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Page No 3:

Question 1:

Express -35 as a rational number with denominator
(i) 20
(ii) −30
(iii) 35
(iv) −40

Answer:

If ab is a fraction and m is a non-zero integer, then ab=a×mb×m.

Now,

(i) -35=-3×45×4=-1220

(ii) -35=-3×-65×-6=18-30

(iii)-35=-3×75×7=-2135

(iv)-35=-3×-85×-8=24-40

Page No 3:

Question 2:

Express -4298 as a rational number with denominator 7.

Answer:

If ab is a rational number and m is a common divisor of a and b, then ab=a÷mb÷m.

∴ -4298=-42÷1498÷14=-37

Page No 3:

Question 3:

Express -4860 as a rational number with denominator 5.

Answer:

If ab is a rational integer and m is a common divisor of a and b, then ab=a÷mb÷m.

∴​ -4860=-48÷1260÷12=-45

Page No 3:

Question 4:

Express each of the following rational numbers in standard form:
(i) -1230
(ii) -1449
(iii) 24-64
(iv) -36-63

Answer:

A rational number ab is said to be in the standard form if a and b have no common divisor other than unity and b>0.
Thus,

(i) The greatest common divisor of 12 and 30 is 6.
   
     ∴ -1230=-12÷630÷6=-25 (In the standard form)

(ii)The greatest common divisor of 14 and 49 is 7.    
     
    ∴ -1449=-14÷749÷7=-27 (In the standard form)

(iii) 24-64=24×(-1)-64×-1=-2464
   
     The greatest common divisor of 24 and 64 is 8.          
    
     ∴ -2464=-24÷864÷8=-38 (In the standard form)

(iv) -36-63=-36×(-1)-63×-1=3663
 
     The greatest common divisor of 36 and 63 is 9.     
  
      ∴ 3663=36÷963÷9=47 (In the standard form)

Page No 3:

Question 5:

Which of the two rational numbers is greater in the given pair?
(i) 38 or 0
(ii) -29 or 0
(iii) -34 or 14
(iv) -57 or -47
(v) 23 or 34
(vi) -12 or -1

Answer:

We know:
(i) Every positive rational number is greater than 0.
(ii) Every negative rational number is less than 0.

Thus, we have:

(i)38 is a positive rational number.
    ∴ 38>0

(ii)-29 is a negative rational number.
    ∴ -29<0

(iii) -34 is a negative rational number.
    ∴ -34<0
    Also,
    14 is a positive rational number.
    ∴ 14>0
    Combining the two inequalities, we get:
   -34<14

(iv)Both -57 and -47 have the same denominator, that is, 7.
    So, we can directly compare the numerators.

    ∴ -57<-47

(v)The two rational numbers are 23 and 34.
    The LCM of the denominators 3 and 4 is 12.
    Now,
   23=2×43×4=812
    Also, 
   34=3×34×3=912
    Further
   812<912

    ∴23<34

(vi)The two rational numbers are -12 and -1.
    We can write -1=-11.
    The LCM of the denominators 2 and 1 is 2.
    Now,
    -12=-1×12×1=-12
    Also,
    -11=-1×21×2=-22
    ∵ -21<-11
    ∴ -1<-12

Page No 3:

Question 6:

Which of the two rational numbers is greater in the given pair?
(i) -43 or -87
(ii) 7-9 or -58
(iii) -13 or 4-5
(iv) 9-13 or 7-12
(v) 4-5 or -710
(vi) -125 or -3

Answer:

1. The two rational numbers are -43and-87.

The LCM of the denominators 3 and 7 is 21.

Now,
 
-43=-4×73×7=-2821

Also,

-87=-8×37×3=-2421

Further,
 
-2821<-2421

∴ -43<-87

2. ​The two rational numbers are 7-9and-58.

The first fraction can be expressed as 7-9=7×-1-9×-1=-79.

The LCM of the denominators 9 and 8 is 72.

Now, 

-79=-7×89×8=-5672

Also,

-58=-5×98×9=-4572

Further,
 
-5672<-4572

∴​ 7-9<-58

3. ​The two rational numbers are -13and4-5 .

4-5=4×-1-5×-1=-45

The LCM of the denominators 3 and 5 is 15.

Now, 

-13=-1×53×5=-515

Also,

-45=-4×35×3=-1215

Further,
 
-1215<-515

∴ 4-5<-13

4. The two rational numbers are 9-13and7-12.

Now,  9-13=9×-1-13×-1=-913 and 7-12=7×-1-12×-1=-712 

The LCM of the denominators 13 and 12 is 156.

Now, 

-913=-9×1213×12=-108156

Also,

-712=-7×1312×13=-91156

Further,
 
-108156<-91156

∴ 9-13<7-12

5. The two rational numbers are 4-5 and -710.

∴​ 4-5=4×-1-5×-1=-45

The LCM of the denominators 5 and 10 is 10.

Now,
 
-45=-4×25×2=-810

Also,

-710=-7×110×1=-710

Further,
 
-810<-710

∴ -45<-710, or, 4-5<-710

6. The two rational numbers are -125and -3.
 -3 can be written as -31.

The LCM of the denominators is 5.

Now,
 
-31=-3×51×5=-155

Because -155<-125, we can conclude that -3<-125.

Page No 3:

Question 7:

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)We will write each of the given numbers with positive denominators.

One number = -37 
Other number =6-13=6×(-1)-13×(-1)=-613

 LCM of 7 and 13 = 91

 ∴ -37=-3×137×13=-3991 

And,

-613=-6×713×7=-4291-613=-6×713×7=-4291-613=-6×713×7=-4291

Clearly,

-39>-41

∴ ​-3991 >-4291

Thus,

-37>6-13

(ii) We will write each of the given numbers with positive denominators.

One number = 5-13=5×(-1)-13×(-1)=-513 

Other number =-3591

 LCM of 13 and 91 = 91

 ∴ -513=-5×713×7=-3591 and -3591

Clearly,
 
-35=-35

 ∴ -3591 =-3591

Thus,
 
-513=-3591 
 

(iii) We will write each of the given numbers with positive denominators.

One number = -2
 
We can write -2 as-21.
Other number =-135

 LCM of 1 and 5 = 5

 ∴​ -21=-2×51×5=-105 and -135=-13×15×1=-135

Clearly,

-10>-13

  ∴ -105>-135

Thus,
 
-21>-135 
 
-2>-135

(iv) We will write each of the given numbers with positive denominators.

One number = -23 
Other number =5-8=5×(-1)-8×(-1)=-58

 LCM of 3 and 8 = 24

∴ ​-23=-2×83×8=-1624 and -58=-5×38×3=-1524

Clearly,

-16<-15
 
∴ -1624<-1524

Thus,
 
-23<-58 
 
-23<5-8

(v) -3-5=-3×-1-5×-1=35

35 is a positive number.

Because every positive rational number is greater than 0, 35>00<35.

(vi) We will write each of the given numbers with positive denominators.

One number = -89

Other number = -910

 LCM of 9 and 10 = 90

∴​-89=-8×109×10=-8090 and -910=-9×910×9=-8190

Clearly,

-81<-80

∴​-8190<-8090

Thus,
 
-910<-89 

Page No 3:

Question 8:

Arrange the following rational numbers in ascending order:
(i) 4-9, -512, 7-18, -23
(ii) -34, 5-12, -716, 9-24
(iii) 3-5, -710, -1115, -1320
(iv) -47, -914, 13-28, -2342

Answer:

(i) We will write each of the given numbers with positive denominators.

We have:

4-9=4×(-1)-9×(-1)=-49 and7-18=7×(-1)-18×(-1)=-718

Thus, the given numbers are -49, -512, -718 and -23.

LCM of 9, 12, 18 and 3 is 36.


Now, 

-49=-4×49×4=-1636

-512=-5×312×3=-1536

-718=-7×218×2=-1436

-23=-2×123×12=-2436

Clearly, 

-2436<-1636<-1536<-1436

∴ ​-23<-49<-512<-718 
 
That is

-23<4-9<-512<7-18

(ii) We  will write each of the given numbers with positive denominators.

We have:

5-12=5×(-1)-12×(-1)=-512 and9-24=9×(-1)-24×(-1)=-924

Thus, the given numbers are -34, -512, -716 and -924.

LCM of  4, 12, 16 and 24 is 48.

Now,
 
-34=-3×124×12=-3648

-512=-5×412×4=-2048

-716=-7×316×3=-2148

-924=-9×224×2=-1848

Clearly, 

-3648<-2148<-2048<-1848

∴​ -34<-716<-512<-924 

That is

-34<-716<5-12<9-24

(iii) We will write each of the given numbers with positive denominators.

We have:

3-5=3×(-1)-5×(-1)=-35

Thus, the given numbers are -35, -710, -1115 and -1320.

LCM of 5, 10, 15 and 20 is 60.

Now, 

-35=-3×125×12=-3660

-710=-7×610×6=-4260

-1115=-11×415×4=-4460

-1320=-13×320×3=-3960

Clearly,
 
-4460<-4260<-3960<-3660

∴​ -1115<-710<-1320<-35.

That is 

-1115<-710<-1320<3-5

(iv) We will write each of the given numbers with positive denominators.

We have:

13-28=13×(-1)-28×(-1)=-1328

Thus, the given numbers are -47, -914, -1328 and -2342.

LCM of 7, 14, 28 and 42 is 84.

Now, 

-47=-4×127×12=-4884

-914=-9×614×6=-5484

-1328=-13×328×3=-3984

-2342=-23×242×2=-4684

Clearly, 

-5484<-4884<-4684<-3984

∴​ -914<-47<-2342<-1328.
 
That is

-914<-47<-2342<13-28


 

Page No 3:

Question 9:

Arrange the following rational numbers in descending order:
(i) -2, -136, 8-3, 13
(ii) -310, 7-15, -1120, 14-30
(iii) -56, -712, -1318, 23-24
(iv) -1011, -1922, -2333, -3944

Answer:

(i) We will first write each of the given numbers with positive denominators. We have:

   8-3=8×(-1)-3×(-1)=-83

Thus, the given numbers are -2, -136, -83 and 13

LCM of 1, 6, 3 and 3 is 6

Now,
 
-21=-2×61×6=-126

-136=-13×16×1=-136

-83=-8×23×2=-166

and 

13=1×23×2=26

Clearly,Thus,
 
26>-126>-136>-166

∴​ 13>-2>-136>-83. i.e 13>-2>-136>8-3


(ii) We will first write each of the given numbers with positive denominators. We have:

   7-15=7×(-1)-15×(-1)=-715 and 17-30=17×(-1)-30×(-1)=-1730 

Thus, the given numbers are -310, -715, -1120 and -1730

LCM of 10, 15, 20 and 30 is 60

Now,
 
-310=-3×610×6=-1860 

-715=-7×415×4=-2860

-1120=-11×320×3=-3360

and 

-1730=-17×230×2=-3460

Clearly,
 
-1860>-2860>-3360>-3460

∴ -310>-715>-1120>-1730. i.e -310>7-15>-1120>17-30

(iii) We will first write each of the given numbers with positive denominators. We have:

   23-24=23×(-1)-24×(-1)=-2324 

Thus, the given numbers are -56, -712, -1318and-2324

LCM of 6, 12, 18 and 24 is 72

Now, 

-56=-5×126×12=-6072

-712=-7×612×6=-4272

-1318=-13×418×4=-5272

and 

-2324=-23×324×3=-6972

Clearly,
 
-4272>-5272>-6072>-6972

∴​ -712>-1318>-56>-2324. i.e -712>-1318>-56>23-24

(iv) The given numbers are -1011, -1922, -2333 and -3944

LCM of 11, 22, 33 and 44 is 132

Now, 

-1011=-10×1211×12=-120132

-1922=-19×622×6=-114132

-2333=-23×433×4=-92132

and 

-3944=-39×344×3=-117132

Clearly,
 
-92132>-114132>-117132>-120132

∴ -2333>-1922>-3944>-1011

Page No 3:

Question 10:

Which of the following statements are true and which are false?
(i) Every whole number is a rational number.
(ii) Every integer is a rational number.
(iii) 0 is a whole number but it is not a rational number.

Answer:

1. True
A whole number can be expressed as ab, with b=1 and a0. Thus, every whole number is rational.

2. True
Every integer is a rational number because any integer can be expressed as ab, with b=1 and 0>a0. Thus, every integer is a rational number.

3. False
0=ab, for a=0 and b0. Thus, 0 is a rational and whole number.



Page No 5:

Question 1:

Represent each of the following numbers on the number line:
(i) 13
(ii) 27
(iii) 134

(iv) 225

(v) 312

(vi) 557

(vii) 423

(viii) 8

Answer:

(i)


(ii)

 
(iii)


(iv)


(v)

(vi)


(vii)


(viii)

Page No 5:

Question 2:

Represent each of the following numbers on the number line:
(i) -13
(ii) -34
(iii) -123
(iv) -317
(v) -435
(vi) -256
(vii) −3
(viii) -278

Answer:

(i)

(ii)


(iii)


(iv)


(v)


(vi)


(vii)


(viii)

Page No 5:

Question 3:

Which of the following statements are true and which are false?
(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) The rational numbers 13 and -52 are on opposite sides of 0 on the number line.
(iv) The rational number -18-13 lies to the left of 0 on the number line.

Answer:

(i) True
A negative number always lies to the left of 0 on the number line.

(ii) False
A negative number always lies to the left of 0 on the number line.

(iii) True
Negative and positive numbers always lie on the opposite sides of 0 on the number line.

(iv) False
The negative sign cancels off and the number becomes 1813; it lies to the right of 0 on the number line.



Page No 10:

Question 1:

Add the following rational numbers:
(i) -25 and 45
(ii) -611 and -411
(iii) -118 and 58
(iv) -73 and 13
(v) 56 and -16
(vi) -1715 and -115

Answer:

1. -25 +45=-2+45=25


2. -611+-411=-6+(-4)11=-6-411=-1011


3. -118+58=-11+58=-68=-3×24×2=-34


4. -73+13=-7+13=-63=-3×23=-2


5. 56+-16=5+(-1)6=46=2×23×2=23


6. -1715+-115=-17+(-1)15=-17-115=-1815=-6×35×3=-65

Page No 10:

Question 2:

Add the following rational numbers:
(i) 34 and -35
(ii) 58 and -712
(iii) -89 and 116
(iv) -516 and 724
(v) 7-18 and 827
(vi) 1-12 and 2-15
(vii) -1 and 34
(viii) 2 and -54
(ix) 0 and -25

Answer:

1. The denominators of the given rational numbers are 4 and 5.

LCM of 4 and 5 is 20.

Now, 

34=3×54×5=1520 and -35=-3×45×4=-1220

∴ 34+-35=1520+-1220=15+(-12)20=15-1220=320

2.​ The denominators of the given rational numbers are 8 and 12.

LCM of 8 and 12 is 24.

Now, 

58=5×38×3=1524 and -712=-7×212×2=-1424

∴​ 58+-712=1524+-1424=15+(-14)24=15-1424=124

3. ​The denominators of the given rational numbers are 9 and 6.

LCM of 9 and 6 is 18.

Now, 

-89=-8×29×2=-1618 and 116=11×36×3=3318

∴​ -89+116=-1618+3318=-16+3318=-16+3318=1718

4.​ The denominators of the given rational numbers are 16 and 24.

LCM of 16 and 24 is 48.

Now, 

-516=-5×316×3=-1548 and 724=7×224×2=1448

∴​ -516+724=-1548+1448=-15+1448=-148

5. We will first write each of the given numbers with positive denominators.

7-18=7×(-1)-18×(-1)=-718

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

LCM of 18 and 27 is 54.

Now, 

-718=-7×318×3=-2154 and 827=8×227×2=1654

∴ 7-18+827=-2154+1654=-21+1654=-554

6. ​We will first write each of the given numbers with positive denominators.

1-12=1×(-1)-12×(-1)=-112 and 2-15=2×(-1)-15×(-1)=-215

​The denominators of the given rational numbers are 12 and 15.

LCM of 12 and 15 is 60.

Now, 

-112=-1×512×5=-560 and -215=-2×415×4=-860

∴ 1-12+2-15=-560+-860=-5+(-8)60=-5-860=-1360

7. We can write -1 as-11.

The denominators of the given rational numbers are 1 and 4.

LCM of 1 and 4 is 4.

Now, 

-11=-1×41×4=-44 and 34=3×14×1=34

∴ -1+34=-44+34=-4+34=-14

8. ​We can write 2 as21.

The denominators of the given rational numbers are 1 and 4.

LCM of 1 and 4 is 4.

Now, 

21=2×41×4=84 and -54=-5×14×1=-54

∴ 2+(-5)4=84+(-5)4=8+(-5)4=8-54=34

9. ​We can write 0 as01.

The denominators of the given rational numbers are 1 and 5.

LCM of 1 and 5 is 5, that is, (1 × 5).

Now,
 
01=0×51×5=05=0 and -25=-2×15×1=-25

∴ 0+(-2)5=05+(-2)5=0+(-2)5=0-25=-25

Page No 10:

Question 3:

Verify the following:
(i) -125+27=27+-125
(ii) -58+-913=-913+-58
(iii) 3+-712=-712+3
(iv) 2-7+12-35=12-35+2-7

Answer:

1. LHS = -125+27

LCM of 5 and 7 is 35.

-12×75×7+2×57×5=-8435+1035=-84+1035=-7435

RHS = 27+-125

LCM of 5 and 7 is 35.

2×57×5 +-12×75×7=1035+-8435=10-8435=-7435

∴ -125+27=27+-125

2. ​LHS = -58+-913

LCM of 8 and 13 is 104.

-5×138×13+-9×813×8=-65104+-72104=-65+(-72)104=-65-72104=-137104

RHS = -913+-58

LCM of 13 and 8 is 104.

-9×813×8 +-5×138×13=-72104+-65104=-72-65104=-137104

∴ -58+-913=-913+-58

3. ​LHS = 31+-712

LCM of 1 and 12 is 12.

3×121×12+-7×112×1=3612+-712=36+(-7)12=36-712=2912

RHS = -712+31

LCM of 12 and 1 is 12.

-7×112×1 +3×121×12=-712+3612=-7+3612=2912

3+-712=-712+3

4. LHS = ​2-7+12-35

We will write the given numbers with positive denominators.

2-7=2×(-1)-7×(-1)=-27 and 12-35=12×(-1)-35×(-1)=-1235

LCM of 7 and 35 is 35.

-2×57×5+-12×135×1=-1035+-1235=-10+(-12)35=-10-1235=-2235

RHS = 12-35+2-7

We will write the given numbers with positive denominators.

12-35=12×(-1)-35×(-1)=-1235 and 2-7=2×(-1)-7×(-1)=-27

LCM of 35 and 7 is 35.

-2×57×5 +-12×135×1=-1035+-1235=-10+(-12)35=-10-1235=-2235

∴​ 2-7+12-35=12-35+2-7

Page No 10:

Question 4:

Verify the following:
(i) 34+-25+-710=34+-25+-710
(ii) -711+2-5+-1322=-711+2-5+-1322
(iii) -1+-23+-34=-1+-23+-34

Answer:

1.
LHS =  34+-25+-710

15-820+-710=720+-710=720+-1420=7+(-14)20=-720

RHS =  34+-25+-710

34+-410+-710=34+-4-710=34+-1110=34+-1110=1520+-2220=15-2220=-720

∴​ 34+-25+-710=34+-25+-710


2.
LHS =  -711+2-5+-1322

We will first make the denominator positive.

-711+2×(-1)-5×(-1)+-1322=-711+-25+-1322

-711+-25+-1322=-3555+-2255+-1322=-35-2255+-1322=-5755+-1322=-114110+-65110=-114-65110=-179110

RHS = -711+2-5+-1322

We will first make the denominator positive.

-711+2×(-1)-5×(-1)+-1322=-711+-25+-1322

-711+-25+-1322=-711+-44110+-65110=-711+-44+(-65)110=-711+-109110=-70110+-109110=-70-109110=-179110

∴​ -711+2-5+-1322=-711+2-5+-1322


3.
LHS = -1+-23+-34

-11+-23+-34=-11+-812+-912=-11+-8-912=-11+-1712=-11+-1712=-1×121×12+-17×112×1=-12+(-17)12=-12-1712=-2912

RHS = -1+-23+-34

-11+-23+-34=-33+-23+-34=-3-23+-34=-53+-34=-53+-34=-2012+-912=-20-912=-2912

∴ -1+-23+-34=-1+-23+-34

Page No 10:

Question 5:

Fill in the blanks.
(i) -317+-125=-125+......
(ii) -9+-218=......+-9
(iii) -813+37+-134=......+37+-134
(iv) -12+712+-911=-12+712+......
(v) 19-5+-311+-78=19-5+......+-78
(vi) -167+......=......+-167=-167

Answer:

(i) Addition is commutative, that is, a+b=b+a.

Hence, the required solution is -317+-125=-125+-37.
 
(ii) Addition is commutative, that is, a+b=b+a.

Hence, the required solution is -9+-218=-218+-9.

(iii) Addition is associative, that is, a+b+c=a+b+c.

Hence, the required solution is -813+37+-134=-813+37+-134.

(iv) Addition is associative, that is, a+b+c=a+b+c.

Hence, the required solution is -12+712+-911=-12+712+-911.

(v) Addition is associative, that is, a+b+c=a+b+c.

Hence, the required solution is19-5+-311+-78=19-5+-311+-78.

(vi) 0 is the additive identity, that is, 0+a=a.

Hence, the required solution is -167+0=0+-167=-167.



Page No 11:

Question 6:

Find the additive inverse of each of the following:
(i) 13
(ii) 239
(iii) −18
(iv) -178
(v) 15-4
(vi) -16-5
(vii) -311
(viii) 0
(ix) 19-6
(x) -8-7

Answer:

The additive inverse of ab is -ab. Therefore, ab+-ab=0
(i) Additive inverse of 13is-13.

(ii) Additive inverse of  239is-239.

(iii) Additive inverse of -18 is 18.

(iv) Additive inverse of -178is178.

(v) In the standard form, we write 15-4as-154.

    Hence, its additive inverse is 154.

(vi) We can write:
 
-16-5=-16×(-1)-5×(-1)=165

    Hence, its additive inverse is -165.

(vii) Additive inverse of -311is311.

(viii) Additive inverse of 0 is 0.

(ix) In the standard form, we write 19-6as-196.

     Hence, its additive inverse is 196.

(x) We can write:
 
-8-7=-8×(-1)-7×(-1)=87

Hence, its additive inverse is -87.

Page No 11:

Question 7:

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

Answer:

(i) 13-34=13+Additive inverse of34            

                     = 13+-34=412+-912=4-912=-512


(ii)  13--56=13+Additive inverse of-56            

                       = 13+56 (Because the additive inverse of -56is56)

                       =26+56=2+56=76


(iii) -35--89=-35+Additive inverse of-89            

                          = -35+89 (Because the additive inverse of -89is89)

                          =-2745+4045=-27+4045=1345


(iv) -1--97=-1+Additive inverse of-97            

                        =-11+97 (Because the additive inverse of -97is97)

                        =-77+97=-7+97=27


(v) 1--1811=1+Additive inverse of-1811            

                       =11+1811 (Because the additive inverse of -1811is1811)

                       = 1111+1811=11+1811=2911


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

                       =0+139 (Because the additive inverse of -139is139)

                       =139

(vii) -65--3213=-65+Additive inverse of-3213            

                             =-65+3213 (Because the additive inverse of -3213is3213)

                             =  -7865+16065=-78+16065=8265


(viii) -47--71=-47+Additive inverse of-71            

                           = -47+71 (Because the additive inverse of -71is71)

                           = -47+497=-4+497=457

Page No 11:

Question 8:

Using the rearrangement property find the sum:
(i) 43+35+-23+-115
(ii) -83+-14+-116+38
(iii) -1320+1114+-57+710
(iv) -67+-56+-49+-157

Answer:

(i)
 43+-23+35+-115
4-23+3-115
=23+-85=1015+-2415=10-2415=-1415.


(ii)
-83+-116+-14+38

=-166+-116+-28+38

=-16-116+-2+38

=-276+18=-10824+324=-10524
=358


(iii)
-1320+710+1114+-57
=-1320+1420+1114+-1014
=-13+1420+11-1014
=120+114=7140+10140=7+10140=17140=17140.


(iv)
-67+-157+-56+-49

=-67+-157+-1518+-818

=-6-157+-15-818

=-217+-2318=-31+-2318=-5418+-2318=-54-2318=-7718

Page No 11:

Question 9:

The sum of two rational numbers is −2. If one of the numbers  is -145, find the other.

Answer:

Let the other number be x.Now,x+-145=-2x-145=-2x=-2+145x=(-2)×5+145x=-10+145x =45

Page No 11:

Question 10:

The sum of two rational numbers is -12. If one of the numbers is 56, find the other.

Answer:

Let the other number be x.Now,x+56=-12x=-12-56x=-3-56x=-86x=-43

Page No 11:

Question 11:

What number should be added to -58 so as to get -32?

Answer:

Let the required number be x

Now,

-58+x =-32

-58+x+58=-32+58      (Adding 58 to both the sides)

x=-32+58x=-128+58x=-12+58x=-78

Hence, the required number is -78.

Page No 11:

Question 12:

What number should be added to −1 so as to get 57?

Answer:

Let the required number be x.
 
Now,

-1+x=57
-1+x+1=57+1     (Adding 1 to both the sides)

x=5+77x=127
Hence, the required number is 127.

Page No 11:

Question 13:

What number should be subtracted from -23 to get -16?

Answer:

Let the required number be x.
 
Now,

-23-x=-16
-23-x+x=-16+x         (Adding x to both the sides)
-23=-16+x
-23+16=-16+x+16    (Adding 16 to both the sides)
-46+16=x
-4+16=x
-36=x-1×32×3=x-12=x

Hence, the required number is-12.

Page No 11:

Question 14:

(i) Which rational number is its own additive inverse?
(ii) Is the difference of two rational numbers a rational number?
(iii) Is addition commutative on rational numbers?
(iv) Is addition associative on rational numbers?
(v) Is subtraction commutative on rational numbers?
(vi) Is subtraction associative on rational numbers?
(vii) What is the negative of a negative rational number?

Answer:

1. Zero is a rational number that is its own additive inverse.

2. Yes
Consider ab-cd (with a, b, c and d as integers), where b and d are not equal to 0.

ab-cd  implies adbd-bcbd  implies ad-bcbd
Since ad, bc and bd are integers since integers are closed under the operation of multiplication and ad-bc is an integer since integers are closed under the operation of subtraction, then  ad-bcbd 
since it is in the form of one integer divided by another and the denominator is not equal to 0
Since, b and d were not equal to 0

Thus, ab-cd is a rational number.

​3. Yes, rational numbers are commutative under addition. If a and b are rational numbers, then the commutative law under addition is a+b=b+a.

4. Yes, rational numbers are associative under addition. If a, b and c are rational numbers, then the associative law under addition is a+(b+c)=(a+b)+c.

5. No, subtraction is not commutative on rational numbers. In general, for any two rational numbers, (a-b)  (b - a).

6. Rational numbers are not associative under subtraction. Therefore, a-(b-c)(a-b)-c.

7. Negative of a negative rational number is a positive rational number.



Page No 16:

Question 1:

Find each of the following products:
(i) 35×-78
(ii) -92×54
(iii) -611×-53
(iv) -23×67
(v) -125×10-3
(vi) 25-9×3-10
(vii) 5-18×-920
(viii) -1315×-2526
(ix) 16-21×145
(x) -76×24
(xi) 724×-48
(xii) -135×-10

Answer:

(i)

35×-78=3×(-7)5×8=-2140

(ii)

-92×54=(-9)×52×4=-458

(iii)

-611×-53=(-6)×(-5)11×3=3033

Simplifying the above rational number, we get:

3033=30÷333÷3=1011

(iv)

-23×67=(-2)×63×7=-1221

Simplifying the above rational number, we get:

-1221=-12÷321÷3=-47

(v)

-125×10-3=(-12)×105×(-3)=-120-15=12015

Simplifying the above rational number, we get:

12015=120÷315÷3=405=8

(vi)

25-9×3-10=25×3(-9)×(-10)=7590

Simplifying the above rational number, we get:

7590=75÷1590÷15=56

(vii)

5-18×-920=5×(-9)-18×20=-45-360=45360

Simplifying the above rational number, we get:

45360=45÷45360÷45=18

(viii)

-1315×-2526=(-13)×(-25)15×26=325390

Simplifying the above rational number, we get:

325390=325÷5390÷5=6578=65÷1378÷13=56

(ix)

16-21×145=16×14(-21)×5=224-105

Simplifying the above rational number, we get:

224-105=224÷7(-105)÷7=32-15=32×-1-15×-1=-3215

(x)

-76×24=(-7)×246=-1686

Simplifying the above rational number, we get:

-1686=(-168)÷26÷2=843=-84÷33÷3=-28

(xi)

724×(-48)=7×(-48)24=-33624

Simplifying the above rational number, we get:

-33624=-336÷2424÷24=-14

(xii)

-135×(-10)=(-13)×(-10)5=1305

Simplifying the above rational number, we get:

1305=130÷55÷5=26

Page No 16:

Question 2:

Verify each of the following:
(i) 37×-59=-59×37
(ii) -87×139=139×-87
(iii) -125×7-36=7-36×-125
(iv) -8×-1312=-1312×-8

Answer:

(i)

37×-59=-59×37

 LHS=3×(-5)7×9        =-1563Simplifying, we get:-1563=-15÷363÷3=-521

  RHS=-59×37=(-5)×39×7=-1563Simplifying, we get:=-15÷363÷3=-521

LHS = RHS


(ii)

-87×139=139×-87LHS =-87×139 =(-8)×137×9 =-10463 RHS=139×-87 =13 ×(-8)9×7 =-10463 LHS=RHS


(iii)

-125×7-36=7-36×-125

  LHS =-125×7-36=(-12)×75×(-36)=84180Simplifying, we get: =84÷12180÷12=715

 RHS=7-36×-125=7×(-12)(-36)×5=84180Simplifying, we get:=84÷12180÷12=715

LHS = RHS


(iv)
-8 ×-1312=-1312×(-8)

 LHS =-8 ×-1312=(-8)×(-13)12=10412Simplifying, we get: =104÷412÷4=263

RHS=-1312×(-8)=(-13)×(-8)12=10412Simplifying, we get: =104÷412÷4=263

LHS = RHS

Page No 16:

Question 3:

Verify each of the following:
(i) 57×1213×718=57×1213×78
(ii) -1324×-125×3536=-1324×-125×3536
(iii) -95×-103×21-4=-95×-103×21-4

Answer:

(i)

57×1213×718=57×1213×718

LHS=57×1213×718=5×127×13×718=6091×718=4201638=1039


RHS=57×1213×718=57×12×713×18=57×84234=4201638=1039

∴ ​57×1213×718=57×1213×718


(ii)

-1324×-125×3536=-1324×-125×3536

 LHS=-1324×-125×3536=-1324×(-12)×355×36=-1324×-420180=54604320=9172


 RHS=-1324×-125×3536=(-13)×(-12)24×5×3536=156120×3536=156×35120×36=54604320=9172

 ∴ ​-1324×-125×3536=-1324×-125×3536


(iii)

-95×-103×21-4=-95×-103×21-4

  LHS=-95×-103×21-4=(-9)×(-10)5×3×21-4=9015×21-4=90×2115×(-4)=-189060=-632


  RHS=-95×-103×21-4=-95×(-10)×213×(-4)=-95×21012=(-9)×2105×12=-189060=-632

∴ (-95×-103)×21-4=-95×(-103×21-4)

Page No 16:

Question 4:

Fill in the blanks:
(i) -2317×1835=1835×......
(ii) -38×-719=-719×......
(iii) 157×-2110×-56=......×-2110×-56
(iv) -125×415×25-16=-125×415×......

Answer:

(i)

-2317×1835=1835×-2317            (a×b=b×a)

(ii)

-38×-79=-79×-38             (a×b=b×a)

(iii)

(157×-2110)×-56=157×(-2110×-56)     [a×(b×c)=(a×b)×c)]

(iv)

-125×(415×25-16)=(-125×415)×25-16     [a×(b×c)=(a×b)×c]

Page No 16:

Question 5:

Find the multiplicative inverse (i.e., reciprocal) of:
(i) 1325
(ii) -1712
(iii) -724
(iv) 18
(v) −16
(vi) -3-5
(vii) −1
(viii) 02
(ix) 2-5
(x) -18

Answer:

(i)Reciprocal of 1325 is 2513.(ii)Reciprocal of -1712 is 12-17, that is, -1217.(iii) Reciprocal of -724 is 24-7, that is, -247.(iv)Reciprocal of 18 is 118. (v)Reciprocal of-16 is 1-16, that is, -116.(vi)Reciprocal of -3-5 is -5-3, that is, 53.(vii)Reciprocal of-1 is -1.(viii)Reciprocal of 02 does not exist as 20=.(ix)Reciprocal of 2-5 is -52.(x)Reciprocal of -18 is -8.



Page No 17:

Question 6:

Find the value of:
(i) 58-1
(ii) -49-1
(iii) -7-1
(iv) 1-3-1

Answer:

We know that  a-1=1a or a-1×a=1

(i)58-1=85 58×58-1=1(ii)-49-1=9-4=-94-49×-49-1=1(iii)(-7)-1=1-7=-17-7×(-7)-1=1

(iv) (-3)-1(-3)-1=1-3=-13 (-3)-1×-3 = 1

Page No 17:

Question 7:

Verify the following:
(i) 37×56+1213=37×56+37×1213
(ii) -154×37+-125=-154×37+-154×-125
(iii) -83+-1312×56=-83×56+-1312×56
(iv) -167×-89+-76=-167×-89+-167×-76

Answer:

 (i)LHS=37×56+1213=37×65 +7278=37×13778=137182RHS=37×56+1213×37=3×57×6+12×313×7=1542+3691=195+216546=411546=137182

∴ ​37×(56+1213)=(37×56)+(37×1213)

(ii)LHS=-154×(37+-125)=-154×(15-8435)=-154×-6935=(-15)×(-69)140=1035140=20728RHS=(-154×37)+(-154×-125)=(-15)×34×7+(-15)×(-12)4×5=-4528+18020=-225+1260140=1035140=20728 -154×(37+-125)=(-154×37)+(-154×-125)

(iii)

(-83+-1312)×56=(-83×56)+(-1312×56)LHS=(-83+-1312)×56=-32-1312×56=-4512×56=-45×512×6=-22572=-225÷972÷9=-258RHS=(-83×56)+(-1312×56)=-8×53×6+(-13)×512×6=-4018+-6572=-160-6572=-22572=-225÷972÷9=-258 (-83+-1312)×56=(-83×56)+(-1312×56)

(iv)

-167×(-89+-76)=(-167×-89)+(-167×-76)LHS=-167×(-89+-76)=-167×(-16-2118)=-167×-3718=592126=29663RHS=(-167×-89)+(-167×-76)=12863+11242=256+336126=592126=29663 -167×(-89+-76)=(-167×-89)+(-167×-76)

Page No 17:

Question 8:

Name the property of multiplication illustrated by each of the following statements:
(i) -158×-127=-127×-158
(ii) -23×79×-95=-23×79×-95
(iii) -34×-56+78=-34×-56+-34×78
(iv) -169×1=1×-169=-169
(v) -1115×15-11=15-11×-1115=1
(vi) -75×0=0

Answer:

  1. Commutative property
  2. Associative property
  3. Distributive property
  4. Property of multiplicative identity
  5. Property of multiplicative inverse
  6. Multiplicative property of 0

Page No 17:

Question 9:

Fill in the blanks:
(i) The product of a rational number and its reciprocal is .......
(ii) Zero has ....... reciprocal.
(iii) The numbers ....... and ....... are their own reciprocals.
(iv) zero is ....... the reciprocal of any number.
(v) The reciprocal of a, where a ≠ 0, is .......
(vi) The reciprocal of 1a, where a ≠ 0, is .......
(vii) The reciprocal of a positive rational rational number is .......
(viii) The reciprocal of a negative rational number is .......

Answer:

(i) 1
(ii) no
(iii) 1; -1
(iv) not
(v) 1a
(vi) a
(vii) positive
(viii) negative



Page No 19:

Question 1:

Simplify:
(i) 49÷-512
(ii) -8÷-716
(iii) -127÷-18
(iv) -110÷-85
(v) -1635÷-1514
(vi) -6514÷137

Answer:

(i)49÷-512=49×12-5=4×129×-5=48-45=-4845=-1615(ii)-8÷-716=-8×16-7=8×167=1287(iii)-127÷(-18)=-127×1-18=12126=12÷3126÷3=442=4÷242÷2=221(iv)-110÷-85=-110×5-8=580=5÷580÷5=116(v)-1635÷-1514=-1635×14-15=224525(vi)-6514÷137=-6514×713=-52

Page No 19:

Question 2:

Verify whether the given statement is true or false:
(i) 135÷2610=2610÷135
(ii) -9÷34=34÷-9
(iii) -89÷-43=-43÷-89
(iv) -724÷3-16=3-16÷-724

Answer:

(i)135÷2610=2610÷135LHS135÷2610=135×1026=130130=1RHS2610÷135=2610×513=130130=1TRUE(ii)-9 ÷34=34÷(-9) LHS -9÷34 =-9×43 =-363 =-12 RHS 34÷(-9) =34×1-9 =3-36 =-112 FALSE iii)-89÷-43=-43÷-89 LHS -89÷-43 =-89×3-4 =2436 =23 RHS -43÷-89 =-43×9-8 =3624 =32 FALSE (iv)-724÷3-16=3-16÷-724 LHS -724×-163 =11272 RHS 3-16÷-724 =3-16×24-7 =72112 FALSE

Page No 19:

Question 3:

Verify whether the given statement is true or false:
(i) 59÷13÷52=59÷13÷52
(ii) -16÷65÷-910=-16÷65÷-910
(iii) -35÷-1235÷114=-35÷-1235÷14

Answer:

(i)(59÷13)÷52=59÷(13÷52)LHS(59÷13)÷52=(59×31)×25=5×3×29×1×5=3045=23RHS59÷13÷52=59÷13×25=59÷215=59×152 =7518=256LHSRHSFALSE

​(ii)
[(-16)÷65]÷-910=(-16)÷[65÷-910]LHS=[(-16)÷65]÷-910=[(-16)×56]×10-9=(-16)×5×106×(-9)=80054=40027RHS(-16)÷(65÷-910)=(-16)÷(65×10-9)=-16÷-6045=-16×-4560=-16×-34=484=12LHS RHSFALSE

(iii)
(-35÷-1235)÷114=-35÷(-1235÷14)LHS=(-35×35-12)×14=(-3)×35×145×(-12)=147060=492RHS=-35÷(-1235÷14)=-35÷(-1235×41)=-35÷(-12×435)=-35÷(-4835)=-35×35-48=3×355×48=105240=716LHSRHSFALSE

Page No 19:

Question 4:

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

Answer:

Let the number be x.Now,x×(-12)=-9x=-9÷(-12)x=(-9)×1-12x=-9-12x=34

Page No 19:

Question 5:

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

Answer:

Let the number be x.Now,x×-43=-169x=-169÷-43x=-169×3-4x=-16×39×(-4)x=4836x=43.

Page No 19:

Question 6:

By what rational number should we multiply -1556 to get -57?

Answer:

Let the number be x.Now,x×-1556=-57x=-57÷-1556x=-57×56-15x=280105x=280÷5105÷5x=5621x=56÷721÷7x=83

Page No 19:

Question 7:

By what rational number should -839 be multiplied to obtain 126?

Answer:

Let the number be x.Now,x×-839=126x=126÷-839x=126×39-8x=39-208x=39×-1-208×-1x=-39208x=-39÷13208÷13x=-316

Page No 19:

Question 8:

By what number should -338 be divided to get -112?

Answer:


Let the number be x.Now,-338÷x=-112-338×1x=-1121x=-112÷-3381x=-112×8-331x=88661x=43x=34                          (Reciprocal of 43)

Page No 19:

Question 9:

Divide the sum of 135 and -127 by the product of -317 and 1-2.

Answer:

135+-127÷-317×1-2=91-6035÷-31-14=3135÷3114=3135×1431=1435=14÷735÷7=25

Page No 19:

Question 10:

Divide the sum of 6512 and 83 by their difference.

Answer:

 6512+83÷6512-83=6512+3212÷6512-3212=9712÷3312=9712×1233=9733

Page No 19:

Question 11:

Fill in the blanks:
(i) 98+...=-32
(ii) ...÷-75=1019
(iii) ...÷-3=-415
(iv) -12÷...=-65

Answer:

(i)Let 98÷x=-3298×1x=-321x=-32÷981x=-32×891x=-24181x=-43x=-34             [Reciprocal of -43]

(ii)Let  x÷-75=1019x×5-7=1019 x=1019÷5-7x=1019×-75x=-7095x=-1419

(iii)Let x÷(-3)=-415  x × 1-3=-415x=-415×(-3)x=1215x=45

(iv)Let (-12)÷x=-65(-12)×1x=-651x=-65÷(-12)1x=-65×1-121x=110x=10 

Page No 19:

Question 12:

(i) Are rational numbers always closed under division?
(ii) Are rational numbers always commutative under division?
(iii) Are rational numbers always associative under division?
(iv) Can we divide 1 by 0?

Answer:


​(i)  No, rational numbers are not closed under division in general.
 
a0=; it is not a rational number.

(ii) No

 ab÷cd=ab×dc=adbc Also, cd÷ab=cd×ba=cbda Thus,  ab÷cdcd÷ab

Therefore, division is not commutative.

(iii) No, rational numbers are not associative under division. 

ab÷cd÷efab÷cd÷ef

(iv) No, we cannot divide 1 by 0. The answer will be, which is not defined.



Page No 21:

Question 1:

Find a rational number between 14 and 13.

Answer:

Required number=12(14+13)=12(3+412)=(12×712)=724

Page No 21:

Question 2:

Find a rational number between 2 and 3.

Answer:

Required Number=12×(2+3)                               =52

Page No 21:

Question 3:

Find a rational number between -13 and 12.

Answer:

Required number=12×-13+12=12×-2+36=12×16=112

Page No 21:

Question 4:

Find two rational numbers between −3 and −2.

Answer:

Required number=12×-3-2=12(-5)=-52We know:-3<-52<-2Rational number between -3 and -52=12×-3-52=12(-6-52)=12×-112=-114Thus, the required numbers are -52 and -114.

Page No 21:

Question 5:

Find three rational numbers between 4 and 5.

Answer:


Rational number between 4 and 5:12(4+5)=92Rational number between 4 and 92:12(4+92)=12(8+92)=12(172)=174Rational number between 92and 5:12(92+5)=12(9+102)=194We know:4<174<92<194<5Thus, the three rational numbers are 174, 92 and 194.

Page No 21:

Question 6:

Find three rational numbers between 23 and 34.

Answer:

Rational number between 23 and 34:12(23+34)=12(8+912)=1724We know:23<1724 <34Rational number between 23 and 1724:12(23+1724)=12(16+1724)=12(3324)=3348=33÷348÷3=1116Rational number between  1724 and 34:12(1724 + 34)=12(17+1824)=12(3524)=3548We know:23<1116<1724<3548<34Thus, the three rational numbers are 1116, 1724 and 3548.

Page No 21:

Question 7:

Find 10 rational numbers between -34 and 56.

Answer:

LCM of 4 and 6 is 12.Now,-34=-3×34×3=-912 And,56=5×26×2=1012Rational numbers lying between -34 and 56: -812, -712, -612, -512, -412,...112, 212, 312, 412, 512, 612, 712, 812, 912

We can take any 10 out of these.

Page No 21:

Question 8:

Find 12 rational numbers between −1 and 2.

Answer:

We may write: -1=-1010 and 2=2010Rational numbers between -1 and 2:-910, -810, -710, -610, -510, -410,...,1410, 1510, 1610, 1710, 1810 and 1910We can take any 12 numbers out of these.

Page No 21:

Question 1:

From a rope 11 m long, two pieces of lengths 235m and 3310m are cut off. What is the length of the remaining rope?

Answer:

Length of the rope when two pieces of lengths 235 m and 3310 m are cut off = Total length of the rope - Length of the two cut off pieces
11-235+3310
Now,

235+33102+35+3+310                     =135+3310
LCM of 5 and 10 is 10, i.e., 5×1×2.

 We have:13×2+33×110=26+3310=5910

∴​ 235+3310=5910
Length of the remaining rope =11-5910

                                            =110-5910=5110=5110 m

Therefore, the length of the remaining rope is 5110 m.
 

Page No 21:

Question 2:

A drum full of rice weighs 4016kg. If the empty drum weighs 1334kg, find the weight of rice in the drum.

Answer:

Weight of rice in the drum = Weight of the drum full of rice - Weight of the empty drum

                                       =4016-1334=40+16-13+34=2416-554=2416+Additive inverse of 554=482-16512=31712=26512 kg
Therefore, the weight of rice in the drum is 26512 kg.

Page No 21:

Question 3:

A basket contains three types of fruits weighing 1913kg in all. If 819kg of these be apples, 316kg be oranges and the rest pears, what is the weight of the pears in the basket?

Answer:

Weight of pears in the basket = Weight of the basket containing three types of fruits - (Weight of apples + Weight of oranges)
 =1913-819+316
Now,

819+3168+19+3+16                        =739+196

LCM of 9 and 6 is 18, that is, 3×3×2.

We have:73×2+19×318=146+5718=20318

∴​ 819+316=20318
Now,
Weight of pears in the basket = 1913-20318
                                            =19+13-20318=583-20318=583+Additive inverse of20318=348-20318=14518=8118 kg
 ​Therefore, the weight of the pears in the basket is 8118 kg.



Page No 22:

Question 4:

On one day a rickshaw puller earned Rs 160. Out of his earnings he spent Rs 2635 on tea and snacks, Rs 5012 on food and Rs 1625 on repairs of the rickshaw. How much did he save on that day?

Answer:

Total earning = ₹160
Money spent on tea and snacks = ₹2635
Money spent on food = ₹5012
Money spent on repairs = ₹1625
Let the savings be ₹x.
Money spent on tea and snacks + Money spent on food + Money spent on repairs + Savings = Total earning
So, 2635 + 5012 + 1625 + x = 160
2635+5012+1625+x=1601335+1012+825+x=160266+505+16410+x=16093510+x=160x=160-93510
x=1600-93510x=66510=6612
So, the savings are ₹6612.

Page No 22:

Question 5:

Find the cost of 325 metres of cloth at Rs 6334 per metre.

Answer:

Cost of 1 m of cloth = ₹6334
So, cost of 325 m of cloth
= 6334 × 325
=2554×175=21634
So, the cost of 325 m of cloth is ₹21634.

Page No 22:

Question 6:

A car is moving at an average speed of 6025 km/hr. How much distance will it cover in 614 hours?

Answer:

Speed = 6025 km/h
Time = 614 h
We know that
Speed=DistanceTimeSpeed×Time=DistanceDistance=6025×614Distance=3025×254Distance=37712 km
Hence, the distance covered in 614 h is 37712km.

Page No 22:

Question 7:

Find the area of a rectangular park which is 3635 m long and 1623 m broad.

Answer:

Area of the rectangular park = Length of the park × Breadth of the park     (∵ Area of rectangle = Length × Breadth)

=3635×1623=36+35×16+23=1835×503=183×505×3=915015=610 m2

Therefore, the area of the rectangular park is 610 m2.

Page No 22:

Question 8:

Find the area of a square plot of land whose each side measures 812 metres.

Answer:

Area of the square plot = Side × Side = Side2 = a2  (Because the area of the square is a2, where a is the side of the square)
                                                                                                   
 =812×812=8+12×8+12=172×172=17×172×2=2894=7214 m2
Therefore, the area of the square plot is 7214 m2.

Page No 22:

Question 9:

One litre of petrol costs ₹ 6334 . What is the cost of 34 litres of petrol?

Answer:

Cost of 1 litre of petrol = ₹6334
Cost of 34 litres of petrol = 6334 × 34 = 2554×34=216712
So, the cost of 34 litres of petrol is ₹216712.

Page No 22:

Question 10:

An aeroplane covers 1020 km in an hour. How much distance will it cover in 416 hours?

Answer:

Distance covered by the aeroplane in 416 hours = 416×1020
                                                                         =4+16×1020=256×1020=256×10201=25×10206×1=255006=4250 km

Therefore, the distance covered by the aeroplane is 4250 km.

Page No 22:

Question 11:

The cost of 312 metres of cloth is ₹ 16614 . what is the cost of one metre of cloth?

Answer:

Cost of 312 m of cloth = ₹16614
So, the cost of 1 m of cloth = 16614312=665472=6654×27=952=4712
Hence, the cost of 1 m of cloth is ₹ 4712.

Page No 22:

Question 12:

A cord of length 7112 m has been cut into 26 pieces of equal length. What is the length of each piece?

Answer:

Length of each piece of the cord = 7112÷26
                                                 =71+12÷26=1432÷26=1432÷261=1432×126=143×12×26=14352=94=234 m

Hence, the length of each piece of the cord is 234 metres.

Page No 22:

Question 13:

The area of a room is 6514m2. If its breadth is 5716 metres, what is its length?

Answer:

Area of a room = Length × Breadth
Thus, we have: 
 6514=Length×5716Length=6514÷5716
            =65+14÷5+716=2614÷8716=2614×1687=261×164×87=4176348=12 m

Hence, the length of the room is 12 metres.

Page No 22:

Question 14:

The product of two fractions is 935. If one of the fractions is 937, find the other.

Answer:

Let the other fraction be x.

Now, we have:

937×x=935      x=935÷937             =9+35÷9+37             =485÷667             =485×766             =48×75×66             =336330             =5655             =1155          
Hence, the other fraction is 1155.

Page No 22:

Question 15:

In a school, 58 of the students are boys. If there are 240 girls, find the number of boys in the school.

Answer:

If 58of the students are boys, then the ratio of girls is 1-58, that is, 38.

Now, let x be the total number of students.

Thus, we have:

38x=240  x=240÷38

         =240×83=2401×83=240×81×3=19203=640

Hence, the total number of students is 640.
Now,
Number of boys = Total number of students - Number of girls
                         =640-240=400

Hence, the number of boys is 400.

Page No 22:

Question 16:

After reading 79 of a book, 40 pages are left. How many pages are there in the book?

Answer:

Ratio of the read book = 79
Ratio of the unread book = 1-79

                                      =29
Let x be the total number of pages in the book.

Thus, we have:
                         
29×x=40 x=40÷29

        =40×92=401×92=40×91×2=3602=180

Hence, the total number of pages in the book is 180.

Page No 22:

Question 17:

Rita had Rs 300. She spent 13 of her money on notebooks and 14 of the remainder on stationery items. How much money is left with her?

Answer:

Amount of money spent on notebooks = 300×13

                                                          =3001×13=3003=100

∴ Money left after spending on notebooks = 300-100
                                                                =200
Amount of money spent on stationery items from the remainder = 200×14
                                                                                               =2001×14=2004=50

∴ Amount of money left with Rita = 200-50
                                                   =Rs 150

Page No 22:

Question 18:

Amit earns ₹ 32000 per month. He spends 14 of his income of food; 310 of the remainder on house rent and 521 of the remainder on the education of children. How much money is still left with him?

Answer:

Amit's income per month = ₹32,000
Money spent on food = 14 of 32,000=14×32,000=8,000
Remaining amount = ₹32,000 − ₹8,000 = ₹24,000
Money spent on house rent = 310×24,000=Rs 7,200
Money left = ₹24,000 − ₹7,200 = ₹16,800
Money spent on education of children = 521×16,800=4,000
Amount of money still left with him = ₹16,800 − ₹4,000 = ₹12,800

Page No 22:

Question 19:

If 35 of a number exceeds its 27 by 44, find the number.

Answer:

Let x be the required number.
We know that 35 of the number exceeds its 27 by 44.
That is,

35×x=27×x+44
  35×x-27×x=44   35-27×x=4435+Additive inverse of 27×x=44                            21-1035×x=44
                                       1135×x=44
                                                 x=44÷1135

                                                   =44×3511=441×3511=44×351×11=154011=140

Hence, the number is 140.

Page No 22:

Question 20:

At a cricket test match 27 of the spectators were in a covered place while 15000 were in open. Find the total number of spectators.

Answer:

Ratio of spectators in the open =1-27
                                               =57
Total number of spectators in the open = x
Then,57×x=15000
                                                          x=15000÷57

                                                                 =15000×75=150001×75=15000×71×5=105005=21000

Hence, the total number of spectators is 21,000

Page No 22:

Question 1:

Tick (✓) the correct answer
-516+712=?
(a) -748
(b) 124
(c) 1348
(d) 13

Answer:

(c) 1348
The denominators of the given rational numbers are 16 and 12, respectively.
LCM of 16 and 12 is 4×4×3, that is,48
Now, we have:
-516+712=3×-5+4×748

                      =-15+2848=1348



Page No 23:

Question 2:

Tick (✓) the correct answer
8-15+4-3=?
(a) 2815
(b) -2815
(c) -45
(d) -415

Answer:

(b) -2815
8-15=-815 and4-3=-43

Now, we have:

8-15+4-3=-815+-43

LCM of 15 and 3 is 3×5×1, that is,15

-815+-43=1×-8+5×-415
                   =-8+-2015=-2815

Page No 23:

Question 3:

Tick (✓) the correct answer
7-26+1639=?
(a) 1178
(b) -1178
(c) 1139
(d) -1139

Answer:

7-26=-726

Now, we have:

7-26+1639=-726+1639

LCM of 26 and 39 is 1014, that is, 29×1×36.

(a) 1178
-726+1639=39×-7+26×161014
                      =-273+4161014=1431014=1178

Page No 23:

Question 4:

Tick (✓) the correct answer
3+5-7=?
(a) -167
(b) 167
(c) -267
(d) -87

Answer:

(b) 167

3=31 and 5-7=-57

Now, we have:
           
3+5-7=31+-57

LCM of 1 and 7 is 7

31+-57=7×3+1×-57
                    =21+-57=167

Page No 23:

Question 5:

Tick (✓) the correct answer
31-4+-58=?
(a) 678
(b) 578
(c) -578
(d) -678

Answer:

(d) -678
 31-4=-314

We have:
         
31-4+-58=-314+-58

LCM of 4 and 8 is 8, that is, 4×1×2.

-314+-58=2×-31+1×-58
                         =-62+-58=-678

Page No 23:

Question 6:

Tick (✓) the correct answer
What should be added to 712 to get -415?
(a) 1720
(b) -1720
(c) 720
(d) -720

Answer:

(b) -1720
Let the required number be x

Now,
  
712+x=-415

x=-415+-712

=4×-4+5×-760=-16+-3560=-5160=-1720
 

Page No 23:

Question 7:

Tick (✓) the correct answer
23+-45+715+-1120=?
(a) -15
(b) -415
(c) -1360
(d) -730

Answer:

(c) -1360
Using the commutative and associative laws, we can arrange the terms in any suitable manner. Using this rearrangement property, we have:

23+-45+715+-1120=23+715+-45+-1120
 

                                         =(10+7)15+[-16+-11]20=1715+-2720=[68+-81]60=-1360

Page No 23:

Question 8:

Tick (✓) the correct answer
The sum of two numbers is -43. If one of the numbers is −5, what is the other?
(a) -113
(b) 113
(c) -193
(d) 193

Answer:

(b) 113
Let the other number be x

Now,

x+-5=-43
x=-43+Additive inverse of -5x=-43+5

       =-43+51=-4+153=113
 

Page No 23:

Question 9:

Tick (✓) the correct answer
What should be added to -57 to get -23?
(a) -2921
(b) 2921
(c) 121
(d) -121

Answer:

(c) 121
Let the required number be x

Now,

-57+x=-23
x=-23+Additive inverse of -57x=-23+57
       =-14+1521=121
 

Page No 23:

Question 10:

Tick (✓) the correct answer
What should be subtracted from -53 to get 56?
(a) 52
(b) 32
(c) 54
(d) -52

Answer:

(d) -52
Let the required number be x

Now,

-53-x=56
x=-53-56

       =-10-56=-156=-52
Thus, the required number is -52
 

Page No 23:

Question 11:

Tick (✓) the correct answer
-37-1=?
(a) 73
(b) -73
(c) 37
(d) none of these

Answer:

(b) -73

 -37-1Reciprocal of-37

The reciprocal of -37 is 7-3, i.e., -73
 

Page No 23:

Question 12:

Tick (✓) the correct answer
The product of two rational numbers is -2881. If one of the numbers is 1427 then the other one is
(a) -23
(b) 23
(c) 32
(d) -32

Answer:

(a) -23
Let the other number be x

Now,

x×1427=-2881

x=-2881÷1427

       =-2881×2714=-28×2781×14=-28×2781×14=-2×39×1=-69=-23
Thus, the other number is -23

Page No 23:

Question 13:

Tick (✓) the correct answer
The product of two numbers is -1635. If one of the numbers is -1514, the other is
(a) -25
(b) 815
(c) 3275
(d) -83

Answer:

(c) 3275
Let the other number be x

Now,

x×-154=-1635
x=-1635÷-1514

       =-1635×14-15=-16×14-35×15=16×1435×15 =224525 =3275

Thus, the other number is 3275
 



Page No 24:

Question 14:

Tick (✓) the correct answer
What should be subtracted from -35 to get −2?
(a) -75
(b) -135
(c) 135
(d) 75

Answer:

(d) 75
Let the required number be x

Now,

-35-x=-2-35=-2+xx=-35+2x =-3+105 x=75
Thus, the required number is 75
 

Page No 24:

Question 15:

Tick (✓) the correct answer
The sum of two rational numbers is −3. If one of them is -103 then the other one is
(a) -133
(b) -193
(c) 13
(d) 133

Answer:

(c) 13
Let the other number be x

Now,

x+-103=-3x=-3+Additive inverse of -103x=-3+103
     =-31+103=-9+103=13
Thus, the other number is 13

Page No 24:

Question 16:

Tick (✓) the correct answer
Which of the following numbers is in standard form?
(a) -1226
(b) -4971
(c) -916
(d) 28-105

Answer:

(b) -4971 and (c) -916

The numbers -4971 and -916 are in the standard form because they have no common divisor other than 1 and their denominators are positive. 

Page No 24:

Question 17:

Tick (✓) the correct answer
-916×815=?
(a) -310
(b) -415
(c) -925
(d) -25

Answer:

(a) -310

-916×815=-9×816×15

                      =-72240=-310

Page No 24:

Question 18:

Tick (✓) the correct answer
-59÷23=?
(a) -52
(b) -56
(c) -1027
(d) -65

Answer:

(d) -56

-59÷23=-59×32

                =-5×39×2=-1518=-56

Page No 24:

Question 19:

Tick (✓) the correct answer
49÷?=-815
(a) -3245
(b) -85
(c) -910
(d) -56

Answer:

(d) -56

Let 49÷ab=-815

Now,

    49×ba=-815ba=-815×94

=-65

ab=5-6

=-56
Hence, the missing number is -56.

Page No 24:

Question 20:

Tick (✓) the correct answer
Additive inverse of -59 is
(a) -95
(b) 0
(c) 59
(d) 95

Answer:

(c) 59

Additive inverse of -59 is 59.

Page No 24:

Question 21:

Tick (✓) the correct answer
Reciprocal of -34 is
(a) 43
(b) 34
(c) -43
(d) 0

Answer:

(c) -43
 Reciprocal of -34 is 4-3, i.e., -43.

Page No 24:

Question 22:

Tick (✓) the correct answer
A rational number between -23 and 14 is
(a) 512
(b) -512
(c) 524
(d) -524

Answer:

(d) -524
Rational number between -23 and 14 = 12-23+14
                                                           =12-8+312=12×-512=-524

Page No 24:

Question 23:

Tick (✓) the correct answer
The reciprocal of a negative rational number
(a) is a positive rational number
(b) is a negative rational number
(c) can be either a positive or a negative rational number
(d) does not exist

Answer:

(b) is a negative rational number

The reciprocal of a negative rational number is a negative rational number.



Page No 27:

Question 1:

Find the additive inverse of:
(i) 7-10
(ii) 85.

Answer:

(i) 7-10=7×-1-10×-1=-710

Additive inverse of -710 is 710.

(ii) Additive inverse of 85 is -85.

Page No 27:

Question 2:

The sum of two rational numbers is −4. If one of them is -115, find the other.

Answer:

Let the other number be x. Thus, we have:x+-115=-4x-115=-4x=-4+Additive inverse of -115x=-4+115x=-41+115x=-4×5+11×15x=-20+115x=-95

Page No 27:

Question 3:

What number should be added to -35 to get 23?

Answer:

Let the required number be x.Thus, we have:x+(-3)5=23x-35=23x=23+35=2×5+3×315x=10+915x=1915

Page No 27:

Question 4:

What number should be subtracted from -34 to get -12?

Answer:

Let the required number be x.Thus, we have:-34-x=-12-34+12=xx=2-34x=-14

Page No 27:

Question 5:

Find the multiplicative inverse of:
(i) -34
(ii) 114.

Answer:

(i) Multiplicative inverse of -34 is 4-3, i.e., -43.(ii) Multiplicative inverse of 114 is 411. 

Page No 27:

Question 6:

The product of two numbers is −8. If one of them is −12, find the other.

Answer:

Let the other number be x. Thus, we have:-12 × x=-8  x=(-8)÷(-12)x=-8×1-12x=812x=23

Page No 27:

Question 7:

Evaluate:
(i) -35×107
(ii) -58-1
(iii) -6-1

Answer:

(i)-35×107=-3×105×7=-3035=-6×57×5=-67(ii)(-58)-1=1-58=1×8-5=8-5=8×-1-5×-1=-85(iii)(-6)-1=1-6=1×-1-6×-1=-16

Page No 27:

Question 8:

Name the property of multiplication shown by each of the following statements:
(i) -125×34=34×-125
(ii) -815×1=-815
(iii) -23×78×-57=-23×78×-57
(iv) -23×0=0
(v) 25×-45+-310=25×-45+25×-310

Answer:

(i) Commutative law of multiplication

(ii) Existence of  multiplicative identity

(iii) Associative law of multiplication

(iv) Multiplicative property of 0

(v) Distributive law of multiplication over addition

Page No 27:

Question 9:

Find two rational numbers lying between -13 and 12.

Answer:

Required number=12×-13+12=12×-2+36=12×16=112-13<112<12Rational number between -13 and 112:12×-13+112=12×1-412=12×-312=-324=-3÷324÷3=-18Thus, 112 and -18 are the two rational numbers between -13 and 12.

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Question 10:

Mark (✓) against the correct answer
What should be added to -35 to get -13?
(a) 45
(b) 815
(c) 415
(d) 25

Answer:

 (c) 415

Let the number be x
Now,

-35+x=-13x=-13+Additive inverse of -35x=-13+35x=-1×53×5+3×35×3x=-515+915x=-5+915x=415

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Question 11:

Mark (✓) against the correct answer
What should be subtracted from -23 to get 34?
(a) -1112
(b) -1312
(c) -54
(d) -1712

Answer:

 (d) -1712

Let the number be x.
Now,

-23-x=34-1×23+x=3423+x=-34x=-34+Additive inverse of 23 x=-34+-23 x=-34+-23 x=-3×34×3+-2×43×4 x-912+-812 x=-1712

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Question 12:

Mark (✓) against the correct answer
-54-1=?
(a) 45
(b) -45
(c) 54
(d) 35

Answer:

 (b) -45

We have:

-54-1=1-54=1×4-5=4-5=4×-1-5×-1=-45

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Question 13:

Mark (✓) against the correct answer
The product of two numbers is -14. If one of them is -310, then the other is
(a) 56
(b) -56
(c) 43
(d) -85

Answer:

 (a) 56

Let the required number be x.
Now,

-310×x=-14x=-14÷-310x=-14×10-3x=1012=56



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Question 14:

Mark (✓) against the correct answer
-56÷-23=?
(a) -54
(b) 54
(c) -45
(d) 45

Answer:

(b)​ 54

We have:
-56÷-23=-56×3-2=1512=54

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Question 15:

Mark (✓) against the correct answer
43÷?=-52
(a) -85
(b) 85
(c) -815
(d) 815

Answer:

(c) -815

We have:43÷x=-5243×1x=-521x=-52431x=-52×341x=-158x=8-15=8×-1-15×-1=-815

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Question 16:

Mark (✓) against the correct answer
Reciprocal of -79 is
(a) 97
(b) -97
(c) 79
(d) none of these

Answer:

(b) -97
Reciprocal of -79=-79-1Now, we have:1-79=9-7=9×-1-7×-1=-97

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Question 17:

A rational number between -23 and 12 is
(a) -16
(b) -112
(c) -56
(d) 56

Answer:

(b) -112

Number between -23 and 12=12×-23+12=12×-2×23×2+1×32×3=12×-46+36=12×-4+36=-112

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Question 18:

Fill in the blanks.
(i) 258÷.......=-10.
(ii) -89×.......=-23.
(iii) -1+.......=-29.
(iv) 23-.......=115.

Answer:

(i)Let the number be x.Now, we have:258÷x=-10258×1x=-101x=-10÷2581x=-10×8251x=-8025x=25-80x=25×-1-80×-1x=-2580x=-25÷580÷5x=-516

(ii)
Let the number be x.Now, we have:-89×x=-23x=-23÷-89x=-23×9-8x=1824=18÷624÷6x=34

(iii)
Let the blank space be x.Now, we have:(-1)+x=-29x=-29+1x=-2+99x=79

(iv)
Let the blank space be x.Now, we have:23-x=115-x=115-23-x=1-1015-x=-915x=915=35

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Question 19:

Write 'T' for true and 'F' for false for each of the following:
(i) Rational numbers are always closed under subtraction.
(ii) Rational numbers are aways closed under division.
(iii) 1 ÷ 0 = 0.
(iv) Subtraction is commutative on rational numbers.
(v) --78=78.

Answer:

(i) T

​If ab and cd are rational numbers, then ab-cd=ad-bcbd is also a rational number because ad, bc and bd are all rational numbers.

(ii) F

​Rational numbers are not always closed under division. They are closed under division only if the denominator is non-zero.

(iii) F

1÷0 cannot be defined.

(iv) F

​Let ab and cd represent rational numbers. 

Now, we have:

ab-cd=ad-bcbd
cd-ab=bc-adbd

∴ ab-cdcd-ab

(v) T
`
--78=-1×-78=-1×-78=78



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