) 14
(ii) -7
(iii) -49
(iv) 1470
Solution 7:
(i) (169/(-294))
To get 14 as a denominator of the given number we have to divide both numerator and denominator by -21
Then, (169/(-294)) ÷ (21/(-21)) = ((-8)/14)
Hence (169/(-294)) as a rational number with denominator 14 is ((-8)/14)
(ii) (169/(-294))
To get -7 as a denominator of the given number we have to divide both numerator and denominator by 42
Then, (169/(-294)) ÷ (42/42) = (4/(-7))
Hence (169/(-294)) as a rational number with denominator -7 is (4/(-7))
(iii) (169/(-294))
To get -49 as a denominator of the given number we have to divide both numerator and denominator by 6
Then, (169/(-294)) ÷ (6/6) = (28/(-49))
Hence (169/(-294)) as a rational number with denominator -49 is (28/(-49))
(iv) (169/(-294))
To get 1407 as a denominator of the given number we have to multiply both numerator and denominator by -5
Then, (169/(-294)) × ((-5)/(-5)) = ((-840)/1470)
Hence (169/(-294)) as a rational number with denominator 1470 is ((-840)/1470)
Question :8. Write ((-14)/42) in a form so that the numerator is equal to:
(i) -2
(ii) 7
(iii) 42
(iv) -70
Solution 8:
(i) ((-14)/42)
To get -2 as a numerator of the given number we have to divide both numerator and denominator by 7
Then, ((-14)/42) ÷ (7/7) = ((-2)/6)
Hence ((-14)/42) as a rational number with numerator -2 is ((-2)/6)
(ii) ((-14)/42)
To get7 as a numerator of the given number we have to divide both numerator and denominator by -2
Then, ((-14)/42) ÷ ((-2)/(-2)) = (7/(-21))
Hence ((-14)/42) as a rational number with numerator -14 is (7/(-21))
(iii) ((-14)/42)
To get 42 as a numerator of the given number we have to multiply both numerator and denominator by -3
Then, ((-14)/42) × ((-3)/(-3)) = (42/(-126))
Hence ((-14)/42) as a rational number with numerator 42 is (42/(-126))
(iv) ((-14)/42)
To get -70 as a numerator of the given number we have to multiply both numerator and denominator by 5
Then, ((-14)/42) × (5/5) = ((-70)/210)
Hence ((-14)/42) as a rational number with numerator -70 is ((-70)/210)
Question 9: Select those rational numbers which can be written as a rational number with numerator 6: (1/22), (2/3), (3/4), (4/(-5)), ((-6)/7), ((-7)/8)
Solution 9:
Rational number with numerator 6 are:
=(1/22)
Multiplying by 6, (1/22) written as
((1 ×6)/(22 ×6)) = (6/132)
=(2/3)
Multiplying by 3, (2/3) written as
((2 ×3)/(3 ×3)) = (6/9)
=(3/4)
Multiplying by 2, (3/4) written as
((3×2)/(4×2) ) = (6/8)
= ((-6)/7)
Multiplying by -1, ((-6)/7) written as
((-6 ×(-1))/(7 ×(-1))) = (6/(-7))
Hence, rational numbers can be written as a rational number with numerator 6 are (1/22), (2/3), (3/4), and ((-6)/7)
Question :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 10:
Rational numbers that can be written as rational number with denominator 4 are:
(7/8) ÷ (2/2)= (3.5/4)
On dividing both denominator and denominator by 2
(64/16) ÷ (4/4)= (16/4)
On dividing both denominator and numerator by 4
(36/(-12)) ÷ ((-3)/(-3))= ((-12)/4)
On dividing both denominator and numerator by -3
(5/(-4)) ÷ ((-1)/(-1))= ((-5)/4)
On multiplying both denominator and numerator by -1
(140/28) ÷ (7/7)= (20/4)
On dividing both numerator and denominator by 7
Question 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 11:
(i) (3/4) and (5/12) : multiply by 3
(3/4) = ((3 × 3))/((4 × 3)) = (9/12)
Equivalent forms are (9/12) and (5/12) having same denominators
(ii) (2/3), (7/6) and (11/12) : multiply by 4
(2/3) = ((2 × 4))/((3 × 4)) = (8/12)
(7/6) = ((7 × 2))/((6 × 2)) = (14/12)
Equivalent forms are (8/12), (14/12) and (11/12) having same denominators
(iii) (5/7), (3/8), (9/14) and (20/21)
(5/7) multiply by 24
= ((5 × 24))/((7 × 24)) = (120/168)
(3/8) multiply by 21
= ((3 × 21))/((8 × 21)) = (63/168)
(9/14) multiply by 12
= ((9 × 12))/((14 × 12)) = (160/168)
(20/21) multiply by 8
= ((20 × 8))/((21×8)) = (160/168)
Forms are (120/168), (63/168), (108/168) and (160/168) having same denominators.
Exercise 4.3
Question 1: Determine whether the following rational numbers are in the lowest form or not:
(i) (65/84)
(ii) ((-15)/32)
(iii) (24/128)
(iv) ((-56)/(-32))
Solution 1:
(i) (65/84) = 65 and 84 have no common factor their HCF is 1.
Thus, (65/84) is in its lowest form.
(ii) ((-15)/32) = -15 and 32 have no common factor i.e., their HCF is 1.
Thus, ((-15)/32) is in its lowest form.
(iii) (24/128) = HCF of 24 and 128 is not 1.
Thus, given rational number is not in its simplest form.
(iv) ((-56)/(-32)) = HCF of 56 and 32 is 8 and also not equal to 1.
Thus, given rational number is not in its simplest form.
Question 2: Express each of the following rational numbers to the lowest form:
(i) (4/22)
(ii) ((-36)/180)
(iii) (132/(-428))
(iv) ((-32)/(-56))
Solution 2:
(i) (4/22) HCF of 4 and 22 is 2
Divide the given number by its HCF
(4 ÷ 2/22 ÷ 2) = (2/11)
Hence, (2/11) is the simplest form of (4/22)
(ii) ((-36)/180) HCF of 36 and 180 is 36
Divide the given number by its HCF
(-36 ÷ 36/180 ÷ 36) = ((-1)/5)
Hence, ((-1)/5) is the simplest form of ((-36)/180)
(iii) (132/(-428)) HCF of 132 and 428 is 4
Divide the given number by its HCF
(132 ÷ (4/(-428)) ÷ 4) = (33/(-107))
Hence, (33/(-107)) is the simplest form of (132/(-428))
(iv) ((-32)/(-56)) HCF of 32 and 56 is 8
Divide the given number by its HCF
(-32 ÷ (8/(-56)) ÷ 8) = (4/7)
Hence, (4/7) is the simplest form of ((-32)/(-56))
Question :3. Fill in the blanks:
(i) ((-5)/7) = (…/35) = (…/49)
(ii) ((-4)/(-9)) = (…/18) = (12/…)
(iii) (6/(-13)) = ((-12)/…) = (24/…)
(iv) ((-6)/…) = (3/11) = (…/(-55))
Solution 3:
(i) ((-5)/7) = (…/35) = (…/49)= ((-5)/7) = ((-25)/35) = ((-35)/49)
Description:
((-5)/7) = (…/35) = (…/49)
On multiplying by 5:
((-5)/7) × (5/5) = ((-25)/35)
On multiplying by 7
((-5)/7) × (7/7) = ((-35)/49)
(ii) ((-4)/(-9)) = (…/18) = (12/…)= ((-4)/(-9)) = (8/18) = (12/27)
Description:
((-4)/(-9)) = (…/18) = (12/…)
On multiplying by -2
((-4)/(-9)) × ((-2)/(-2)) = (8/18)
on multiplying by -3
((-4)/(-9)) × ((-3)/(-3)) = (12/27)
(iii)(6/(-13)) = ((-12)/…) = (24/…)= (6/(-13)) = ((-12)/26) = (24/(-52))
Description:
(6/(-13)) = ((-12)/…) = (24/…)
On multiplying by -2
(6/(-13)) × ((-2)/(-2)) = ((-12)/26)
On multiplying by 4
(6/(-13)) × (4/4) = (24/(-52))
(iv) ((-6)/…) = (3/11) = (…/(-55))= ((-6)/(-22)) = (3/11) = ((-15)/(-55))
Description:
((-7)/…) = (3/11) = (…/(-55))
0n multiplying by -2
(3/11) × ((-2)/(-2)) = ((-6)/(-22))
On multiplying by -5
(3/11) × ((-5)/(-5)) = ((-15)/(-55))
Exercise 4.4
Question 1: Write each of the following rational numbers in the standard form:
(i) (2/10)
(ii) ((-8)/36)
(iii) (4/(-16))
(iv) ((-15)/(-35))
(v) (299/(-161))
(vi) ((-63)/(-210))
(vii) (68/(-119))
(viii) ((-195)/275)
Solution 1:
(i) (2/10) HCF of 2 and 10 is 2
Dividing the fraction by HCF i.e. 2:
(2/10) ÷ (2/2) = (1/5)
Hence, (1/5) is the standard form of (2/10).
(ii) ((-8)/36) HCF of 8 and 36 is 4
Dividing the fraction by HCF i.e. 4:
((-8)/36) ÷ (4/4) = ((-2)/9)
Hence ((-2)/9) is the standard form of ((-8)/36).
(iii) (4/(-16))
Denominator is negative here, so we multiply fraction by -1
(4/(-16)) × ((-1)/(-1)) = (4/(-16))
HCF of 4 and 16 is 4
Dividing the fraction by HCF i.e. 4:
(4/(-16)) ÷ (4/4) = ((-1)/4)
Hence ((-1)/4) is the standard form of (4/(-16)).
(iv) ((-15)/(-35))
Denominator is negative here, so we multiply fraction by -1
((-15)/(-35)) × ((-1)/(-1)) = (15/35)
HCF of 15 and 35 is 4
Dividing the numerator and denominator by HCF i.e. 5:
((-15)/(-35)) ÷ (5/5) = (3/7)
Hence (3/7) is the standard form of ((-15)/(-35))
(v) (299/(-161))
Denominator is negative here, so we multiply fraction by -1
(299/(-161)) × ((-1)/(-1)) = ((-299)/161)
The HCF of 299 and 161 is 23
Dividing the numerator and denominator by HCF i.e. 23:
(299/(-161)) ÷ (23/23) = ((-13)/7)
Hence ((-13)/7) is the standard form of (299/(-161))
(vi) ((-63)/(-210))
HCF of 63 and 210 is 21
Dividing the numerator and denominator by HCF i.e. 21:
((-63)/(-210)) ÷ (21/21) = ((-3)/(-10)) = (3/10)
Hence (3/10) is the standard form of ((-63)/(-210))
(vii) (68/(-119))
Here denominator is negative so we have multiply both fraction by -1
(68/(-119)) × ((-1)/(-1)) = ((-68)/119)
The HCF of 68 and 119 is 17
Dividing the fraction by HCF i.e. 17:
((-68)/119) ÷ (17/17) = ((-4)/7)
Hence, ((-4)/7) is the standard form of (68/(-119))
(viii) ((-195)/275)
The HCF of 195 and 257 is 5
Dividing the fraction by HCF i.e. 5:
((-195)/275) ÷ (5/5) = ((-39)/5)
Hence, ((-39)/5) is the standard form of ((-195)/275)
Exercise 4.5
Question :1. Which of the following rational numbers are equal?
(i) ((-9)/12) and (8/(-12))
(ii) ((-16)/20) and (20/(-25))
(iii) ((-7)/21) and (3/(-9))
(iv) ((-8)/(-14)) and (13/21)
Solution 1:
(i) ((-9)/12) and (8/(-12))
By dividing the fraction by their HCF i.e. 3
Then the standard form of ((-9)/12) is ((-3)/4)
Dividing the fraction of given number by their HCF i.e. by 4
The standard form of (8/(-12)) = ((-2)/3)
Since, the standard forms of two rational numbers are not same. Hence, they are not equal.
(ii) ((-16)/20) and (20/(-25))
Multiplying fraction of ((-16)/20) by the denominator of (20/(-25)) i.e. -25.
((-16)/20) × ((-25)/(-25)) = (400/(-500))
Multiplying the fraction of (20/(-25)) by the denominator of
((-16)/20) i.e. 20
(20/(-25)) × (20/20) = (400/(-500))
Clearly, the numerators of the above obtained rational numbers are equal.
Hence, the given rational numbers are equal
(iii) ((-7)/21) and (3/(-9))
Multiplying fraction of ((-7)/21) by the denominator of (3/(-9))
i.e. -9.
((-7)/21) × ((-9)/(-9)) = (63/(-189))
Now multiply the fraction of (3/(-9)) by the denominator of
((-7)/21) i.e. 21
(3/(-9)) × (21/21) = (63/(-189))
Clearly, the numerators of the above obtained rational numbers are equal.
Hence, the given rational numbers are equal
(iv) ((-8)/(-14)) and (13/21)
Multiplying fraction of ((-8)/(-14)) by the denominator of (13/21)
i.e. 21
((-8)/(-14)) × (21/21) = ((-168)/(-294))
Now multiply the fraction of (13/21) by the denominator of
((-8)/(-14)) i.e. -14
(13/21) × ((-14)/(-14)) = ((-182)/(-294))
Clearly, the numerators of the above obtained rational numbers are not equal.
Hence, the given rational numbers are also not equal
Question 2: In each of the following pairs represent a pair of equivalent rational numbers, find the values of x.
(i) (2/3) and (5/x)
(ii) ((-3)/7) and (x/4)
(iii) (3/5) and (x/(-25))
(iv) (13/6) and ((-65)/x)
Solution 2:
(i) (2/3) and (5/x)
They are equivalent rational number so (2/3) = (5/x)
x = ((5 × 3))/2
x = (15/2)
(ii) ((-3)/7) and (x/4)
they are equivalent rational number so ((-3)/7) = (x/4)
x = ((-3 × 4))/7
x = ((-12)/7)
(iii) Given (3/5) and (x/(-25))
They are equivalent rational number so (3/5) = (x/(-25))
x = ((3 × -25))/5
x = ((-75)/5)
x = -15
(iv) (13/6) and ((-65)/x)
They are equivalent rational number so (13/6) = ((-65)/x)
x = 6/13 x (- 65)
x = 6 x (-5)
x = -30
Question 3: In each of the following, fill in the blanks so as to make the statement true:
(i) A number which can be expressed in the form p/q, where p and q are integers and q is not equal to zero, is called a _______.
(ii) If the integers p and q have no common divisor other than 1 and q is positive, then the rational number (p/q) is said to be in the _______.
(iii) Two rational numbers are said to be equal, if they have the same _______ form
(iv) If m is a common divisor of a and b, then (a/b) = (a ÷ m)/ _______.
(v) If p and q are positive Integers, then p/q is a _______ rational number and (p/(-q)) is a ______ rational number.
(vi) The standard form of -1 is _______.
(vii) If (p/q) is a rational number, then q cannot _______.
(viii) Two rational numbers with different numerators are equal, if their numerators are in the same _______ as their denominators.
Solution 3:
(i) A number which can be expressed in the form p/q, where p and q are integers and q is not equal to zero, is called a Rational number.
(ii) If the integers p and q have no common divisor other than 1 and q is positive, then the rational number (p/q) is said to be in the Standard form.
(iii) Two rational numbers are said to be equal, if they have the same Standard form
(iv) If m is a common divisor of a and b, then (a/b) = (a ÷ m)/ b ÷ m.
(v) If p and q are positive Integers, then p/q is a Positive rational number and (p/(-q)) is a negative rational number.
(vi) The standard form of -1 is((-1)/1) .
(vii) If (p/q) is a rational number, then q cannot Zero .
(viii) Two rational numbers with different numerators are equal, if their numerators are in the same Ratio as their denominators.
Question 4: In each of the following state if the statement is true (T) or false (F):
(i) The quotient of two integers is always an integer.
(ii) Every integer is a rational number.
(iii) Every rational number is an integer.
(iv) Every traction is a rational number.
(v) Every rational number is a fraction.
(vi) If (a/b) is a rational number and m any integer, then (a/b) = ((a x m))/((b x m))
(vii) Two rational numbers with different numerators cannot be equal.
(viii) 8 can be written as a rational number with any integer as denominator.
(ix) 8 can be written as a rational number with any integer as numerator.
(x) (2/3) is equal to (4/6).
Solution 4:
(i) The quotient of two integers is always an integer.
(ii) Every integer is a rational number.
(iii) Every rational number is an integer.
(iv) Every traction is a rational number.
(v) Every rational number is a fraction.
(vi) If (a/b) is a rational number and m any integer, then (a/b) = ((a x m))/((b x m))
(vii) Two rational numbers with different numerators cannot be equal.
(viii) 8 can be written as a rational number with any integer as denominator.
(ix) 8 can be written as a rational number with any integer as numerator.
(x) (2/3) is equal to (4/6).
(i) False
Description:
The quotient of two integers is not necessary to be an integer
(ii) True
Description:
Every integer can be expressed in the form of p/q, where q is not zero.
(iii) False
Description:
Every rational number is not necessary to be an integer
(iv) True
Description:
According to definition of rational number i.e. every integer can be expressed in the form of p/q, where q is not zero.
(v) False
Description:
It is not necessary that every rational number is a fraction.
(vi) True
Description:
If (a/b) is a rational number and m any integer, then (a/b) = ((a x m))/((b x m)) is one of the rule of
rational numbers
(vii) False
Description:
They can be equal, when simplified further.
(viii) False
Description:
8 can be written as a rational number but we can’t write 8 with any integer as denominator.
(ix) False
Description:
8 can be written as a rational number but we can’t with any integer as numerator.
(x) True
Description:
When convert it into standard form they are equal
Exercise 4.6
Question :1. Draw the number line and represent following rational number on it:
(i) (2/3)
(ii) (3/4)
(iii) (3/8)
(iv) ((-5)/8)
(v) ((-3)/16)
(vi) ((-7)/3)
(vii) (22/(-7))
(viii) ((-31)/3) (-31/3)
Solution 1:
(i) We know that (2/3) is greater than 2 and less than 3.
∴ it lies between 2 and 3. It can be represented on number line as,
Question :2. Which of the two rational numbers in each of the following pairs of rational number is greater?
(i) ((-3)/8), 0
(ii) (5/2), 0
(iii) ((-4)/11), (3/11)
(iv) ((-7)/12), (5/(-8))
(v) (4/(-9)), ((-3)/(-7))
(vi) ((-5)/8), (3/(-4))
(vii) (5/9), ((-3)/(-8))
(viii) (5/(-8)), ((-7)/12)
Solution 2:
(i) Given ((-3)/8), 0
We know that every positive rational number is greater than zero and every negative
rational number is smaller than zero. Thus, – ((-3)/8) > 0
(ii) Given (5/2), 0
We know that every positive rational number is greater than zero and every negative rational number is smaller than zero. Thus, (5/2) > 0
(iii) Given ((-4)/11), (3/11)
We know that every positive rational number is greater than zero and every negative rational number is smaller than zero, also the denominator is same in given question now we have to compare the numerator, thus((-4)/11) < (3/11).
Question :3. Which of the two rational numbers in each of the following pairs of rational numbers is smaller?
(i) ((-6)/(-13)), (7/13)
(ii) (16/(-5)), 3
(iii) ((-4)/3), (8/(-7))
(iv) ((-12)/5), (-3)
Solution 3:
(i) Given ((-6)/(-13)), (7/13)
Here denominator is same Hence compare the numerator,
Thus ((-6)/(-13)) < (7/13)
(ii) Given (16/(-5)), 3
We know that 3 is a whole number with positive sign
Hence (16/(-5)) < 3
(iii) Given ((-4)/3), (8/(-7))
Consider ((-4)/3)
Multiply both numerator and denominator by 7 then we get
((-4)/3) × (7/7) = ((-28)/21)…… (1)
Now consider (8/(-7))
Multiply both numerator and denominator by 3 we get
(8/(-7)) × (3/3) = ((-24)/21)…… (2)
The denominator is same in equation (1) and (2) now we have to compare the numerator, thus ((-4)/3) < (8/(-7))
(iv) Given ((-12)/5), (-3)
Now consider ((-3)/1)
Multiply both numerator and denominator by 5 we get
((-3)/1) × (5/5) = ((-15)/5)
The denominator is same in above equation, now we have to compare the numerator, thus ((-12)/5) > (-3)
Question :4. Fill in the blanks by the correct symbol out of >, =, or <:
(i) ((-6)/7) …. (7/13)
(ii) ((-3)/5) …. ((-5)/6)
(iii) ((-2)/3) …. (5/(-8))
(iv) 0 …. ((-2)/5)
Solution 4:
(i) ((-6)/7) < (7/13)
Description:
Because every positive number is greater than a negative number.
(ii) ((-3)/5) > ((-5)/6)
Description:
Consider ((-3)/5)
Multiply both numerator and denominator by 6 then we get
((-3)/5) × (6/6) = ((-18)/30)…… (1)
Now consider (-5/6)
Multiply both numerator and denominator by 5 we get
((-5)/6) × (5/5) = ((-25)/30)…… (2)
The denominator is same in equation (1) and (2) now we have to compare the numerator, thus ((-3)/5) >((-5)/6)
(iii) ((-2)/3) < (5/(-8))
Description:
Consider ((-2)/3)
Multiply both numerator and denominator by 8 then we get
((-2)/3) × (8/8) = ((-16)/24)…… (1)
Now consider (5/(-8))
Multiply both numerator and denominator by 3 we get
(5/(-8)) × (3/3) = (15/(-24))…… (2)
The denominator is same in equation (1) and (2) now we have to compare the numerator, thus ((-2)/3) < (5/(-8))
(iv) 0 > ((-2)/5)
Description:
Because every positive number is greater than a negative number
Question :5. Arrange the following rational numbers in ascending order:
(i) (3/5), ((-17)/(-30)), (8/(-15)), ((-7)/10)
(ii) ((-4)/9), (5/(-12)), (7/(-18)), (2/(-3))
Solution 5:
(i) Given (3/5), ((-17)/(-30)), (8/(-15)), ((-7)/10)
The LCM of 5, 30, 15 and 10 is 30
Multiplying the numerators and denominators to get the denominator equal to the LCM i.e. 30
Consider (3/5)
Multiply both numerator and denominator by 6, then we get
(3/5) × (6/6) = (18/30) ….. (1)
Consider (8/(-15))
Multiply both numerator and denominator by 2, then we get
(8/(-15)) × (2/2) = (16/(-30)) ….. (2)
Consider ((-7)/10)
Multiply both numerator and denominator by 3, then we get
((-7)/10) × (3/3) = ((-21)/30) ….. (3)
In the above equation, denominators are same
Now on comparing the ascending order is:
((-7)/10) < (8/(-15)) < ((-17)/(-30)) < (3/5)
(ii) Given ((-4)/9), (5/(-12)), (7/(-18)), (2/(-3))
The LCM of 9, 12, 18 and 3 is 36
Multiplying the numerators and denominators to get the denominator equal to the LCM i.e. 36
Consider ((-4)/9)
Multiply both numerator and denominator by 4, then we get
((-4)/9) × (4/4) = ((-16)/36) ….. (1)
Consider (5/(-12))
Multiply both numerator and denominator by 3, then we get
(5/(-12)) × (3/3) = (15/(-36)) ….. (2)
Consider (7/(-18))
Multiply both numerator and denominator by 2, then we get
(7/(-18)) × (2/2) = (14/(-36)) ….. (3)
Consider (2/(-3))
Multiply both numerator and denominator by 12, then we get
(2/(-3)) × (12/12) = (24/(-36)) ….. (4)
In the above equation, denominators are same
Now on comparing the ascending order is:
(2/(-3)) < ((-4)/9) < (5/(-12)) < (7/(-18))
Question :6. Arrange the following rational numbers in descending order:
(i) (7/8), (64/16), (39/(-12)), (5/(-4)), (140/28)
(ii) ((-3)/10),(17/(-30)),(7/(-15)), ((-11)/20)
Solution 6:
(i) Given (7/8), (64/16), (39/(-12)), (5/(-4)), (140/28)
The LCM of 8, 16, 12, 4 and 28 is 336
Multiplying the numerators and denominators to get the denominator equal to the LCM i.e. 336
Consider (7/8)
Multiply both numerator and denominator by 42, then we get
(7/8) × (42/42) = (294/336) ….. (1)
Consider (64/16)
Multiply both numerator and denominator by 21, then we get
(64/16) × (21/21) = (1344/336) ….. (2)
Consider (39/(-12))
Multiply both numerator and denominator by 28, then we get
(39/(-12)) × (28/28) = ((-1008)/336) (-1008/336) ….. (3)
Consider (5/(-4))
Multiply both numerator and denominator by 84, then we get
(5/(-4)) × (84/84) = ((-420)/336) ….. (4)
In the above equation, denominators are same
Now on comparing the descending order is:
(140/28) > (64/16) > (7/8) > (5/(-4)) > (39/(-12))
(ii) Given ((-3)/10),(17/(-30)),(7/(-15)), ((-11)/20)
The LCM of 10, 30, 15 and 20 is 60
Multiplying the numerators and denominators to get the denominator equal to the LCM i.e. 60
Consider ((-3)/10)
Multiply both numerator and denominator by 6, then we get
((-3)/10) × (6/6) = ((-18)/60) ….. (1)
Consider (17/(-30))
Multiply both numerator and denominator by 2, then we get
(17/(-30)) × (2/2) = (34/(-60)) ….. (2)
Consider (7/(-15))
Multiply both numerator and denominator by 4, then we get
(7/(-15)) × (4/4) = (28/(-60)) ….. (3)
In the above equation, denominators are same
Now on comparing the descending order is:
((-3)/10) > (7/(-15)) > ((-11)/20) > (17/(-30))
Question :7. Which of the following statements are true:
(i) The rational number (29/23) lies to the left of zero on the number line.
(ii) The rational number ((-12)/(-17)) lies to the left of zero on the number line.
(iii) The rational number (3/4) lies to the right of zero on the number line.
(iv) The rational number ((-12)/(-5)) and ((-7)/17) are on the opposite side of zero on the number line.
(v) The rational number ((-2)/15) and (7/(-31)) are on the opposite side of zero on the number line.
(vi) The rational number ((-3)/(-5)) is on the right of ((-4)/7) on the number line.
Solution 7:
(i) False
Description:
It lies to the right of zero because it is a positive number.
(ii) False
Description:
It lies to the right of zero because it is a positive number.
(iii) True
Description:
Always positive number lie on the right of zero
(iv) True
Description:
Because they are of opposite sign
(v) False
Description:
Because they both are of same sign
(vi) True
Description:
They both are of opposite signs and positive number is greater than the negative number. Thus, it is on the right of the negative number.