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Page No 36:

Question 1:

Carry out the following additions of rational numbers.

(i)  536+642  

(ii)  123+245 

(iii)  1117+1319

(iv) 2311+1377

Answer:


 i 536+642
At first, we will calculate the LCM of 36 and 42. The prime factorisation is 36 and 42 is,
36 = 2 × 2 × 3 × 3
42 = 2 × 3 × 7
Now, LCM of 36 and 42 = 2 × 2 × 3 × 3 × 7 = 252
536+642=5×736×7+6×642×6=35252+36252=35+36252=71252
ii 123+245=1×3+23+2×5+45=53+145
Now, LCM of 3 and 5 is 15.
53+145=5×53×5+14×35×3=2515+4215=6715=4715
iii 1117+1319
Now, LCM of 17 and 19 is 323.
1117+1319=11×1917×19+13×1719×17=209323+221323=430323
iv 2311+1377=2×11+311+1×77+377=2511+8077
Now, LCM of 11 and 77 is 77.
2511+8077=25×711×7+80×177×1=17577+8077=25577=32477

Page No 36:

Question 2:

Carry out the following subtarctions involving rational numbers.

(i) 711-37

(ii) 1336-240

(iii) 123-356

(iv) 412-313

Answer:


i 711-37
Now, LCM of 11 and 7 is 77.
711-37=7×711×7-3×117×11=4977-3377=49-3377=1677
ii 1336-240
Now, LCM of 36 and 40 is 360.
1336-240=13×1036×10-2×940×9=130360-18360=130-18360=112360=112÷8360÷8    since, HCF of 112 and 360 is 8=1445
iii 123-356=1×3+23-3×6+56=53-236
Now, LCM of 3 and 6 is 6.
53-236=5×23×2-23×16×1=106-236=10-236=-136
iv 412-313=4×2+12-3×3+13=92-103
Now, LCM of 2 and 3 is 6.
92-103=9×32×3-10×23×2=186-206=18-206=-26=-13

Page No 36:

Question 3:

Multiply the following rational numbers.

(i) 311×25

(ii) 125×415

(iii) -89×34

(iv) 06×34

Answer:


i 311×25=3×211×5=655
ii 125×415=12×45×15=4875=48÷375÷3    Since, HCF of 48 and 75 is 3=1625
iii -89×34=-8×39×4=-2436=-24÷1236÷12    Since, HCF of 24 and 36 is 12=-23
iv 06×34=0×36×4=024=0

Page No 36:

Question 4:

Write the multiplicative inverse.

(i) 25 

(ii) -3 8 

(iii)  -1739 

(iv) 7

(v) -713

Answer:

It is known that, the multiplicative inverse of any rational number a is the reciprocal of the rational number i.e., 1a.
(i) Multiplicative inverse of 25=125=52
(ii) Multuplicative inverse of -38=1-38=-83
(iii) Multiplicative inverse of -1739=1-1739=-3917
(iv) Multiplicative inverse of 7 = 17
(v) The given number is -713.
Now, -713=-7+13=-21+13=-223
Multiplicative inverse of -223=1-223=-322
 

Page No 36:

Question 5:

Carry out the divisions of rational numbers.

(i) 4012÷104

(ii) -1011÷-1110

(iii) -78÷-36

(iv) 23÷-4

(v) 215÷536

(vi) -513÷726

(vii) 911÷-8

(viii) 5÷25

Answer:


i 4012÷104=4012×410=40×412×10=160120=160÷40120÷40    Since, HCF of 160 and 120 is 40=403
ii -1011÷-1110=-1011×-1011=-10×-1011×11=100121
iii -78÷-36=-78×-63=-7×-68×3=4224=42÷624÷6    Since, HCF of 42 and 24 is 6   =74
iv 23÷-4=23×-14=2×-13×4=-212=-2÷212÷2    Since, HCF of 2 and 12 is 2=-16
v 215÷536=2×5+15÷5×6+36=115÷336=115×633=11×65×33=66165=66÷33165÷33    Since, HCF of 66 and 165 is 33  =25



Page No 38:

Question 1:

Write three rational numbers that lie between the two given numbers.

(i) 27, 67 

(ii) 45, 23

(iii)  -23, 45

(iv) 79, -59 

(v) -34, +54

(vi)  78, -53

(vii) 57, 117

(viii) 0, -34

Answer:

(i) The given numbers are 27 and 67.
We know that,
2 < 3 < 4 < 5 < 6
 27<37<47<57<67
Hence, 3 rational numbers between 27 and 67 are :
37,47 and 57.

(ii) The given numbers are 45 and 23.
Let us convert these numbers into fractions with equal denominators.
45=4×65×6=243023=2×103×10=2030
We know that,
20 < 21 < 22 < 23 < 24
 2030<2130<2230<2330<243023<2130<2230<2330<45
Hence, 3 rational numbers between 23 and 45 are :
2130,2230 and 2330.

(iii) The given numbers are -23 and 45.
Let us convert each of given numbers into fractions with equal denominators.
-23=-2×53×5=-101545=4×35×3=1215
We know that,
−10 < −9 < −8 < −7 <...........< 1 < 2 < 3 < 4 <..........< 12
 -23<-915<-815<-715<.....<115<215<315<415<.....<415
Hence, 3 rational numbers between -23 and 45 are:
-915,-715 and 415.

(iv) The given numbers are 79 and -59.
We know that,
−5 < −4 < −3 < −2 < −1 < 0 <.....< 6 < 7
 -59<-49<-39<-29<-19<0<.....<69<79
Hence, 3 rational numbers between -59 and 79 are:
-49, 0 and 69.

(v) The given numbers are -34 and 54.
We know that,
−3 < −2 < −1 < 0 < 1 < 2 < 3 < 4 < 5
 -34<-24<-14<0<14<24<34<44<54
Hence, 3 rational numbers between -34 and 54 are:
-24, -14 and 34.

(vi) The given numbers are 78 and -53.
Let us convert each of the given numbers into fractions with equal denominators.
78=7×38×3=2124-53=-5×83×8=-4024
We know that,
−40 < −39 <....< −13 < −12 <......<11 < 12 <....17 <.... 21
 -4024<-3924<.....<-1324<-1224<....<1124<1224<....<1724<....<2124 -53<-3924<......<-1324<-1224<....<1124<1224<....<1724<....<78
Hence, 3 rational numbers between -53 and 78 are:
-1324, 1124 and 1724.

(vii) The given numbers are 57 and 117.
We know that, 
5 < 6 < 7 < 8 < 9 < 10 < 11
57<67<77<87<97<107<117
Hence, 3 rational numbers between 57 and 117 are :
67,87 and 97

(viii) The given numbers are 0 and -34.
Let us convert each of the given numbers into fractions with equal denominators.
0=0×81×8=08-34=-3×24×2=-68
We know that,
−6 < −5 < −4 < −3 < −2 < −1 < 0
 -68<-58<-48<-38<-28<-18<08 -34<-58<-48<-38<-28<-18<0
Hence, 3 rational numbers between -68 and 0 are:
-58, -28 and -18.



Page No 41:

Question 1:

Write the following rational numbers in decimal form.

(i) 134

(ii) -78

(iii) 735 

(iv) 512

(v) 227

(vi) 43

(vii) 79

Answer:

(i) The given number is 134.


 134=3.25
The decimal form of 134 is 3.25.
(ii) The given number is -78.

 78=0.875
The decimal form of -78 is − 0.875.
(iii) The given number is 735.
735=7×5+35=35+35=385

 385=7.6
The decimal form of 735 is 7.6.
(v) The given number is 227.

 227=3.142857142857....=3.142857
The decimal form of 227 is 3.142857.
(vi) The given number is 43.

 43=1.33....=1.3
The decimal form of 43 is 1.3.
(vii) The given number is 79.

 79=0.77....=0.7
The decimal form of 79 is 0.7.



Page No 42:

Question 1:

Simplify:
50 × 5 ÷ 2 + 24

Answer:


50 × 5 ÷ 2 + 24
= 250 ÷ 2 + 24
= 125 + 24
= 149

Page No 42:

Question 2:

Simplify:
(13 × 4) ÷ 2 – 26

Answer:


(13 × 4) ÷ 2 – 26
= 52 ÷ 2 – 26
= 26 – 26
= 0

Page No 42:

Question 3:

Simplify:
140 ÷ [( – 11) × ( – 3) – ( – 42) ÷ 14 – 1]

Answer:


140 ÷ [( – 11) × ( – 3) – ( – 42) ÷ 14 – 1]
= 140 ÷ [33 – ( – 42) ÷ 14 – 1]
= 140 ÷ [33 + 42 ÷ 14 – 1]
= 140 ÷ [33 + 3 – 1]
= 140 ÷ [36 – 1]
= 140 ÷ 35
= 4

Page No 42:

Question 4:

Simplify:
{(220 – 140) + [10 × 9 + ( – 2 × 5)]} – 100

Answer:


{(220 – 140) + [10 × 9 + (– 2 × 5)]} – 100
= {80 + [10 × 9 + (– 10)]} – 100
= {80 + [10 × 9  –  10]} – 100
= {80 + [90 – 10]} –  100
= {80 + 80} –  100
= 160 – 100
= 60

Page No 42:

Question 5:

Simplify:
35+38÷64

Answer:


 35+38÷64=35+38×46=35+14=4×3+5×120    Since, LCM of 5 and 4 is 20=12+520=1720



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