Exercise 4.1
Page-60Question 1:
What are rational numbers? Give examples of five positive and five negative rational numbers. Is there any rational number which is neither positive nor negative? Name it.
Answer 1:
The numbers that are in the form of pq, where p and q are integers and q ≠0, are called rational numbers.
For example:
Five positive rational numbers:
57,−3−4,78,−14−15,59
Five negative rational numbers:
−37,−38,8−9,−1925,8−25
Yes, there is a rational number that is neither positive nor negative, i.e. zero (0).
Question 2:
Which of the following are rational numbers?
(i) 5-8
(ii) -611
(iii) 715
(iv) -8-12
(v) 6
(vi) −3
(vii) 0
(viii) 01
(ix) 10
(x) 00
Answer 2:
i) 5−8 is a rational number because it is in the form of pq, where p and q are integers and q≠0. ii)−611 is a rational number because it is in the form of pq, where p and q are integers and q≠0. iii)−1315is a rational number because it is in the form of pq, where p and q are integers and q≠0. iv)−8−12is a rational number because it is in the form of pq, where p and q are integers and q≠0. v) 6 is a rational number because it is in the form of pq, where p and q are integers and q≠0. vi) -3 is a rational number because it is in the form of pq, where p and q are integers and q≠0. vii) 0 = 01is a rational number because it is in the form of pq, where p and q are integers and q≠0. viii)01 is a rational number because it is in the form of pq, where p and q are integers and q≠0. ix)10is not a rational number because, here, q = 0.x)00 is not a rational number because, here, q = 0.
Question 3:
Write down the numerator and the denominator of each of the following rational numbers:
(i) 819
(ii) 5-8
(iii) -1315
(iv) -8-11
(v) 9
Answer 3:
(i) 819
Numerator = 8
Denominator =19
(ii)5-8
Numerator = 5
Denominator = −8
(iii) -135
Numerator = −13
Denominator =15
(iv)-8-11
Numerator = −8
Denominator = −11
(v) 9
i.e 91
Numerator = 9
Denominator = 1
Question 4:
Write each of the following integers as a rational number. Write the numerator and the denominator in each case.
(i) 5
(ii) −3
(iii) 1
(iv) 0
(v) −23
Answer 4:
(i) 5
The rational number will be 51.
Numerator = 5
Denominator = 1
(ii) -3
The rational number will be -31.
Numerator = -3
Denominator = 1
(iii)1
The rational number will be 11.
Numerator = 1
Denominator = 1
(iv) 0
The rational number will be 01.
Numerator =0
Denominator = 1
(v) -23
The rational number will be -231.
Numerator = -23
Denominator = 1
Question 5:
Which of the following are positive rational numbers?
(i) 3-5
(ii) -1115
(iii) -5-8
(iv) 3753
(v) 03
Answer 5:
Positive rational numbers:
(iii) -5-8
(iv) 3753
(vi) 8 because 8 can be written as 81, where 1≠0.
0 is neither positive nor negative.
Question 6:
Which of the following are negative rational numbers?
(i) -15-14
(ii) 0
(iii) -57
(iv) 4-9
(v) −6
(vi) 1-2
Answer 6:
Negative rational numbers:
(iii) -57
(iv) 4-9
(v) -6
(vi) 1-2
Question 7:
Find four rational numbers equivalent to each of the following.
(i) 611
(ii) -38
(iii) 7-15
(iv) 8
(v) 1
(vi) −1
Answer 7:
(i) Following are the four rational numbers that are equivalent to 611.
6×211×2,6×311×3,6×411×4 and 6×511×5i.e. 1222,1833,2444 and 3055
(ii) Following are the four rational numbers that are equivalent to -38.
-3×28×2,-3×38×3, -3×48×4 and -3×58×5
i.e. -616, -924, -1232 and -1540
(iii) Following are the four rational numbers that are equivalent to 7-15.
7×2−15×2, 7×3−15×3, 7×4−15×4 and 7×5−15×5i.e. 14−30, 21−45, 28−60 and 35−75
(iv) Following are the four rational numbers that are equivalent to 8, i.e. 81.
8×21×2, 8×31×3, 8×41×4 and 8×51×5i.e. 162, 243, 324 and 405
(v) Following are the four rational numbers that are equivalent to -1, i.e. 11.
1×21×2, 1×31×3, 1×41×4 and 1×51×5i.e. 22,33, 44 and 55
(vi) Following are the four rational numbers that are equivalent to -1, i.e. -11.
−1×21×2, −1×31×3, −1×41×4 and −1×51×5i.e. −22,−33, −44 and −55
Question 8:
Write each of the following as a rational number with positive denominator.
(i) 12-17
(ii) 1-2
(iii) -8-19
(iv) 11-6
Answer 8:
(i) 12×(−1)(−17)×(−1)=−1217
(ii) 1×(−1)(−2)×(−1)=−12
(iii) −8−19=−8×(−1)(−19)×(−1)=819
(iv) 11×(−1)-6×(−1)=−116
Question 9:
Express 58 as a rational number with numerator
(i) 15
(ii) −10
Answer 9:
(i) Numerator of 58 is 5.
5 should be multiplied by 3 to get 15.
Multiplying both the numerator and the denominator by 3:
5×38×3=1524 58=1524
(ii) Numerator of 58 is 5.
5 should be multiplied by −2 to get −10.
Multiplying both the numerator and the denominator by −2:
5×(−2)8×(−2)=−10−16 58=−10−16
Question 10:
Express 47 as a rational number with denominator
(i) 21
(ii) −35
Answer 10:
(i) Denominator of 47 is 7.
7 should be multiplied by 3 to get 21.
Multiplying both the numerator and the denominator by 3:
4×37×3= 1221
4×37×3= 47
(ii)
Denominator of 47 is 7.
7 should be multiplied by -5 to get −35.
Multiplying both the numerator and the denominator by −5:
4×(−5)7×(−5)=−20−35 47=−20−35
Question 11:
Express -1213 as a rational number with numerator
(i) −48
(ii) 60
Answer 11:
(i) Numerator of -1213 is −12.
−12 should be multiplied by 4 to get 48.
Multiplying both the numerator and the denominator by 4:
−12×413×4=−4852−1213=−4852
(ii) Numerator of -1213 is −12.
−12 should be multiplied by −5 to get 60
Multiplying its numerator and denominator by -5:
−12×(−5)13×(−5)=60−65−1213=60−65
Question 12:
Express -811 as a rational number with denominator
(i) 22
(ii) −55
Answer 12:
(i) Denominator of-811 is 11.
Clearly, 11×2= 22
Multiplying both the numerator and the denominator by 2:
−8×211×2=−1622−811=−1622
(ii) Denominator of-811 is 11.
Clearly, 11×5=55
Multiplying both the numerator and the denominator by 5:
−8×511×5=−4055 −811=−4055
Question 13:
Express 14-5 as a rational number with numerator
(i) 56
(ii) −70
Answer 13:
(i) Numerator of 14-5 is 14.
Clearly, 14×4=56
Multiplying both the numerator and the denominator by 4:
14×4−5×4=56−20
14-5=56−20
(ii) −70
Numerator of 14-5 is 14.
Clearly, 14×(−5)=−70
Multiplying both the numerator and the denominator by -5:
14×(−5)(−5)×(−5)=−7025
14-5=−7025
Question 14:
Express 13-8 as a rational number with denominator
(i) −40
(ii) 32
Answer 14:
(i) Denominator of 13-8 is −8.
Clearly, (−8)×5= −40
Multiplying both the numerator and the denominator by 5:
13×5−8×5=65−4013−8=65−40
(ii) Denominator of 13-8 is −8.
Clearly, (−8)×(−4)= 32
Multiplying both the numerator and the denominator by −4:
13×(−4)−8×(−4)=−5232
13-8=−5232
Question 15:
Express -3624 as a rational number with numerator
(i) −9
(ii) 6
Answer 15:
(i) Numerator of -3624 is -36.
Clearly, (−36) ÷ 4= (−9)
Dividing both the numerator and the denominator by 4:
−36÷424÷4=−96
(ii) Numerator of -3624 is −36.
Clearly, (−36) ÷ ( −6) = 6
Dividing both the numerator and the denominator by -6:
−36÷(−6)24÷(−6)=6-4
-3624= 6-4
Question 16:
Express 84-147 as a rational number with denominator
(i) 7
(ii) −49
Answer 16:
(i) Denominator of 84-147 is −147.
∴ −147 ÷(−21)=7
Dividing both the numerator and the denominator by -21:
84÷(−21)−147÷(−21)=−4784−147=−47
(ii)Denominator of 84-147 is −147.
−147÷3=−49
Dividing both the numerator and the denominator by 3:
84÷3−147÷3=28−49
84−147=28−49
Question 17:
Write each of the following rational numbers in standard form:
(i) 3549
(ii) 8-36
(iii) -2745
(iv) -14-49
(v) 91-78
(vi) -68119
(vii) -87116
(viii) 299-161
Answer 17:
(i) 3549
H.C.F. of 35 and 49 is 7.
Dividing the numerator and the denominator by 7:
35÷749÷7=57
So, 3549 is equal to 57 in the standard form.
(ii)8-36
Denominator is -36, which is negative.
Multiplying both the numerator and the denominator by -1:
8×(-1)-36×(-1)=-836
H.C.F. of 8 and 36 is 4.
Dividing its numerator and denominator by 4:
-8÷436÷4=-29
So, 8-36 is equal to -29 in the standard form.
(iii) -2745
H.C.F. of 27 and 45 is 9.
Dividing its numerator and denominator by 9:
−27÷945÷9=−35
Hence, −2745 is equal to -35 in the standard form.
(iv) -14-49The denominator is negative. Multiplying its numerator and denominator by -1:-14×(-1)-49×(-1)=1449
H.C.F. of 14 and 49 is 7.
Dividing both the numerator and the denominator by 7.
14÷749÷7=27Hence, -14-49 is equal to 27 in the standard form.
(v) 91-78The denominator is negative. Multiplying its denominator and denominator by -1:91×(-1)-78×(-1)=-9178
H.C.F. of 91 and 78 is 13.
Dividing both the numerator and the denominator by 13:
-91÷1378÷13=-76Hence, 91-78 is equal to -76 in the standard form.
(vi) -68119
H.C.F. of 68 and 119 is 17.
Dividing both the numerator and the denominator by 17:
-68÷17119÷17=-47Hence, -68119 is equal to -47 in the standard form.
(vii) -87116
H.C.F. of 87 and 116 is 29.
Dividing both the numerator and the denominator by 29:
-87÷29116÷29=-34Hence, -87116 is equal to -34in the standard form.
(viii) 299-161
The denominator is negative.
Multiplying both the numerator and denominator by -1:
299×(-1)-161×(-1) =-299161
H.C.F. of 299 and 161 is 23.
Dividing both the numerator and the denominator by 23:
-299÷23161÷23=-137Hence, 299-161 is equal to -137in the standard form.
Question 18:
Fill in the blanks:
(i) -95=......20=27......=-45......
(ii) -611=-18......=......44
Answer 18:
(i)
−9×45×4=−3620−9×(-3)5×(-3)=27−15−9×55×5=−4525∴
(ii)
Question 19:
Which of the following are pairs of equivalent rational numbers?
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Answer 19:
(i)
We have:
(−13)×(−21) = 273
And 7×39=273
(ii)
We have:
3×16=48
And (−8) ×(−6) =48
∴ 3×16 =(−8)×(−6)
(iii)
We have:
9×(−16)= −144
And 4×(-36)= −144
9×(−16) = 4×(−36)
Therefore, they are equivalent rational numbers.
(iv)
We have:
7×60 =420
And 15×(-28)= −420
∴ 7×60 ≠15×(−28)
Therefore, the rational numbers are not equivalent.
(v)
We have:
3 ×4=12
And 12×(−1)= −12
12 ≠ −12
Therefore, the rational numbers are not equivalent.
(vi)
We have:
2×2=4
And 3×3=9
2×2≠3×3
Therefore, the rational numbers are not equivalent.
Question 20:
Find x such that:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Answer 20:
(i)
=> −x =5×8
=> x= −40
(ii)
=> (−3)x=7×6
=> x=
=> x=−14
(iii)
=> 5x=3×(−25)
=> x=
=>x = (−15)
(iv)
=> 13x=6×(−65)
=> x=
=> x= 6×(−5)
=> x = −30
(v)
=>
=> x= (−4)
vi)
=>
=>
=>
x= (−24)
Question 21:
Which of the following rational numbers are equal?
(i)
(ii)
(iii)
Answer 21:
(i)
8×15 =120
And ( −10)×(−12)=120
8×15 =(−10) ×(−12)
Therefore, the rational numbers are equal.
ii)
(−3)×(−21) =63
And 7× 9=63
∴ (−3)×(−21) =7×9
Therefore, the rational numbers are equal.
(iii)
(−8) × 21 = −168
And 15 ×(−14) = − 210
(−8) × 21 ≠ 15 × 14
Therefore, the rational numbers are not equal.
Question 22:
State whether the given statement is true of false:
(i) Zero is the smallest rational number.
(ii) Every integer is a rational number.
(iii) The quotient of two integers is always a rational number.
(iv) Every fraction is a rational number.
(v) Every rational number is a fraction.
Answer 22:
(i) False
For example, −1 is smaller than zero and is a rational number.
(ii)True
All integers can be written with the denominator 1.
(iii) False
Though 0 is an integer, when the denominator is 0, it is not a rational number.
For example, is not a rational number.
(iv)True
(v) False
−1 is a rational number but not a fraction.
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