Rs Aggarwal 2019 2020 Solutions for Class 8 Math Chapter 1 Rational Numbers are provided here with simple step-by-step explanations. These solutions for Rational Numbers are extremely popular among Class 8 students for Math Rational Numbers Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rs Aggarwal 2019 2020 Book of Class 8 Math Chapter 1 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rs Aggarwal 2019 2020 Solutions. All Rs Aggarwal 2019 2020 Solutions for class Class 8 Math are prepared by experts and are 100% accurate.

Page No 3:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

Answer:

(i)


(ii)

 
(iii)


(iv)


(v)

(vi)


(vii)


(viii)

Page No 5:

Answer:

(i)

(ii)


(iii)


(iv)


(v)


(vi)


(vii)


(viii)

Page No 5:

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:

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:

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:

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:

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:

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:

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:

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:

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:

Answer:

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

Page No 11:

Answer:

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

Page No 11:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

Answer:

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



Page No 19:

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:

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:

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:

Answer:

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

Page No 19:

Answer:

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

Page No 19:

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:

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:

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:

Answer:

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

Page No 19:

Answer:

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

Page No 19:

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:

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:

Answer:

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

Page No 21:

Answer:

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

Page No 21:

Answer:

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

Page No 21:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

Answer:

(b) -73

 -37-1Reciprocal of-37

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

Page No 23:

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:

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:

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:

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:

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:

Answer:

(a) -310

-916×815=-9×816×15

                      =-72240=-310

Page No 24:

Answer:

(d) -56

-59÷23=-59×32

                =-5×39×2=-1518=-56

Page No 24:

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:

Answer:

(c) 59

Additive inverse of -59 is 59.

Page No 24:

Answer:

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

Page No 24:

Answer:

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

Page No 24:

Answer:

(b) is a negative rational number

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



Page No 27:

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:

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:

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:

Answer:

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

Page No 27:

Answer:

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

Page No 27:

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:

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:

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:

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.

Page No 27:

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

Page No 27:

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

Page No 27:

Answer:

 (b) -45

We have:

-54-1=1-54=1×4-5=4-5=4×-1-5×-1=-45

Page No 27:

Answer:

 (a) 56

Let the required number be x.
Now,

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



Page No 28:

Answer:

(b)​ 54

We have:
-56÷-23=-56×3-2=1512=54

Page No 28:

Answer:

(c) -815

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

Page No 28:

Answer:

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

Page No 28:

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

Page No 28:

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

Page No 28:

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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