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Page No 60:

Question 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:

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:

57,34,78,1415,59

Five negative rational numbers:

 37,38,89,1925,825

Yes, there is a rational number that is neither positive nor negative, i.e. zero (0).   

Page No 60:

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:

i) 58 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)812is a rational number because it is in the form of   pq, where p and q are integers and q0. v) 6  is a rational number because it is in the form of  pq, where p and q are integers and q0. vi) -3 is a rational number because it is in the form of  pq, where p and q are integers and q0. vii) 0 = 01is a rational number because it is in the form of pq, where p and q are integers and q0. viii)01 is a rational number because it is in the form of   pq, where p and q are integers and q0. ix)10is not a rational number because, here, q = 0.x)00 is not a rational number because, here, q = 0.      

Page No 60:

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:

(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

Page No 60:

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:

(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

Page No 60:

Question 5:

Which of the following are positive rational numbers?

(i) 3-5
(ii) -1115
(iii) -5-8
(iv) 3753
(v) 03

Answer:

Positive rational numbers:
(iii) -5-8

(iv) 3753
(vi) 8 because 8 can be written as 81, where 10.

0 is neither positive nor negative.

Page No 60:

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:

Negative rational numbers:

(iii) -57

(iv) 4-9

(v)  -6

(vi) 1-2

Page No 60:

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:

(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.
7×215×2, 7×315×3, 7×415×4 and 7×515×5i.e14302145, 2860 and 3575

(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

 



Page No 61:

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:

(i) 12×(1)(17)×(1)=1217

(ii) 1×(1)(2)×(1)=12

(iii) 819=8×(1)(19)×(1)=819

(iv) 11×(1)-6×(1)=116

Page No 61:

Question 9:

Express 58 as a rational number with numerator

(i) 15
(ii) −10

Answer:

(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)=1016 58=1016

Page No 61:

Question 10:

Express 47 as a rational number with denominator
(i) 21
(ii) −35

Answer:

(i) Denominator of 47 is 7.
7 should be multiplied by 3 to get 21.
Multiplying both the numerator and the denominator by 3:

4×37×31221

4×37×347 

(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)=2035 47=2035

Page No 61:

Question 11:

Express -1213 as a rational number with numerator
(i) −48
(ii) 60

Answer:

(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=48521213=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)=60651213=6065

Page No 61:

Question 12:

Express -811 as a rational number with denominator

(i) 22
(ii) −55

Answer:

(i) Denominator of-811 is 11.
​Clearly, 11×2= ​22

Multiplying both the numerator and the denominator by 2:

8×211×2=1622811=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

Page No 61:

Question 13:

Express 14-5 as a rational number with numerator

(i) 56
(ii) −70

Answer:

(i) Numerator of 14-5 is 14.
Clearly, 14×4=56

Multiplying both the numerator and the denominator by 4:

14×45×4=5620


14-5=5620

(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

Page No 61:

Question 14:

Express 13-8 as a rational number with denominator

(i) −40
(ii) 32

Answer:

(i) Denominator of 13-8 is −8.
​
Clearly, (
−8)×5= −40
Multiplying both the numerator and the denominator by 5:
13×58×5=6540138=6540


(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

Page No 61:

Question 15:

Express -3624 as a rational number with numerator

(i) −9
(ii) 6

Answer:

(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


-36246-4

Page No 61:

Question 16:

Express 84-147 as a rational number with denominator

(i) 7
(ii) −49

Answer:

(i) Denominator of 84-147 is 147.
​∴ −147 ÷(−21)=7
Dividing both the numerator and the denominator by -21:

84÷(21)147÷(21)=4784147=47

(ii)
Denominator of 
84-147 is 147.
 −147÷3=−49
Dividing both the numerator and the denominator by 3:

  84÷3147÷3=2849

 84147=2849

Page No 61:

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:

(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. 

Page No 61:

Question 18:

Fill in the blanks:

(i) -95=......20=27......=-45......
(ii) -611=-18......=......44

Answer:

(i)

 9×45×4=36209×(-3)5×(-3)=27159×55×5=452595=3620=2715=4525


(ii) 
6×311×3=-18336×411×4=-2444 611=1833=-2444

Page No 61:

Question 19:

Which of the following are pairs of equivalent rational numbers?

(i) -137,39-21
(ii) 3-8,-616
(iii) 94,-36-16
(iv) 715,-2860
(v) 312,-14
(vi) 23,32

Answer:

(i) 137,3921
We have:
(−13)×(−21) = 273

And 7×39=273

(13)×(21) =7×39or 137=3921Hence, 137 and 3921 are equivalent rational numbers.                 

(ii) 3-8, -616
We have:

3×16=48

And (−8) ×(−6) =48

∴ 3×16 =(−8)×(−6)

3-8 =-616

(iii)94, -36-16
 
We have:

9×(−16)= −144

And 4×(-36)= −144


 9×(−16) = 4×(−36)

94=-36-16
Therefore, they are equivalent rational numbers.

(iv)715, -2860 

We have:

7×60 =420
And 15×(-28)= −420


∴ 7×60 ≠15×(−28)
Therefore, the rational numbers are not equivalent.

(v) 
312, -14

We have:
3 ×4=12
And 12×(−1)= −12

12 ≠ −12
Therefore, the rational numbers are not equivalent.

(vi) 23,32

We have:

2×2=4

And 3×3=9


2×2≠3×3

Therefore, the rational numbers are not equivalent.           

Page No 61:

Question 20:

Find x such that:

(i) -15=8x
(ii) 7-3=x6
(iii) 35=x-25
(iv) 136=-65x
(v) 16x=-4
(vi) -48x=2

Answer:

(i)−15=8x

=> −x =5×8
=> x= −40
 
(ii)7-3=x6
=> (
−3)x=7×6

=> 
x=(7×6)(-3)
=>  x=−14
 
 
(iii) 35=x−25
=>    5x=3×(−25)

=>   x=3×(−25)5
=>x  = (−15)

(iv)136=−65x

=> 13x=6×(−65)

=>  x=(−65)13

=>  x= 6×(−​5)

=>  x = −​30

(v)16x=-4
    => x =16(−4)
     =>  x= (−4)

vi)−48x=2
=> 
−482=x1
=>2x=(-48)×1
=>x=-482
x= (−24)

Page No 61:

Question 21:

Which of the following rational numbers are equal?

(i) 8-12and-1015
(ii) -39and7-21
(iii) -8-14and1521

Answer:

(i)8-12 and−1015


8×15 =120
And ( −10)×(−12)=120

8×15 =(−10) ×(−12)

 8-12 =−1015

Therefore, the rational numbers are equal.

ii)−39, 7-21

(−3)×(−21) =63
And 7× 9=63


∴ (−3)×(−21) =7×9

−39= 7-21

Therefore, the rational numbers are equal.

(iii) −8-14,1521 


(−8) × 21 = −168
And 15 ×(
−​14) = − ​210


(−8) × 21 ≠ 15 × 14

Therefore, the rational numbers are not equal.

Page No 61:

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:

(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, 10 is not a rational number. 

(iv)True
(v) False
−1 is a rational number but not a fraction.



Page No 66:

Question 1:

Represent each of the following rational numbers on the number line:

(i) 13
(ii) 27
(iii) 73
(iv) 227
(v) 378
(vi) -13
(vii) -34
(viii) -127
(ix) 36-5
(x) -439

Answer:

(i)


(ii)
 


(iii) (7/3)=2+(1/3)



(iv) 

227 can be written as 3 + 17. So, we need to move to the right of point 3. Then, we need to move 17 distance more to the right.  




(v) 378 can be written as 4+58.  So, we need to move to the right of point 4. Then, we need to move 58 distance more to the right.




(vi) 



(vii) 


(viii)

-127 can be written as -1-57. So, we need to move to the left of point -1. Then, we need to move 57 distance more to the left. 


(ix) 

36-5 can be written as -7-15. So, we need to move to the left of point -7. Then, we need to move 15 distance more to the left.  






(x) -439 can be written as -4-79. So, we need to move to the left of point -4. Then, we need to move 79 distance more to the left. 

Page No 66:

Question 2:

Which of the two rational numbers is greater in each of the following pairs?

(i) 56 or 0
(ii) -35 or 0
(iii) 58or 38
(iv) 79 or -59
(v) -611 or 5-11
(vi) -154 or -174

Answer:

(i)56. This is because 0 can be written as  06  and 06<56.(ii)35 <0. This is because 0 can be written as 05  and 3<0.(iii)58 >38. This is because  5  >3.(iv)79 >59. This is because 7 > 5.(v)611 <511. This is because 6 < 5.(vi)154 > 174,15 > 17

Page No 66:

Question 3:

Which of the two rational numbers is greater in each of the following pairs?

(i) 59 or -3-8
(ii) 4-3 or -87
(iii) -125 or -3
(iv) 7-9 or -58
(v) 4-5 or -78
(vi) 9-13 or 7-12

Answer:

(i)59,38(3)×(1)(8)×(1)=38L.C.M. of 9 and 8  is 72. 5×89×8=4072   3×98×9=277227<4038<59So, 59 is greater                                                                                   
(ii)43 ,87We will convert each negative denominator into positive. 4×-13×-1=-43L.C.M. of 3 and 7 is 21.-4×(7)(3)×(7)=2821 (8)×37×3=2421(24)>(28)87>4(3)So, 87 is greater.


(iii)125,3 L.C.M. of 5 and 1 is 5.12×15×1=1253×51×5=155  12>15125>3125 is greater. 
(iv)79,58L.C.M. of 9 and 8 is 72.7×89×8=56725×98×9=4572 56<4579<58                                                                          

(v) 4-5,78We will convert each negative denominator into positive. 4×-1-5×-1=-45L.C.M. of 5 and 8 is 40.-4×85×8=32407×58×5=3540 32>3545>78                            


(vi)913,712We will convert each negative denominator into positive. 9×-113×-1=-9137×-112×-1=-712      L.C.M. of 13 and 12 is 156.     -9×1213×(12)=108156-7×13(12)×13=91156 108<91913<712

Page No 66:

Question 4:

Fill in the blanks with the correct symbol out of >, = and <:

(i) -37......6-13
(ii) 5-13......-3591
(iii) -2......-135
(iv) -23......5-8
(v) 0......-3-5
(vi) -89......-910

Answer:

(i)37>613L.C.M. of 7 and 13 is 91. 3×137×13=39916×713×7=4291 3991>4291(ii)513=3591L.C.M. of 13 and 91 is 91. 5×(7)13×(7)=3591(iii)2>135L.C.M of 1 and 5 is 5. 2×51×5=10513×15×1=135 105>135                                                                                    
(iv) 23<58L.C.M. of 8 and 3 is 24. 2×83×8=16245×38×3=1524 1624<1524                                                

(v) 0<35L.C.M. of 1 and 5 is 5.0×11×5=053×15×1=35 05<35                                            

(vi) 89>910L.C.M. of 9 and 10 is 90. 8×109×10=80909×910×9=8190 8090>8190                                               

Page No 66:

Question 5:

Arrange the following rational numbers in ascending order:

(i) 25,710,815,1330
(ii) -34,5-12,-716,9-24
(iii) -310,7-15,-1120,17-30
(iv) 23,34,5-6,-712

Answer:

(i)25,710,815,1330L.C.M. of 5, 10, 15 and 30 is 30                                       



2×65×6=12307×310×3=21308×215×2=1630 13×130×1=1330Required order: 25<1330<815<710

(ii)34,5-12,716,9-24First, we need to convert each negative denominator into positive.34,5×-1-12×-1,716,9×-1-24×-134,-512,716,-924


L.C.M. of 4, 12, 16 and 24 is 48.3×124×12=36485×412×4=20487×316×3=21489×224×2=1848Required order: 34<716<512<924
(iii)310,7-15,1120,1730First, we need to convert the negative denominators to make them positive.310,7×-1-15×-1,1120,17×-130×-1310,-715,1120,-1730



L.C.M of 10,15,20,30=603×610×6=18607×415×4=286011×320×3=336017×230×2=3460Therefore, 3460<3360<2860<1860i.e1730<1120<715<310(iv)23,34,56,712First, we need to convert the negative denominators to positive ones.23,34,5×-16×-1,71223,34,-56,712



L.C.M of 3,4,6,12=122×43×4=8123×34×3=9125×26×2=10127×112×1=712Therefore, the correct order is 56<712<23<34.

Page No 66:

Question 6:

Arrange the following rational numbers in descending order:

(i) -25,7-10,-1115,19-30
(ii) -2,-136,8-3,13
(iii) -49,5-12,-718,2-3
(iv) 17-30,11-15,-710,35

Answer:

(i)25,710,1115,19-30First, we need to convert each negative denominator into positive.25,7×-110×-1,1115,19×-1-30×-125,-710,1115,-1930


L.C.M. of 5, 10, 15 and 30 is 30.2×65×6=1230,-7×310×3=2130,11×215×2=2230,-19×130×1=1930,Correct order:25>1930>710>1115(ii)2,136,8-3,13First, we need to convert each negative denominator into positive.2,136,8×-1-3×-1,132,136,-83,13



L.C.M. of 6, 3 and 3 is 6.2×61×6=126,13×16×1=136,8×23×2=166,1×23×2=26,Correct order: 13>2>136>83
(iii)49,5-12,718,2-3First, we need to convert each negative denominator into positive.49,5×-1-12×-1,718,2×-1-3×-149,-512,718,-23


L.C.M. of 9, 12, 18 and 3 is 36.4×49×4=1636,5×312×3=1536,7×218×2=1436,2×123×12=2436Correct order: 718>512>49>23(iv)1730,1115,710,35First, we need to convert each negative denominator into positive.17×-130×-1,11×-115×-1,710,35-1730,-1115,710,35


L.C.M. of 30, 15, 10 and 5 is 30.-17×130×1=1730,-11×215×2=2230,7×310×3=2130,3×65×6=1830,Correct order: 35>1730>710>1115

Page No 66:

Question 7:

Which of the following statements are true?

(i) -35 lies to the left of 0 on the number line.
(ii) -127 lies to the right of 0 on the number line.
(iii) 13 and -52 lie on opposite sides of 0 on the number line.
(iv) -18-13 lies to the left of 0 on the number line.
(v) -5-8 lies on the right of -57 on the number line.

Answer:

(i)True. This is because negative rational numbers lie to the left of 0 on the number line.(ii)False. This is because negative rational numbers do not lie to the right of 0.(iii)True. This is because positive rational numbers lie to the right, while negative rational numbers to the left of 0.(iv)False. This is because it is positive, which means it lies to the right of 0.(v)True. This is because a positive rational number lies to the right of a negative rational number.



Page No 67:

Question 8:

Find five rational numbers between −3 and −2.

Answer:

L.C.M. of 2 and 3 is 6. -3=-3×61×6=-186-2=-2×61×6=-126 Therefore ,-176,-166,-156,-146 and -136 are the five rational number between -3 and -2.

Page No 67:

Question 9:

Find five rational numbers between −1 and 1.

Answer:

-1=-1×51×5, 1=1×51×5-55and55Hence, the five rational numbers between 1 and 1 are -45,-35,-25,-15 and 15.

Page No 67:

Question 10:

Find five rational numbers between -35 and -12.

Answer:

35 and 12L.C.M. of 5 and 2 is 10.3×25×2=6×410×4=24×240×2=4880,1×52×5=5×410×4=20×240×2=4080,Hence, the five rational numbers between 35 and 12 are 4580,4480,4380,4280 and 4180.



Page No 69:

Question 1:

Add the following rational numbers:

(i) 127  and 37
(ii) -25  and 15
(iii) 3-8  and 18
(iv) -511 and 7-11
(v) 9-13  and -11-13
(vi) -29 and -59
(vii) -179 and -119
(viii) -37  and 5-7

Answer:

(i)
127+37=12+37=157

(ii)
-25+15=-2+15=-15

(iii)

3-8×-1-1=-38

-38+18=-3+18=-28

(iv)

7-11×-1-1=-711-511+-711=-5+(-7)11=-5-711=-1211


(v)

-11-13×-1-1=1113

=-913+1113=-9+1113=213

(vi)

-29+-59=-2-59=-79

(vii)

(-17)9+(-11)9=-17-119=-289

(viii)
5-7×-1-1=-57


-37+(-5)7=-3-57=-87



Page No 70:

Question 2:

Add the following rational numbers:

(i) -25  and 34
(ii) -59 and 23
(iii) -4 and 12
(iv) -727  and 518
(v) -536 and -712
(vi) 1-9  and 4-27
(vii) -924 and -18
(viii) 27-4 and -158

Answer:

(i)-25+34

The denominators of the given rational numbers are 5 and 4.L.C.M. of 5 and 4 is 20.-25=(-2)×45×4=-82034=3×54×5=1520Now,(-8)20+1520=-8+1520=720

(ii)-59+23

The denominators of the given rational numbers are 9 and 3.                                            



 L.C.M. of 9 and 3 is 9.-59=(-5)×19×1=-5923=2×33×3=69Now, (-5)9+69=-5+69=19
(iii)-4+12


The denominators of the given rational numbers are 1 and 2.                                           L.C.M. of 1 and 2 is 2.-41=(-4)×21×2=-8212=1×12×1=12Now, (-8)2+12=-8+12=-72

(iv)
-727+518


The denominators of the given rational numbers are 27 and 18.

L.C.M. of 27 and 18 is 54.

-727=(-7)×227×2=-1454518=5×318×3=1554Now, (-14)54+1554=-14+1554=154

(v)-536+-712



The denominators of the given rational numbers are 36 and 12.
L.C.M. of 36 and 12 is 36.

-536=(-5)×136×1=-536-712=-7×312×3=-2136Now, (-5)36+(-21)36=-5-2136-2636=-1318     (26 and 36 are divided by 2 .)

(vi)
1-9+4-27

We need a positive denominator.1-9×-1-1=-19 and  4-27×-1-1=-427The denominators of the given rational numbers are 9 and  27

L.C.M. of 9 and 27 is 27-19=(-1)×39×3=-327-427=-4×127×1=-427(-3)27+(-4)27=-3-427=-727


(vii)

-924+-118The denominators of the given numbers are 24 and 18

L.C.M. of 24 and 18 is 72.-924=-9×324×3=-2772-118=-1×418×4=-472Now, -2772+-472=-27+(-4)72= -27-472=-3172 


(viii)27-4+-158

We need a positive denominator.  27-4×-1-1=-274The denominators of the given rational numbers are 4 and 8.  


L.C.M. of 4 and 8 is 8.-274=-27×24×2=-548(-15)8=(-15)×18×1=-158Now, -548+(-15)8=-54-158=-698

Page No 70:

Question 3:

Evaluate:

(i) -35+75+-15
(ii) -127+37+-27
(iii) 11-12+3-8+14
(iv) -169+-512+718
(v) -3+18+-25
(vi) -138+516+-14

Answer:

(i)-35+75+-15L.C.M. of the given rational number is 5.(-3)5+75+(-1)5=-3+7-15=-4+75=35

(ii)
-127+37+-27=(-12)7+37+(-2)7=-12+3-27=-14+37=-117


(iii)  11-12+3-8+14 We need a positive denominator.11-12×-1-1=-1112and3-8×-1-1=-38L.C.M. of the denominators 12, 8 and 4 is 24. -11×212×2=-2224  -3×38×3=-924    1×64×6=624  Now, (-22)24+(-9)24+624  =-22-9+624=-31+624=-2524                                     
(iv)-169+-512+718  L.C.M. of the denominators  9, 12 and 18 is 36.-16×49×4=-6436-5×312×3=-15367×218×2=1436Now, (-64)36+(-15)36+1436=-64-15+1436-79+1436=-6536                                     

(v)-3+18=-25 L.C.M. of the denominators 1, 8 and 5 is 40.-3×401×40=-120401×58×5=540-2×85×8=-16  40Now, (-120)40+540+(-16)40=-120+5-1640= -136+540=-13140                                                   

(vi)-138+516+-14L.C.M. of the denominator 8, 16 and 4 is 16.-13×28×2=-26165×116×1=516-1×4  4×4=-416Now,(-26)16+516+(-4)16=-26+5-416 Now,-30+516=-2516

Page No 70:

Question 4:

Simplify:

(i) -815+2-3
(ii) -710+13-15+2720
(iii) -1+7-9+1112
(iv) -1139+526+2
(v) 2+-12+-34
(vi) -911+23+-34

Answer:

(i)-815+2-3 We need a positive denominator.  2-3×-1-1=-23 Now, L.C.M. of 15 and 3 is 15.-815=-8×115×1=-815-23=-2×5  3×5=-10 15Now, -815+-1015=-8-1015=-1815=-65                                           

(ii)-710+13-15+2720We need a positive denominator.13-15×-1-1=-1315 Now, L.C.M. of 10, 15 and 20 is 60.-710=-7×610×6=-4260-1315=-13×4  15×4=-52602720=27×320×3=8160 Now, -4260+-5260+8160=(-42)+(-52)+(81)60=-94+8160=-1360                                            


(iii)-1+7-9+1112We need a positive denominator.7-9×-1-1=-79Now, L.C.M. of 1, 9 and 12 is 36.-11=-1×361×36=-3636-79=-7×49×4=-28  361112=11×312×3=3336    -3636+-2836+3336=-36-28+3336=-64+3336=-3136=-54                                                 


(iv)-1139+526+21 L.C.M. of 39, 26 and 1 is 78.-1139=-11×239×2=-2278526=5×326×3=157821=2×781×78=15678Now ,-2278+1578+15678=-22+17178=14978                                                                      
 
   (v)2+-12+-34
2+12+3

L.C.M. of 2 and 4 is 4.2=2×41×4=84-12=-1×22×2=-24-34=-3×14×1=-3484+(-2)4+(-3)4=8-2-34=34                                                           

(vi)-911+23+-34L.C.M. of 11, 3 and 4 is 132.-911=-9×1211×12=-10813223=2×443×44=88132-34=-3×334×33=-99132-108132+88132+(-99)132=-108+88-99132=-207+88132=-119132                                                     

Page No 70:

Question 5:

Express each of the following rational numbers as the sum of an integer and a rational number:

(i) 125
(ii) -117
(iii) -259
(iv) -10320

Answer:

(i) 125=225=2+25(ii) -117=-117=-147=-1+-47(iii) -259=-259=-279=-2+-79(iv) -10320=-10320=-5320=-5+-320



Page No 72:

Question 1:

Find the additive inverse of:

(i) 5
(ii) −9
(iii) 314
(iv) -1115
(v) 15-4
(vi) -18-13
(vii) 0
(viii) 1-6

Answer:

(i) Additive inverse of 5 is −5.

(ii) Additive inverse of −9 is 9.

(iii) Additive inverse of 314 is -314.

(iv) Additive inverse of -1115 is 1115.

(v) Additive inverse of 15-4=15×(-1)(-4)×(-1)
                                         
                                          = -154=154

(vi) Additive inverse of -18-13=-18×(-1)(-13)×(-1)
                                              = 1813=-1813

(vii) Additive inverse of​ 0 is 0.

(viii) Additive inverse of   1-6=1×(-1)(-6)×(-1)
                                                = -16=16

Page No 72:

Question 2:

Subtract:

(i) 34 from 13 
(ii) -56 from 13 
(iii) -89 from -35 
(iv) -97 from -1 
(v) -1811 from 1 
(vi) -139 from 0 
(vii) -3213 from -65 
(viii) -7 from -47 
(ix) 59 from -23 
(x) 5 from -35 

Answer:

(i) 13-34=13+ (additive inverse of 34) L.C.M. of 3 and 4 is 12.13+-34=4+(-9)12=-512


(ii)    13-(-5)6=13+additive inverse of -56=13+56 L.C.M. of 3 and 6 is 6.=2+56=76

(iii)       (-3)5--89=(-3)5+additive inverse of -89=(-3)5+89L.C.M. of 5 and 9 is 45.=-27+4045    =1345

(iv)(-1)1-(-9)7=(-1)1+additive inverse of -97=(-1)1+97L.C.M. of 7 and 1 is 7.=-7+97=27

(v)11-(-18)11=11+additive inverse of -1811=(1)1+1811    =11+1811     =2911

(vi)
0--139=0+additive inverse of -139=0+139=139

.(vii)(-6)5-(-32)13=(-6)5+additive inverse of -3213=-65+3213L.C.M. of 5 and 13 is 65=-78+16065 =8265       


(viii) -47-(-7)1=-47+additive inverse of -71=(-4)7+71L.C.M. of 7 and 1 is 7.= -4+497  =457
(ix)-23-59=-23+additive inverse of 59=-23-59 L.C.M. of 3 and 9 is 9.=-6-59 =-119

(x)-35-51=-35+additive inverse of 51=-35-51    L.C.M. of 5 and 1 is 5. =-3-255=-285            

Page No 72:

Question 3:

Evaluate:

(i) 34-45
(ii) -3-47
(iii) 724-1936
(iv) 1415-1320
(v) 49-2-3
(vi) 711--4-11
(vii) -514--27
(viii) -5-8--34

Answer:


(  i)34-45=34-45=34+additive inverse of 45L.C.M. 4 and 5 is 20.   =15-1620=-120                                        


(ii)-31-47=-31+additive inverse of 47=-31-47     =-21-47     =-257

(iii)     724-1936=  724+additive inverse of 1936= 724-1936   L.C.M. of 24 and 36 is 72.=21-3872=-1772                                

(iv)  1415-1320=1415+additive inverse of 1320=1415-1320  L.C.M. of 15 and 20 is 60.=56-3960=1760

(v)49-2(-3)We need a positive denominator.2-3×(-1)(-1)=-2349-(-2)3  =49additive inverse of -23L.C.M. of 3 and 9 is 9.=4+69=109

(vi)711-(-4)(-11)We need a positive denominator.     -4-11×(-1)(-1)=411711-411=711+additive inverse of411 =7-411=311
(vii) -514--27=-514+additive inverse of -27-514+27L.C.M. of 14 and 7 is 14.-5+414=-114




(viii)-5-8--34  We need a positive denominator.    -5-8×-1-1=5858-(-3)4         =58additive inverse of -34L.C.M. of 8 and 4 is 8.=5+68=118    [L.C.M. of 8 and 4 is 8.]

Page No 72:

Question 4:

Subtract the sum of -3611 and 4922 from the sum of 338 and -194.

Answer:

First we will find the sum of  -3611  and 4922.        -3611+4922L.C.M. of 11 and 22 is 22.=-72+4922= -2322Now. we have to find the sum of   338and-194.338+(-19)4L.C.M. of 8 and 4 is 8.=33-388=-58Now, (-5)8-(-23)22=-58+2322L.C.M. of 8 and 22 is 88.=-55+9288=3788

Page No 72:

Question 5:

The sum of two rational numbers is 421. If one of them is 57, find the other.

Answer:

Let the  other number that be x.57+x=421=>x=421-57L.C.M. of 21 and 7 is 21.=>x=4-1521=>x=-1121Hence, the required number is -11  21.

Page No 72:

Question 6:

The sum of two rational numbers is -38. If one of them is 316, find the other.

Answer:

Let the other number be x.                                        316+x=-38=>x=-38-316L.C.M. of 8 and 16 is 16.=>x=-6-316=>x=-916

Page No 72:

Question 7:

The sum of two rational numbers is −3. If one of them is -157, find the other.

Answer:

  Let the other number be x.   -157+x=-3=>x=-3-(-15)7=>x=-21+157=>x = -67Hence, the required number is  -67

Page No 72:

Question 8:

The sum of two rational numbers is -43. If one of them is −5, find the other.

Answer:

 Let the required number be x-5+x=-43=>x=-43+5=>x=-4+153=>x=113Hence, the requied number is 113.

Page No 72:

Question 9:

What should be added to -38 to get 512?

Answer:

 Let the required number be x.

 -38+x=512=>x=512-(-3)8=>x=10+924=>x=1924Hence, the required number is 1924.

Page No 72:

Question 10:

What should be added to -125 to get 3?

Answer:

Let the number that is to be added be x.

 -125+x=3=>x=3-(-12)5=>x=15+125=>x=275Hence, the required number is  275.

Page No 72:

Question 11:

What should be added to -57 to get -23?

Answer:

Let the number that is to be added be x.

-57+x=-23=>x=-23-(-5)7L.C.M. of 3 and 7 is 21.=>x=-14+1521=>x=121Hence, the required number is 121.

Page No 72:

Question 12:

What should be added to 29 to get −1?

Answer:

Let the number that is to be added be x.
29+x=-1=>x=-1-29=>x=-9-29=>x=-119Hence, the required number is -119.



Page No 73:

Question 13:

What should be added to -134+-38 to get 1?

Answer:

Let the required number that is to be added be x.  -134+-38+x=1=>x=1--134+(-3)8L.C.M. of 4 and 8 is 8.=1--26-38=1--298=1+298=8+298=378Hence, the required number is  378.

Page No 73:

Question 14:

What should be subtracted from -34 to get 56?

Answer:

Let the required number that is to be subtracted be  x.  -34-x=56=>-x=56-(-3)4L.C.M. of 6 and 4 is 12.=>-x=10+912=>-x=1912=>-x × -1=1912×-1=>x=-1912Hence, the required number is -1912.

Page No 73:

Question 15:

What should be subtracted from -23 to get -56?

Answer:

Let the required number that is to be subtracted be x.  -23-x=-56=>-x=-56-(-2)3L.C.M. of 6 and 3 is 6.=>-x=-5+46=>-x=-16=>x=16

Page No 73:

Question 16:

What should be subtracted from -34 to get 1?

Answer:

Let the required number that is to be subtracted be x.  -34-x=1=>-x=1-(-3)4=>-x=4+34=>-x=74=>x=-74



Page No 75:

Question 1:

Multiply:

(i) 34 by 57
(ii)  98 by 323
(iii) 76 by 24
(iv) -23 by 67
(v) -125 by 10-3
(vi) 25-9 by 3-10
(vii) -710 by -4021
(viii) -365 by 20-3
(ix) -1315 by -2526

Answer:

(i)34×57=(3×5)(4×7)=1528(ii)9381×32431=(3×4)(1×1)=12(iii)761×2441=7×4=28(iv)231×627=(2×2)7=47(v) We need a positive denominator.   ∴ 103×11=103=12451×10231=4×2=8
(vi)25593×31102=53×12=56(vii)71101×404213=43(viii)361251×20431=12×4=48(ix)131153×255262=13×52=56

Page No 75:

Question 2:

Simplify:

(i) 320×45
(ii) -730×514
(iii) 5-18×-920
(iv) -98×-163
(v) -32×-736
(vi) 16-21×-145

Answer:

(i)3205×415=3×15×5=325(ii)71306×51142=1×16×2=112(iii)51182×91204=1×(1)2×4=18=18(iv)9381×16231=(3)×(2)=6(v) 321×736=328×(7)1×369=8×(7)9=569(vi)We need a positive denominator.1621×11=1621Now, 16213×1425=(16)×(2)3×5=3215

Page No 75:

Question 3:

Simplify:

(i) 724×-48
(ii) -1936×16
(iii) -34×43
(iv) -13×1726
(v) -135×-10
(vi) -916×-6427

Answer:

(i)7241×(482)=7×(2)=14(ii)19369×164=199×4=769(iii)3141×4131=1(iv)13×1726=131×17262=172(v)1351×(102)=26(vi)(91)161×(644)273=43

Page No 75:

Question 4:

Simplify:

(i) 138×1213+-49×3-2
(ii) 1615×-258+-1427×67
(iii) 655×-229-26125×-1039
(iv) -127×-1427--845×916

Answer:

(i)(13182 × 123131) + (4293 × 3121)=32 + 23L.C.M. of 2 and 3 is 6.=9 + 46136(ii)(1615×258) + (1427×67)=(162153×25581) + (14227×671)=[23×(5)1] + [(2)27×61]=(10)3+(124)279=103+49L.C.M. of 3 and 9 is 9.=3049=349(iii)(655×229)(26125×1039)=(62555×22293)(26212525×102393)=[(4)15(4)75]=415+475L.C.M. of 15 and 75 is 75.=20+475=1675(iv)(12471×142279)(81455×91162)=[(4)1×(2)9][15×12]=89+110L.C.M. of 9 and 10 is 90.=80+990=8990

Page No 75:

Question 5:

Find the cost of 313 metres of cloth at Rs 4012 per metre.

Answer:

Cost of 1 meter cloth =Rs 4012Cost of 312 meter cloth = Rs (4012× 312)                                                  =Rs (812×72)                                                    =Rs 5674                                                    =Rs 141.75

Page No 75:

Question 6:

A bus is moving at an average speed of 4623 km/h. How much distance will it cover in 225 huors?

Answer:

Distance covered in 1 hour = 4623 kmDistance coverd in  225 hours = (4623× 225)=(140283×12451)=(28×4)=112 kmHence, the required distance is 112 km.



Page No 78:

Question 1:

Find the multiplicative inverse of reciprocal of each of the following:

(i) 18
(ii) −16
(iii) 1325
(iv) -1712
(v) -619
(vi) -3-5
(vii) −1
(viii) 0

Answer:

(i) Multiplicative inverse of 18 =118ii) Multiplicative inverse of16 =116iii) Multiplicative inverse of 1325=2513iv) Multiplicative inverse of1712=1217v) Multiplicative inverse of619=196vi) Multiplicative inverse of35=53=53vii) Multiplicative inverse of1 =11=1viii) Multiplicative inverse of 0 =  10=infinityHence, it does not exist

Page No 78:

Question 2:

Simplify:

(i) 49÷-512
(ii) -8÷-516
(iii) -127÷-18
(iv) -110÷-85
(v) -1635÷-1514
(vi) -6514÷13-7

Answer:

(i)49÷(512)=493×124(5)=4×43×(5)=1615(ii)8÷(516)=-8×-165=1285(iii)127÷(18)=1227×(1183)=221(iv)110÷(85)=1102×(58)=12×1(8)=116=116(v)1635÷(1514)=16355×142(15)=3275=3275(vi)(6514)÷(137)=(655142)×(7)13=(52)×(11)=52

Page No 78:

Question 3:

Fill in the blanks:

(i) (......) ÷ -75=1019
(ii) (......) ÷ -3=-415
(iii) 98÷(......)=-32
(iv) -12÷......=-65

Answer:

(i)(?..)÷75=1019     (?..)=1019 × 75     (?..)=1419 (ii) (..?..)÷(3)=415      (?..)=415×(3)      (?..)=45(iii)98÷(..?..) =32   98÷(..?..)= (3)2       (..?..)   =98×2(3)       (..?..)=34 (iv) (12)÷(..?..)=65(..?..)=(12)×(56)   (..?..) =10

Page No 78:

Question 4:

Divide the sum of 6512 and 83 by their difference.

Answer:

Sum =6512+83=65+3212=9712Difference=651283=653212=33129712÷3312=97121×12133 = 9733

Page No 78:

Question 5:

By what number should -449 be divided to get -113?

Answer:

Let the required number be x. 449 ÷x = 113=>x=44493 ×3-11=>x=43    
    

Page No 78:

Question 6:

By what number should -815 be multiplied to get 24?

Answer:

Let the required number be x.x ×815=24x=24÷815=243×(1581)=45×(-1)1×(-1)=45

Page No 78:

Question 7:

The product of two rational numbers is 10. If one of the numbers is −8, find the other.

Answer:

Let the other number be x.x × -8=10=>x=10÷(8)          =10×18          =105×184 Other number =54

Page No 78:

Question 8:

The product of two rational numbers is −9. If one of the numbers is −12, find the other.

Answer:

Let the other number be x. x ×(-12) =-9=>x=9÷(12)                                            =9×(112)          =9×(112)           =93124Hence, the other number is 34.

Page No 78:

Question 9:

The product of two rational numbers is -169. If one of the numbers is -43, find the other.

Answer:

Let the other number be x.x ×-43=-169=> x= 169÷(43)            =16493×(3141)              =43Hence, the other number is 43. 

Page No 78:

Question 10:

By what rational number should -839 be multiplied to obtain 526?

Answer:

Let the required number be x.x ×839=526 => x=526÷839             = 5262×(3938)             =15×-116×-1=-1516Hence, the required number is -1516.

Page No 78:

Question 11:

If 24 pairs of trousers of equal size can be prepared with 54 m of cloth, what length of cloth is required for each pair of trousers?

Answer:

Length of the cloth required to prepare 24 trousers = 54 mLength of the cloth required for each pair of trousers= 54 ÷ 24                                                                             =549244                                                                          =94=214mHence, 214  m length of cloth is required for each pair of trousers.

Page No 78:

Question 12:

How many pieces, each of length 334 m, can be cut from a rope of length 30 m?

Answer:

Length of a rope = 30 mNumber of pieces=30 ÷ 334=30 ÷ 154  =230 × 415                                 =8 Hence, the number of pieces would be 8. 

Page No 78:

Question 13:

The cost of 212 metres of cloth is Rs 7834. Find the cost of cloth per metre.

Answer:

Cost of 212 m cloth = Rs 7834Cost of cloth per meter =7834 ÷ 212                                         =3154 ÷ 52                                         =3156342 × 2151                                         =Rs 632=Rs 3112 Cost of the cloth (per metre) = Rs 3112



Page No 79:

Question 1:

Mark (✓) against the correct answer
33-55 in standard form is

(a) 3-5
(b) -35
(c) -3355
(d) none of these

Answer:

 (b) 35                                                                                 


H.C.F. of 33 and 55 is 11-33 ÷ 1155 ÷ 11=35        

Page No 79:

Question 2:

Mark (✓) against the correct answer
-102119 in standard form is

(a) -47
(b) -67
(c) -617
(d) none of these

Answer:

(b) 67



H.C.F. of 102 and 119 is 17102 ÷ 17119 ÷ 17=67 The standard form of 102119 is 67

Page No 79:

Question 3:

Mark (✓) against the correct answer
If x6=7-3,  then the value of x is

(a) −14
(b) 14
(c) 21
(d) −21

Answer:

 The correct option is (a). The value of  x  is 14.[x=7×63=421431=14]

Page No 79:

Question 4:

What should be added to -59 to get 1?
(a) 49
(b) -49
(c) 149
(d) -149

Answer:

The correct option is (c).149 should be added to 59 to get 1.Let the required number be x.x + 59=1x = 1  (5)9=9 + 59=149

Page No 79:

Question 5:

What should be subtracted from -34 to get 56?
(a) 1912
(b) -1912
(c) 112
(d) -112

Answer:

The correct option is (b).Let the number that is to be subtracted  be x.34x=56=>x=56  (34)=>x=56 + 34=>x=(5 × 2) + (3 × 3)12=>x=1912Hence, 1912 should be subtracted from  34 to get 56.

Page No 79:

Question 6:

Mark (✓) against the correct answer
Which is smaller out of 5-6 and -712?
(a) 5-6
(b) -712
(c) cannot be compared

Answer:

  The correct option is (a).5 ×-1-6 ×-1=-56L.C.M.​ of 6 and 12 is 12.-5×26×2=-1012 and 7×112×1=712Hence, 56 is smaller than 712.

Page No 79:

Question 7:

Mark (✓) against the correct answer
Which is larger out of 2-3 and -45?
(a) 2-3
(b) -45
(c) cannot be compared

Answer:

The correct option is (a).2×-13×-1=-23L.C.M. of 3 and 5 is 15.-2×53×5=1015 and 4×35×3=1215Thus, 23 is greater than 45 .

Page No 79:

Question 8:

Mark (✓) against the correct answer
Reciprocal of −6 is

(a) 6
(b) 16
(c) -16
(d) none of these

Answer:

 The correct option is (c).Reciprocal of 6 is 16.

Page No 79:

Question 9:

Mark (✓) against the correct answer
Multiplicative inverse of -23 is
(a) 23
(b) -32
(c) 32
(d) none of these

Answer:

 The correct option is (b).Multiplicative inverse of  23 is  32.

Page No 79:

Question 10:

Mark (✓) against the correct answer
-219-6=?

(a) -819
(b) 819
(c) 419
(d) -419

Answer:

 The correct option is (a).  219  6=199  6 = 19  549=739=819

Page No 79:

Question 11:

Mark (✓) against the correct answer
-613--715=?


(a) -181195
(b) 181195
(c) 1195
(d) -1195

Answer:

 The correct option is (c).    613  [7]15L.C.M. of 13 and 15 is 195.=613  [7]15 =90 + 91195 =1195



Page No 80:

Question 12:

Mark (✓) against the correct answer
-213+435=?

(a) -2415
(b)  2415
(c) -215
(d) 2215

Answer:

  The correct option is (b).213 + 435=73 + 235L.C.M. of 5 and 5 is 15.=35 + 6915 =3415=2415

Page No 80:

Question 13:

Mark (✓) against the correct answer
23-157=?

(a) 1121
(b)  -1121
(c) 521
(d) -521

Answer:

 The correct option is (b).23157=23  127L.C.M. of 3 and 7 is 21.=14  3621 =2221=1121

Page No 80:

Question 14:

Which is greater between -49 and -512?
(a) -49
(b) -512
(c) both are equal

Answer:

The correct option is (b).
 512 is greater than 49.L.C.M. of 9 and 12 is 36.5×312×3=15364×412×4=1636(15)>(16) 512>49

Page No 80:

Question 15:

Mark (✓) against the correct answer
-914+?=-1

(a) 514
(b) -514
(c) 17
(d) -17

Answer:

  The correct option is (b).914+ ?=1?=1(9)14?=14 + 914?=514

Page No 80:

Question 16:

Mark (✓) against the correct answer
54-76--23=?

(a) 34
(b) -34
(c) -712
(d) 712

Answer:

(a) 34    5476(2)3L.C.M. of 4, 6 and 3 is 12.=15  14 + 812=23  1412=93124=34

Page No 80:

Question 17:

Mark (✓) against the correct answer
1÷12=?

(a) 12
(b) 2
(c) 212
(d) 112

Answer:

  (b) 2 1÷12=1×21=2

Page No 80:

Question 18:

Mark (✓) against the correct answer
-314×?=512

(a) -3518
(b) 3518
(c) 73
(d) -73

Answer:

  (a) -3518?=512÷(3)14=512×14(3)=7036=35×-118×-1?=3518

Page No 80:

Question 19:

Mark (✓) against the correct answer
0÷-75=?

(a) not defined
(b) -57
(c) 0
(d) 57

Answer:

 (c) 0 0 ÷ 75 = 0

Page No 80:

Question 20:

Mark (✓) against the correct answer
-38÷0=?

(a) -38
(b) 0
(c) -83
(d) not defined

Answer:

 (d) Not definedThis is because 38 ÷ 0 is not defined.



Page No 82:

Question 1:

Express each of the following rational numbers in standard form:

(i) -209247
(ii) -46115
(iii) 84-147

Answer:

 (i) 209247H.C.F. of 209 and 247 is 19.Dividing both the numerator and the denominator by 19=209 ÷ 19247 ÷ 19=1113                                         


(ii) 46115 H.C.F. of 46 and 115 is 23Dividing both the numerator and the denominator by 23 =46 ÷ 23115 ÷ 23=25                                                       

  (iii)84147Converting the number to a positive denominator:84 × (-1)147 × (-1)=-84147H.C.F. of 84 and 147 is 21Dividing both the numerator and the denominator by 21=-84 ÷ (21)147 ÷ (21)=47                                                      

Page No 82:

Question 2:

List five rational numbers between −2 and −1.

Answer:

-2=-2×61×6=-126-1=-1×61×6=-66 The integers between -12 and -6 are -11,-10,-9,-8,-7.Hence, five rational numbers between 2 and 1 are -116,-106,-96,-86,-76.

Page No 82:

Question 3:

The sum of two rational numbers is −4. If one of them is -116, find the other.

Answer:

Let the required number be x.x + -116=-4=>x=(4)  116         =4 + 116        =24 + 116          =136Hence, the other number is 136.

Page No 82:

Question 4:

What should be added to -78 to get 59?

Answer:

Let the other number be x.x + -78=59=>x =59  (78)        =59 + 78L.C.M. of 9 and 8 is 72.        =40 + 6372        =10372 

Hence , the other number is 10372

Page No 82:

Question 5:

A car is moving at an average speed of 5635 km per hour. How much distance will it cover in 712 hours?

Answer:

Average speed =5635 km/h              =2835 km/hTime =712 h        =152 hDistance = Speed × Time               =2835×1532                =8492              = 42412 kmThe car will cover 42412 km in 712 h.

 

Page No 82:

Question 6:

By what number should -438 be divided to obtain -312?

Answer:

Let the required number be x.438 ÷ x=312=>x =358 × 27         = 54Hence, the required number is 54 .                                                                 

Page No 82:

Question 7:

How many pieces, each of length 334 m, can be cut from a rope of length 45 m?

Answer:

Length of rope = 45 mNumber of pieces =45 ÷ 334                          =45 ÷ 154                        =45 × 415                         =12

Page No 82:

Question 8:

Find the cost of 313 m of cloth at Rs 12112 per metre.

Answer:

Cost of 1 m cloth = Rs 12112   Cost of  313 m cloth = 12112 × 313                                 =243812 × 1053                                 =Rs 405                           

Page No 82:

Question 9:

Mark (✓) against the correct answer
55-66 in standard form is

(a) 5-6
(b) -56
(c) -5566
(d) none of these

Answer:

(b) -56  5566 = 55 × (1)66 × (1)=5566H.C.F. of 55 and 66 is 11.55 ÷ 1166 ÷11=56-56 is the standard form.                                                                

Page No 82:

Question 10:

Mark (✓) against the correct answer
What should be subtracted from -23 to get 34?
(a) -1712
(b) 1712
(c) -1217
(d) 1217

Answer:

 (a) -1712Let the number to be subtracted be x.23 - x =34=> x = 23  34L.C.M. of 3 and 4 is 12.           =8  912            =1712

Page No 82:

Question 11:

Mark (✓) against the correct answer
The product of two numbers is -16. If one of them is -58, the other number is
(a) -415
(b) 415
(c) 154
(d) -154

Answer:

 (b) 415Let the other number be x.58 × x=16    =>x =16÷(58)             =16×(85)            =415             =415Hence , the other number is 415.

Page No 82:

Question 12:

Mark (✓) against the correct answer
The multiplicative inverse of -34 is
(a) 34
(b) 43
(c) -43
(d) none of these

Answer:

 (c) -43Multiplicative iInverse of 34 is 43.

Page No 82:

Question 13:

Mark (✓) against the correct answer
-914+?=-1
(a) 514
(b) -514
(c) 17
(d) -17

Answer:

(b)514Missing number=(1)  -914                             =(1) + 914                            =14 + 914                             =514

Page No 82:

Question 14:

Mark (✓) against the correct answer
7834÷212=?
(a) 3112
(b) 3938
(c) 4013
(d) none of these

Answer:

 (a) 3112        7834 ÷ 212=3154 ÷ 52                      =3156342×25                     =632                     =3112                      

Page No 82:

Question 15:

Mark (✓) against the correct answer
Which is smaller between -56 and -712?
(a) -56
(b) -712
(c) cannot be compared

Answer:

 (a)56      Since L.C.M. of 6 and 12 is 12.          5×26×2=1012 7×112×1=712        56<712



Page No 83:

Question 16:

Fill in the blanks.

(i)......÷-75=-23
(ii) -6514÷......=212
(iii) -38+......=512
(iv) Multiplicative inverse of -134is ...... .

Answer:

(i)(?...)÷75=23       (?)=23×-75                 =1415                  (ii)6514÷(?...)=212           (?...)=6514÷212                    =6514÷52                    =6514×25                    =137(iii)(?...) =512(-38)              =512+38L.C.M. of 12 and 8 is 24.               =10+924              =1924(iv)Multiplicative inverse of -1 34, i.e. 74, is -47.

Page No 83:

Question 17:

Write 'T' for true and 'F' for false for each of the following:

(i) -15-11 lies to the left of 0 on the number line.
(ii) 13 and -32 lie on opposite side of 0 on the number line.
(iii) -813 lies to the left of 0 on the number line.
(iv) -45>-23.
(v) -35 is the largest among -37,-710 and -56.

Answer:

(i)False. This is because 1511=1511, which lies to the right of 0.(ii)True (iii)True(iv)False L.C.M. of 5 and 3 is 15. We know:  4×35×3=1215 and 2×55×3=1015      45<25(v)TrueL.C.M. of 5, 10 and 6 is 30.3×65×6=1830, 7×310×3=2130 and 5×56×5=253035 is the largest among the given fractions.    



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