Rs Aggrawal 2020 2021 Solutions for Class 6 Maths Chapter 1 Number System are provided here with simple step-by-step explanations. These solutions for Number System are extremely popular among Class 6 students for Maths Number System Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rs Aggrawal 2020 2021 Book of Class 6 Maths Chapter 1 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rs Aggrawal 2020 2021 Solutions. All Rs Aggrawal 2020 2021 Solutions for class Class 6 Maths are prepared by experts and are 100% accurate.

#### Page No 5:

(i) Nine thousand eighteen = 9018
(ii) Fifty-four thousand seventy-three = 54073
(iii) Three lakh two thousand five hundred six = 302506
(iv) Twenty lakh ten thousand eight = 2010008
(v) Six crore five lakh fifty-seven = 60500057
(vi) Two crore two lakh two thousand two hundred two = 20202202
(vii) Twelve crore twelve lakh twelve thousand twelve = 121212012
(viii) Fifteen crore fifty lakh twenty thousand sixty-eight = 155020068

#### Page No 5:

(i) 63,005 = Sixty-three thousand five
(ii) 7,07,075 =  Seven lakh seven thousand seventy-five
(iii) 34,20,019 = Thirty-four lakh twenty thousand nineteen
(iv) 3,05,09,012 = Three crore five lakh nine thousand twelve
(v) 5,10,03,604 = Five crore ten lakh three thousand six hundred four
(vi) 6,18,05,008 = Six crore eighteen lakh five thousand eight
(vii) 19,09,09,900 = Nineteen crore nine lakh nine thousand nine hundred
(viii) 6,15,30,807 = Six crore fifteen lakh thirty thousand eight hundred seven
(ix) 6,60,60,060 = Six crore sixty lakh sixty thousand sixty

#### Page No 5:

(i) 15,768 = (1 x 10000) + (5 x 1000) + (7 x 100) + (6 x 10) + (8 x 1)

(ii) 3,08,927 = (3 x 100000) + (8 x 1000) + (9 x 100) + (2 x 10) + (7 x 1)

(iii) 24,05,609 = (2 x 1000000) + (4 x 100000) + (5 x 1000) + (6 x 100) + (9 x 1)

(iv) 5,36,18,493 = (5 x 10000000) + (3 x 1000000) + (6 x 100000) + (1 x 10000) + (8 x 1000) + (4 x 100) + (9 x 10) + (3 x 1)

(v) 6,06,06,006 = (6 x 10000000) + (6 x 100000) + (6 x 1000) + (6 x 1)

(iv) 9,10,10,510 = (9 x 10000000) + (1 x 1000000) + (1 x 10000) + (5 x 100) + (1 x 10)

#### Page No 6:

(i) 6 × 10000 + 2 × 1000 + 5 × 100 + 8 × 10 + 4 x 1 = 62,584

(ii) 5 × 100000 + 8 × 10000 + 1 × 1000 + 6 × 100 + 2 × 10 + 3 × 1 = 5,81,623

(iii) 2 × 10000000 + 5 × 100000 + 7 × 1000 + 9 × 100 + 5 × 1 = 2,05,07,905

(iv) 3 × 1000000 + 4 × 100000 + 6 × 1000 + 5 × 100 + 7 × 1 = 34,06,507

#### Page No 6:

The place value of 9 at ten lakhs place = 90 lakhs = 9000000
The place value of 9 at hundreds place = 9 hundreds = 900
$\therefore$ Required difference = (9000000 ‒ 900) = 8999100

#### Page No 6:

The place value of 7 in 27650934 = 70 lakhs = 70,00,000
The face value of 7 in 27650934 = 7
$\therefore$ Required difference = (7000000 ‒ 7) = 69,99,993

#### Page No 6:

The largest 6-digit number = 999999
The smallest 6-digit number = 100000
$\therefore$ Total number of 6-digit numbers = (999999 ‒ 100000) + 1
= 899999 + 1
= 900000
= 9 lakhs

#### Page No 6:

The largest 7-digit number = 9999999
The smallest 7-digit number = 1000000
∴ Total number of 7-digit numbers = (9999999 - 1000000) + 1
= 8999999 + 1
= 9000000
= Ninety lakhs

#### Page No 6:

One lakh (1,00,000) is equal to one hundred thousand (100 $×$ 1000).
Thus, one hundred thousands make a lakh.

#### Page No 6:

One crore (1,00,00,000) is equal to one hundred lakh (10,000 $×$ 1,000).
Thus, 10,000 thousands make a crore.

#### Page No 6:

The given number is 738.
On reversing the digits of this number, we get 837.
∴ Required difference = 837 ‒ 738 = 99

#### Page No 6:

The number just after 9547999 is 9547999 + 1 = 9548000.

#### Page No 6:

The number just before 9900000 is 9900000 ‒ 1 = 9899999.

#### Page No 6:

The number just before 10000000 is 10000000 ‒ 1 = 9999999.

#### Page No 6:

The 3-digit numbers formed by 2, 3 and 4 by taking each digit only once are 234, 324, 243, 342, 423 and 432.

#### Page No 6:

The smallest number formed by using each of the given digits (i.e, 3,1,0,5 and 7) only once is 10357.

#### Page No 6:

The largest number formed by using each of the given digits only once is 964320.

#### Page No 6:

Representation of the numbers on the international place-value chart:

 Periods Millions Thousands Ones Place Hundred millions Ten millions Millions Hundred thousands Ten thousands Thousands Hundreds Tens Ones HM TM M H Th T Th Th H T O (i) 7 3 5 8 2 1 (ii) 6 0 5 7 8 9 4 (iii) 5 6 9 4 3 8 2 1 (iv) 3 7 5 0 2 0 9 3 (v) 8 9 3 5 0 0 6 4 (vi) 9 0 7 0 3 0 0 6 Crore Ten lakhs Lakhs Ten Thousand Thousand Hundred Tens Ones

The number names of the given numbers in the international system:

(i) 735,821 = Seven hundred thirty-five thousand eight hundred twenty-one
(ii) 6,057,894 = Six million fifty-seven thousand eight hundred ninety-four
(iii) 56,943,821 = Fifty-six million nine hundred forty-three thousand eight hundred twenty-one
(iv) 37,502,093 = Thirty-seven million five hundred two thousand ninety-three
(v) 89,350,064 = Eighty-nine millions three hundred fifty thousand sixty-four
(vi) 90,703,006 = Ninety million seven hundred three thousand and six

#### Page No 6:

 Periods Millions Thousands Ones Place Hundred millions Ten millions Millions Hundred thousands Ten thousands Thousands Hundreds Tens Ones HM TM M H Th T Th Th H T O (i) 3 0 1 0 5 0 6 3 (ii) 5 2 2 0 5 0 0 6 (iii) 5 0 0 5 0 0 5

#### Page No 8:

1003467 $>$ 987965

We know that a 7-digit number is always greater than a 6-digit number. Since 1003467 is a 7-digit number and 987965 is a 6-digit number, 1003467 is greater than 987965.

#### Page No 8:

3572014 $<$ 10235401

We know that a 7-digit number is always less than an 8-digit number. Since 3572014 is a 7-digit number and 10235401 is an 8-digit number, 3572014 is less than 10235401.

#### Page No 8:

Both the numbers have the digit 3 at the ten lakhs places.
Also, both the numbers have the digit 2 at the lakhs places.
However, the digits at the ten thousands place in 3254790 and 3260152 are 5 and 6, respectively.
Clearly, 5 < 6
∴ 3254790 < 3260152

#### Page No 8:

Both have the digit 1 at the crores places.
However, the digits at the ten lakhs places in 10357690 and 11243567 are 0 and 1, respectively.
Clearly, 0 < 1
∴ 10357690 < 11243567

#### Page No 8:

27596381 > 7965412

We know that an 8-digit number is always greater than a 7-digit number. Since 7965412 is a 7-digit number and  27596381 is an 8-digit number, 27596381 is greater than 7965412.

#### Page No 8:

Both the numbers have the same digits, namely 4, 7, 8 and 9, at the crores, ten lakhs, lakhs and ten thousands places, respectively.
However, the digits at the thousands place in 47893501 and 47894021 are 3 and 4, respectively.
Clearly, 3 < 4
∴ 47893501 < 47894021

#### Page No 8:

102345680 is a 9-digit number.

63521047 and 63514759 are both 8-digit numbers.
Both the numbers have the same digits, namely 6, 3 and 5, at the crores, ten lakhs and lakhs places, respectively.
However, the digits at the ten thousands place in 63521047 and 63514759 are 2 and 1, respectively.
Clearly, 2 > 1
∴ 63521047 > 63514759

7355014 and 7354206 are both 7-digit numbers.
Both the numbers have the same digits, namely 7, 3 and 5 at the crores, ten lakhs and lakhs places, respectively.
However, the digits at the ten thousands place in 7355014 and 7354206 are 5 and 4, respectively.
Clearly, 5> 4
∴ 7355014 > 7354206

The given numbers in descending order are:
102345680 > 63521047 > 63514759 > 7355014 > 7354206

#### Page No 8:

23794206 and 23756819 are both 8-digit numbers.
Both the numbers have the same digits, namely 2, 3 and 7 at the crores, ten lakhs and lakhs places, respectively.
However, the digits at the ten thousands place in
23794206 and 23756819 are 9 and 5, respectively.
Clearly, 9 > 5

∴ 23794206  > 23756819

5032790 and 5032786 are both 7-digit numbers.
Both the numbers have the same digits, namely 5, 0, 3, 2 and 7, at the ten lakhs, lakhs, ten thousands, thousands and hundreds places, respectively.
However,
the digits at the tens place in
5032790 and 5032786 are 9 and 8, respectively.
Clearly,   9 > 8
5032790 > 5032786

987876 is a 6-digit number.

The given numbers in descending order are:
23794206  > 23756819 > 5032790 > 5032786 > 987876

#### Page No 8:

16060666 and 16007777 are both 8-digit numbers.
Both the numbers have the same digits, namely 1, 6 and 0, at the crores, ten lakhs and lakhs places, respectively.
However, the digits at the ten thousands place in 16060666 and 16007777 are 6 and 0, respectively.
Clearly, 6 > 0
∴ 16060666 > 16007777

1808090 and 1808088 are both 7-digit numbers.
Both the numbers have the same digits , namely 1, 8, 0, 8 and 0, at the ten lakhs, lakhs, ten thousands, thousands and hundreds places, respectively.
However, the digits at the tens place in 1808090 and 1808088 are 9 and 8, respectively.
Clearly, 9 > 8
∴ 1808090 > 1808088

190909 and 181888 are both 6-digit numbers.
Both the numbers have the same digit, 1, at the lakhs place.
However, the digits at the ten thousands place in 190909 and 181888 are 9 and 8, respectively.
Clearly, 9 > 8
∴ 190909 > 181888

Thus, the given numbers in descending order are:
16060666 > 16007777 > 1808090 > 1808088 >190909 > 181888

#### Page No 8:

1712040, 1704382 and 1702497 are all 7-digit numbers.
The three numbers have the same digits, namely 1 and 7, at the ten lakhs and lakhs places, respectively.
However, the digits at the ten thousands place in
1712040, 1704382 and 1702497 are 1, 0 and 0.
∴ 1712040  is the largest.
Of the other two numbers, the respective digits at the thousands place are 4 and 2.
Clearly, 4 > 2
∴ 1704382 > 1702497

201200, 200175 and 199988 are all 6-digit numbers.
At the lakhs place, we have 2 > 1.
So, 199988 is the smallest of the three numbers.

The other two numbers have the same digits, namely 2 and 0, at the lakhs and ten thousands places, respectively.
However, the digits at the thousands place in
201200 and 200175 are 1 and 0, respectively.
Clearly, 1 > 0
∴ 201200 > 200175

The given numbers in descending order are:
1712040 > 1704382 > 1702497 > 201200 > 200175 > 199988

#### Page No 8:

990357 is 6 digit number.

9873426 and 9874012 are both 7-digit numbers.
Both the numbers have the same digits, namely 9, 8 and 7, at the ten lakhs, lakhs and ten thousands places, respectively.
However, the digits at the thousands place in 9873426 and 9874012
are 3 and 4, respectively.
Clearly, 4 < 7
∴ 9873426 <  9874012

24615019 and  24620010 are both 8-digit numbers.

Both the numbers have the same digits, namely 2, 4 and 6, at the crores, ten lakhs and lakhs places, respectively.
However, the digits at the ten thousands place in 24615019 and 24620010
are 2 and 1, respectively.
Clearly, 1 < 2
∴ 24615019 < 24620010

The given numbers in ascending order are:
990357 < 9873426 <  9874012 < 24615019 < 24620010

#### Page No 8:

5694437 and 5695440 are both 7-digit numbers.
Both have the same digit, i.e., 5 at the ten lakhs place.
Both have the same digit, i.e., 6 at the lakhs place.
Both have the same digit, i.e., 9
at the ten thousands place.
However, the digits at the thousand place in 5694437 and 5695440 are 4 and 5, respectively.
Clearly, 4 < 5
∴ 5694437 < 5695440

56943201, 56943300 and 56944000 are all 8-digit numbers.
They have the same digit, i.e., 5 at the crores place.
They have the same digit, i.e., 6 at the ten lakhs place.
They have the same digit, i.e., 9 at the lakhs place.
They have the same digit, i.e., 4
at the ten thousands place.
However, at the thousands place, one number has 4 while the others have 3 .
∴ 56944000 is the largest.

The other two numbers have 3 and 2 at their hundreds places.
Clearly, 2 <3
∴ 56943201 < 56943300

The given numbers in ascending order are:
5694437 < 5695440 < 56943201 < 56943300 < 56944000

#### Page No 8:

700087 is 6-digit number.

8014257, 8014306 and 8015032 are all 7-digit numbers.
They have the same digits, namely 8, 0 and 1, at the ten lakhs, lakhs and ten thousands places, respectively.
But, at the thousands place, one number has 5 while the other two numbers have 4.
Here, 801503 is the largest.
The other two numbers have 2 and 3 at their hundreds places.
Clearly, 2 < 3
∴ 8014306  < 8015032

10012458 is an 8-digit number.

The given numbers in ascending order are:
700087 <  8014257 <  8014306  < 8015032 < 10012458

#### Page No 8:

893245, 893425 and 980134 are all 6-digit numbers.
Among the three, 980134 is the largest.
The other two numbers have the same digits, namely 8, 9 and 3, at the lakhs, ten thousands and thousands places, respectively.
However, the digits at
the hundreds place in 893245 and 893425 are 2 and 4, respectively.
Clearly, 2 < 4
∴ 893245 < 893425

1020216, 1020304 and 1021403 are all 7-digit numbers.
They have the same digits, namely 1, 0 and 2, at the ten lakhs, lakhs and ten thousands places, respectively.
At the thousands place,
1021403 has 1.
The other two numbers have the digits 2 and 3 at their hundreds places.
Clearly, 2 < 3
∴ 1020216 < 1020304

The given numbers in ascending order are:
893245 < 893425 <  980134 < 1020216 < 1020304 < 1021403

#### Page No 11:

Number of persons who visited the holy shrine in the first year = 13789509
Number of persons who visited the holy shrine in the second year = 12976498
∴ Number of persons who visited the holy shrine during these two years = 13789509 + 12976498 = 26766007

#### Page No 11:

Bags of sugar produced by the first factory in last year = 24809565
Bags of sugar produced by the second factory in last year = 18738576
Bags of sugar produced by the third sugar factory in last year = 9564568
∴ Total number of bags of sugar were produced by the three factories during last year = 24809565 + 18738576 + 9564568
= 53112709

#### Page No 11:

New number = Sum of 37684955 and 3615045
= 37684955 + 3615045
= 41300000

#### Page No 11:

Total number of votes received by the three candidates = 687905 + 495086 + 93756 = 1276747
Number of invalid votes = 13849
Number of persons who did not vote = 25467
∴ Total number of registered voters = 1276747 + 13849 + 25467
= 1316063

#### Page No 11:

People who had only primary education = 1623546
People who had secondary education = 9768678
People who had higher education = 6837954
Illiterate people in the state = 2684536
Children below the age of school admission = 698781
∴ Total population of the state = 1623546 + 9768678 + 6837954 + 2684536 + 698781
= 21613495

#### Page No 11:

Bicycles produced by the company in the first year = 8765435
Bicycles produced by the company in the second year = 8765435 + 1378689
= 10144124

∴ Total number of bicycles produced during these two years = 8765435 + 10144124
= 18909559

#### Page No 11:

Sale receipts of a company during the first year = Rs 20956480
Sale receipts of the company during the second year = Rs 20956480 + Rs 6709570
= Rs 27666050

∴ Total number of sale receipts of the company during these two years = Rs 20956480 + Rs 27666050
= Rs 48622530

#### Page No 11:

Total population of the city = 28756304
Number of males in the city = 16987059
∴ Number of females in the city =  28756304 ‒ 16987059
= 11769245

#### Page No 12:

Required number = 13246510 ‒ 4658642 = 8587868
∴ 13246510 is larger than 4658642 by 8587868.

#### Page No 12:

Required number = 1 crore ‒ 564387
= 10000000 ‒ 5643879
= 4356121

∴ 5643879 is smaller than one crore by 4356121.

#### Page No 12:

11010101 ‒ required number = 2635967

Thus, required number = 11010101 ‒ 2635967
= 8374134

∴ The number 8374134 must be subtracted from 11010101 to get 2635967.

#### Page No 12:

Sum of the two numbers = 10750308
One of the number = 8967519

∴ The other number = 10750308 ‒ 8967519
= 1782789

#### Page No 12:

Initial amount with the man = Rs 20000000
Amount spent on buying a school building = Rs 13607085

∴ Amount left with the man = Rs 20000000 ‒ Rs 13607085
= Rs 6392915

#### Page No 12:

Money need by the society to buy the property = Rs 18536000
Amount collected as membership fee = Rs 7253840
Amount taken on loan from the bank = Rs 5675450
Amount collected as donation = Rs 2937680

∴ Amount of money short = Rs 18536000 ‒ (Rs 7253840 + Rs 5675450 + Rs 2937680)
= Rs 18536000 ‒  Rs 15866970
= Rs 2669030

#### Page No 12:

Initial amount with the man = Rs 10672540
Amount given to his wife = Rs 4836980
Amount given to his son = Rs 3964790

∴ Amount received by his daughter = Rs 10672540 ‒ (Rs 4836980 + Rs 3964790)
= Rs 10672540 ‒ Rs 8801770
= Rs 1870770

#### Page No 12:

Cost of one chair = Rs 1485
Cost of 469 chairs = Rs 1485 $×$ 469
= Rs 696465

∴ Cost of 469 chairs is Rs 696465.

#### Page No 12:

Contribution from one student for the charity program = Rs 625
Contribution from 1786 students = Rs 625 x 1786 = Rs 1116250

∴ Rs 1116250 was collected from 1786 students for the charity program.

#### Page No 12:

Number of screws produced by the factory in one day = 6985
Number of screws produced in 358 days = 6985 x 358
= 2500630

∴ The factory will produce 2500630 screws in 358 days.

#### Page No 12:

We know that
1 year = 12 months
13 years = 13 x 12 = 156 months

Now, we have:
Amount saved by Mr Bhaskar in one month = Rs 8756
Amount saved in 156 months = Rs 8756 $×$ 156 = Rs 1365936

∴ Mr Bhaskar will save Rs 1365936 in 13 years.

#### Page No 12:

Cost of one scooter = Rs 36725
Cost of 487 scooter = Rs 36725 $×$ 487
= Rs 17885075

∴ The cost of 487 scooters will be Rs 17885075.

#### Page No 12:

Distance covered by the aeroplane in one hour = 1485 km
Distance covered in 72 hours = 1485 km $×$ 72 = 106920 km

∴ The distance covered by the aeroplane in 72 hours will be 106920 km.

#### Page No 12:

Product of two numbers = 13421408
One of the number = 364

∴ The other number = 13421408 ÷ 364
= 36872

#### Page No 12:

Cost of 36 flats = Rs 68251500
Cost of one flat = Rs 68251500 ÷ 36
= Rs 1895875

∴ Each flat costs Rs 1895875.

#### Page No 12:

We know that 1 kg = 1000 g
Now, mass of the gas-filled cylinder = 30 kg 250 g = 30.25 kg
Mass of an empty cylinder = 14 kg 480 g = 14.48 kg

∴ Mass of the gas contained in the cylinder = 30.25 kg ‒ 14.48 kg
= 15.77 kg = 15 kg 770 g

#### Page No 12:

We know that 1 m = 100 cm
Length of the cloth = 5 m
Length of the piece cut off from the cloth = 2 m 85 cm

∴ Length of the remaining piece of cloth = 5 m ‒ 2.85 m
= 2.15 m = 2 m 15 cm

#### Page No 12:

We know that 1 m = 100 cm
Now, length of the cloth required to make one shirt = 2 m 75 cm
Length of the cloth required to make 16 such shirts = 2 m 75 cm $×$ 16
= 2.75 m $×$ 16
= 44 m

∴ The length of the cloth required to make 16 shirts will be 44 m.

#### Page No 12:

We know that 1 m = 100 cm
Cloth needed for making 8 trousers = 14 m 80 cm
Cloth needed for making 1 trousers = 14 m 80 cm ÷ 8
= 14 .8 m ÷ 8
= 1.85 m = 1 m 85 cm

∴ 1 m 85 cm of cloth will be required to make one shirt.

#### Page No 12:

We know that 1 kg = 1000 g
Now, mass of one brick = 2 kg 750 g
∴ Mass of 14 such bricks = 2 kg 750 g $×$ 14
= 2.75 kg $×$ 14
= 38.5 kg = 38 kg 500 g

#### Page No 12:

We know that 1 kg = 1000 g
Now, total mass of 8 packets of the same size = 10 kg 600 g
∴ Mass of one such packet = 10 kg 600 g ÷ 8
= 10.6 kg ÷ 8
= 1.325 kg = 1 kg 325 g

#### Page No 12:

Length of the rope divided into 8 equal pieces = 10 m
Length of one piece = 10 m ÷ 8
= 1.25 m = 1 m 25 cm     [∵ 1 m = 100 cm]

#### Page No 14:

(i) In 36, the ones digit is 6 > 5.
∴ The required rounded number = 40

(ii) In 173, the ones digit is 3 < 5.
∴ The required rounded number = 170

(iii) In 3869, the ones digit is 9 > 5.
∴ The required rounded number = 3870

(iv) In 16378, the ones digit is 8 > 5.
∴ The required rounded number = 16380

#### Page No 14:

(i) In 814, the tens digit is 1 < 5.
∴ The required rounded number = 800

(ii) In 1254, the tens digit is 5 = 5
∴ The required rounded number = 1300

(iii) In 43126, the tens digit is 2 < 5
∴ The required rounded number = 43100

(iv) In 98165, the tens digit is 6 > 5
∴ The required rounded number = 98200

#### Page No 14:

(i) In 793, the hundreds digit is 7 > 5
∴ The required rounded number = 1000

(ii) In 4826, the hundreds digit is 8 > 5
∴ The required rounded number = 5000

(iii) In 16719, the hundreds digit is 7 > 5
∴ The required rounded number = 17000

(iv) In 28394, the hundreds digit is 3 < 5
∴ The required rounded number = 28000

#### Page No 14:

(i) In 17514, the thousands digit is 7 > 5
∴ The required rounded number = 20000

(ii) In 26340, the thousands digit is 6 > 5
∴ The required rounded number = 30000

(iii) In 34890, the thousands digit is 4 < 5
∴ The required rounded number = 30000

(iv) In 272685, the thousands digit is 2 < 5
∴ The required rounded number = 270000

#### Page No 14:

57 estimated to the nearest ten = 60
34 estimated to the nearest ten = 30

∴ The required estimation = (60 + 30) = 90

#### Page No 14:

43 estimated to the nearest ten = 40
78 estimated to the nearest ten = 80
∴ The required estimation = (40 + 80) = 120

#### Page No 14:

14 estimated to the nearest ten = 10
69 estimated to the nearest ten = 70
∴ The required estimation = (10 + 70) = 80

#### Page No 14:

86 estimated to the nearest ten = 90
19 estimated to the nearest ten = 20
∴ The required estimation = (90 + 20) = 110

#### Page No 14:

95 estimated to the nearest ten = 100
58 estimated to the nearest ten = 60
∴ The required estimation = (100 + 60) = 160

#### Page No 14:

77 estimated to the nearest ten = 80
63 estimated to the nearest ten = 60
∴ The required estimation = (80 + 60) = 140

#### Page No 14:

356 estimated to the nearest ten = 360
275 estimated to the nearest ten = 280
∴ The required estimation = (360 + 280) = 640

#### Page No 14:

463 estimated to the nearest ten = 460
182 estimated to the nearest ten = 180
∴ The required estimation = (460 + 180) = 640

#### Page No 14:

538 estimated to the nearest ten = 540
276 estimated to the nearest ten = 280
∴ The required estimation = (540 + 280) = 820

#### Page No 14:

236 estimated to the nearest hundred = 200
689 estimated to the nearest hundred = 700
∴ The required estimation = (200 + 700) = 900

#### Page No 14:

458 estimated to the nearest hundred = 500
324 estimated to the nearest hundred = 300
∴ The required estimation = (500 + 300) = 800

#### Page No 14:

170 estimated to the nearest hundred = 200
395 estimated to the nearest hundred = 400
∴ The required estimation = (200 + 400) = 600

#### Page No 15:

3280 estimated to the nearest hundred = 3300
4395 estimated to the nearest hundred = 4400
∴ The required estimation = (3300 + 4400) = 7700

#### Page No 15:

5130 estimated to the nearest hundred = 5100
1410 estimated to the nearest hundred = 1400
∴ The required estimation = (5100 + 1400) = 6500

#### Page No 15:

10083 estimated to the nearest hundred = 10100
29380 estimated to the nearest hundred = 29400
∴ The required estimation = (10100 + 29400) = 39500

#### Page No 15:

32836 estimated to the nearest thousand = 33000
16466 estimated to the nearest thousand = 16000
∴ The required estimation = (33000 + 16000) = 49000

#### Page No 15:

46703 estimated to the nearest thousand = 47000
11375 estimated to the nearest thousand = 11000
∴ The required estimation = (47000 + 11000) = 58000

#### Page No 15:

Number of balls in box A = 54
Number of balls in box B = 79
Estimated number of balls in box A = 50
Estimated number of balls in box B = 80
∴ Total estimated number of balls in both the boxes = (50 + 80) = 130

#### Page No 15:

We have,
53 estimated to the nearest ten = 50
18 estimated to the nearest ten = 20
∴ The required estimation = (50 ‒ 20) = 30

#### Page No 15:

100 estimated to the nearest ten = 100
38 estimated to the nearest ten = 40
∴ The required estimation = (100 ‒ 40) = 60

#### Page No 15:

409 estimated to the nearest ten = 410
148 estimated to the nearest ten = 150
∴ The required estimation = (410 ‒ 150) = 260

#### Page No 15:

678 estimated to the nearest hundred = 700
215 estimated to the nearest hundred = 200
∴ The required estimation = (700 ‒ 200) = 500

#### Page No 15:

957 estimated to the nearest hundred = 1000
578 estimated to the nearest hundred = 600
∴ The required estimation = (1000 ‒ 600) = 400

#### Page No 15:

7258 estimated to the nearest hundred = 7300
2429 estimated to the nearest  hundred = 2400
∴ The required estimation = (7300 ‒ 2400) = 4900

#### Page No 15:

5612 estimated to the nearest hundred = 5600
3095 estimated to the nearest hundred = 3100
∴ The required estimation = (5600 ‒ 3100) = 2500

#### Page No 15:

35863 estimated to the nearest thousand = 36000
27677 estimated to the nearest  thousand = 28000
∴ The required estimation = (36000 ‒ 28000) = 8000

#### Page No 15:

47005 estimated to the nearest thousand = 47000
39488 estimated to the nearest  thousand = 39000
∴ The required estimation = (47000 ‒ 39000) = 8000

#### Page No 15:

38 estimated to the nearest ten = 40
63 estimated to the nearest ten = 60
∴ The required estimation = (40 $×$ 60) = 2400

#### Page No 15:

54 estimated to the nearest ten = 50
47 estimated to the nearest ten = 50
∴ The required estimation = (50 $×$ 50) = 2500

#### Page No 15:

28 estimated to the nearest ten = 30
63 estimated to the nearest ten = 60
∴ The required estimation = (30 $×$ 60) = 1800

#### Page No 15:

42 estimated to the nearest ten = 40
75 estimated to the nearest ten = 80
∴ The required estimation = (40 $×$ 80) = 3200

#### Page No 15:

64 estimated to the nearest ten = 60
58 estimated to the nearest ten = 60
∴ The required estimation = (60 $×$ 60) = 3600

#### Page No 15:

15 estimated to the nearest ten = 20
34 estimated to the nearest ten = 30
∴ The required estimation = (20 $×$ 30) = 600

#### Page No 16:

376 estimated to the nearest hundred = 400
123 estimated to the nearest hundred = 100
∴ The required estimation = (400 $×$ 100) = 40000

#### Page No 16:

264 estimated to the nearest hundred = 300
147 estimated to the nearest hundred = 100
∴ The required estimation = (300 $×$ 100) = 30000

#### Page No 16:

423 estimated to the nearest hundred = 400
158 estimated to the nearest hundred = 200
∴ The required estimation = (400 $×$ 200) = 80000

#### Page No 16:

509 estimated to the nearest hundred = 500
179 estimated to the nearest hundred = 200
∴ The required estimation = (500 $×$ 200) = 100000

#### Page No 16:

392 estimated to the nearest hundred = 400
138 estimated to the nearest hundred = 100
∴ The required estimation = (400 $×$ 100) = 40000

#### Page No 16:

271 estimated to the nearest hundred = 300
339 estimated to the nearest hundred = 300
∴ The required estimation = (300 $×$ 300) = 90000

#### Page No 16:

183 estimated upwards = 200
154 estimated downwards = 100
∴ The required product = (200 $×$ 100) = 20000

#### Page No 16:

267 estimated upwards = 300
146 estimated downwards = 100
∴ The required product = (300 $×$ 100) = 30000

#### Page No 16:

359 estimated upwards = 400
76 estimated downwards = 70
∴ The required product = (400 $×$ 70) =28000

#### Page No 16:

472 estimated upwards = 500
158 estimated downwards = 100
∴ The required product = (500 $×$ 100) = 50000

#### Page No 16:

680 estimated upwards = 700
164 estimated downwards = 100
∴ The required product = (700 $×$ 100) = 70000

#### Page No 16:

255 estimated upwards = 300
350 estimated downwards = 300
∴ The required product = (300 $×$ 300) = 90000

#### Page No 16:

356 estimated downwards = 300
278 estimated upwards = 300
∴ The required product = (300 $×$ 300) = 90000

#### Page No 16:

472 estimated downwards = 400
76 estimated upwards = 80
∴ The required product = (400 $×$ 80) = 32000

#### Page No 16:

578 estimated downwards = 500
369 estimated upwards = 400
∴  The required product = (500 $×$ 400) = 200000

#### Page No 16:

87 ÷ 28 is approximately equal to 90 ÷ 30 = 3.

#### Page No 16:

The estimated quotient for 83 ÷ 17 is approximately equal to 80 ÷ 20 = 8 ÷ 2 = 4.

#### Page No 16:

The estimated quotient of 75 ÷ 23 is approximately equal to 80 ÷ 20 = 8 ÷ 2 = 4.

#### Page No 16:

The estimated quotient of 193 ÷ 24 is approximately equal to 200 ÷ 20 = 20 ÷ 2 = 10.

#### Page No 16:

The estimated quotient of 725 ÷ 23 is approximately equal to 700 ÷ 20 = 70 ÷ 2 = 35.

#### Page No 16:

The estimated quotient of 275 ÷ 25 is approximately equal to 300 ÷ 30 = 30 ÷ 3 = 10.

#### Page No 16:

The estimated quotient of 633 ÷ 33 is approximately equal to 600 ÷ 30 = 60 ÷ 3 = 20.

#### Page No 16:

729 ÷ 29 is approximately equal to 700 ÷ 30 or 70 ÷ 3, which is approximately equal to 23.

#### Page No 16:

858 ÷ 39 is approximately equal to 900 ÷ 40 or 90 ÷ 4, which is approximately equal to 23.

#### Page No 16:

868 ÷ 38 is approximately equal to 900 ÷ 40 or 90 ÷ 4, which is approximately equal to 23.

#### Page No 19:

We may write these numbers as given below:
(i) 2 = II
(ii) 8 = (5 + 3) = VIII
(iii) 14 = (10 + 4) = XIV
(iv) 29 = ( 10 + 10 + 9 ) = XXIX
(v) 36 = (10 + 10 + 10 + 6) = XXXVI
(vi) 43 = (50 - 10) + 3 = XLIII
(vii) 54 = (50 + 4) = LIV
(viii) 61= (50 + 10 + 1) = LXI
(ix) 73 = ( 50 + 10 + 10 + 3) = LXXIII
(x) 81 = (50 + 10 + 10 + 10 + 1) = LXXXI
(xi) 91 =(100 - 10) + 1 = XCI
(xii) 95 = (100 - 10) + 5 = XCV
(xiii) 99 = (100 - 10) + 9 = XCIX
(xiv) 105 = (100 + 5) = CV
(xv) 114 = (100 + 10) + 4 = CXIV

#### Page No 19:

We may write these numbers in Roman numerals as follows:

(i) 164 = (100 + 50 + 10 + 4) = CLXIV
(ii) 195 = 100 + (100 - 10) + 5 = CXCV
(iii) 226 = (100 + 100 + 10 + 10 + 6) = CCXXVI
(iv) 341= 100 + 100+ 100 + (50 -10) + 1 = CCCXLI
(v) 475 = (500 - 100) + 50 + 10 + 10 + 5 = CDLXXV
(vi) 596 = 500 +  (100 - 10) + 6 = DXCVI
(vii) 611= 500 + 100 + 11 = DCXI
(viii) 759 = 500 + 100 + 100 + 50 + 9 = DCCLIX

#### Page No 19:

We can write the given Roman numerals in Hindu-Arabic numerals as follows:

(i) XXVII = 10 + 10 + 7 = 27
(ii) XXXIV = 10 + 10 + 10 + 4 = 34
(iii) XLV = (50 − 10 ) + 5 = 45
(iv) LIV = 50 + 4 = 54
(v) LXXIV = 50 + 10 + 10 + 4 = 74
(vi) XCI = (100 − 10) + 1 = 91
(vii) XCVI = (100 − 10) + 6 = 96
(viii) CXI = 100 + 10 + 1= 111
(ix) CLIV = 100 + 50 + 4 = 154
(x) CCXXIV = 100 + 100 + 10 + 10 + 4 = 224
(xi) CCCLXV = 100 +  100 + 100 + 50 + 10 + 5 = 365
(xii) CDXIV = (500 − 100) + 10 + 4 = 414
(xiii) CDLXIV = (500 − 100) + 50 + 10 + 4 = 464
(xiv) DVI = 500 + 6= 506
(xv) DCCLXVI = 500 + 100 + 100 + 50 + 10 + 6 = 766

#### Page No 19:

(i) VC is wrong because V, L and D are never subtracted.
(ii) IL is wrong because I can be subtracted from V and X only.
(iii) VVII is wrong because V, L and D are never repeated.
(iv) IXX is wrong because X (ten) must be placed before IX (nine).

#### Page No 20:

Option c is correct.

Place value of 6 = 6 lakhs = (6 $×$ 100000) = 600000

#### Page No 20:

Option a is correct.

The face value of a digit remains as it is irrespective of the place it occupies in the place value chart.
Thus, the face value of 4 is always 4 irrespective of where it may be.

#### Page No 20:

Option c is correct.

Place value of 5 = 5 $×$ 10000 = 50000
Face value of 5 = 5

∴ Required difference = 50000 − 5 = 49995

#### Page No 20:

Option b is correct.

The smallest counting number is 1.

#### Page No 20:

Option b is correct.

The largest four-digit number = 9999
The smallest four-digit number = 1000
Total number of all four-digit numbers = (9999 − 1000) + 1
= 8999 + 1
= 9000

#### Page No 20:

Option b is correct.

The largest seven-digit number = 9999999
The smallest seven-digit number = 1000000
Total number of seven-digit numbers = (9999999 − 1000000) + 1
= 8999999 + 1
= 9000000

#### Page No 20:

Option c is correct.

The largest eight-digit number = 99999999
The smallest eight-digit number = 10000000
Total number of eight-digit numbers = (99999999 − 10000000) + 1
= 89999999 + 1
= 90000000

#### Page No 20:

Option b is correct.

The number just before 1000000 is 999999 (i.e., 1000000 − 1).

#### Page No 20:

Option a is correct.

V, L and D are never subtracted. Thus, VX is wrong.

#### Page No 20:

Option c is correct.

I can be subtracted from V and X only. Thus, IC is wrong.

#### Page No 20:

Option b is correct.

V, L and D are never repeated. Thus, XVV is meaningless.

#### Page No 21:

(i) Sixteen crore six lakh twenty-three thousand seven hundred eight
(ii) Fourteen crore twenty-three lakh eight thousand nine hundred fifteen

#### Page No 21:

(i) Eighty million sixty thousand four hundred nine
(ii) Two hundred thirty-four million one hundred fifty thousand three hundred nineteen

#### Page No 21:

We have,
864572 is a 6-digit number.

3903216 and  6940513 are seven-digit numbers.
At the ten lakhs place, one number has 3, while the second number has 6.
Clearly, 3 < 6
∴ 3903216 <  6940513

16531079  and 19430124 are eight-digit numbers.
At the crores place, both the numbers have the same digit, namely 1.
At the ten lakhs place, one number has 6, while the second number has 9.
Clearly, 6 < 9
∴ 16531079  < 19430124

The given numbers in ascending order are:
864572 < 3903216 < 6940513 < 16531079 < 19430124

#### Page No 21:

63240613 and 54796203 are both eight-digit numbers.
At the crores place, one number has 6, while the second number has 5.
Clearly, 5 < 6
∴ 63240613 > 54796203

5125648 and 4675238 are both seven-digit numbers.
However, at the ten lakhs place, one number has 5, while the second number has 4.
Clearly, 4 < 5
∴ 5125648 > 4675238

589623 is a six-digit number.

The given numbers in descending order are:
63240613 > 54796203 > 5125648 > 4675238 > 589623

#### Page No 21:

The largest seven-digit number = 9999999
The smallest seven-digit number  = 1000000
Number of all seven-digits numbers = (9999999 − 1000000) + 1
= 899999 + 1
= 9000000

Hence, there is a total of ninety lakh 7-digit numbers.

#### Page No 21:

The largest number using each of the digits: 1, 4, 6, 8 and 0, is 86410.
The smallest  number using each of the digits: 1, 4, 6, 8 and 0, is 10468.
∴ Required difference = 86410 − 10468
= 75942

#### Page No 21:

(i) CCXLII = 100 + 100 + (50 − 10) + 2 = 242
(ii) CDLXV = (500 − 100) + 50 + 10 + 5 = 465
(iii) LXXVI = 50 + 10 + 10 + 6 = 76
(iv) DCCXLI = 500 + 100 + 100 + ( 50 − 10) + 1 = 741
(v) XCIV = (100 − 10) + 4 = 94
(vi) CXCIX = 100 + (100 − 10) + 9 = 199

#### Page No 21:

(i) 84 = 50 + 30 + 4 = LXXXIV
(ii) 99 = 90 + 9 =  XCIX
(iii) 145 = 100 + (50 − 10) + 5 = CXLV
(iv) 406 = 400 + 6 = CDVI
(v) 519 = 500 +10 + 9 = DXIX

#### Page No 21:

Successor of 999999 = 999999 + 1 = 1000000
Predecessor of 999999 = 999999 − 1 = 999998
∴ Required difference = 1000000 − 999998
= 2

#### Page No 21:

(i) The number is 1046. Its digit at the hundreds place is 0 < 5.
So, the given number is rounded off to the nearest thousand as 1000.

(ii) The number is 973. Its digit at the hundreds place is 9 > 5.
So, the given number is rounded off to the nearest thousand as 1000.

(iii) The number is 5624. Its digit at the hundreds place is 6 > 5.
So, the given number is rounded off to the nearest thousand as 6000.

(iv) The number is 4368. Its digit at the hundreds place is 3 < 5.
So, the given number is rounded off to the nearest thousand as 4000.

#### Page No 21:

Option (a) is correct.

X can be subtracted from L and C only.
i.e., XC = ( 100 − 10 ) = 90

#### Page No 21:

Option (b) is correct.

One lakh (100000) is equal to one hundred thousand (100,000).

#### Page No 21:

Option (b) is correct.

No Roman numeral can be repeated more than three times.

#### Page No 21:

Option (d) is correct.

Between 1 and 100, the digit 9 occurs in 9, 19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98 and 99.
∴ The digit occurs 20 times between 1 and 100.

#### Page No 21:

Option (a) is correct.

7268 will be rounded off to the nearest hundred as 7300.
2427 will be rounded of  to the nearest hundred as 2400.
∴ 7300 − 2400 = 4900

#### Page No 21:

Option (b) is correct.

1 million (1,000,000) = 10 lakh (10 $×$ 1,00,000)

#### Page No 21:

Option (b) is correct.

The number is 1512. Its digit at the tens place is 1 < 5.
So, the given number is rounded off to the nearest hundred as 1500.

#### Page No 21:

Option (c) is correct.

In Roman numerals, V, L and D are never repeated and never subtracted.

#### Page No 21:

Periods:     Crores      Lakhs           Thousands           Hundreds            Tens          Ones
Digits:            8             63                    24                        8                       0                5

Using commas, we write the given number as 8,63,24,805.

#### Page No 21:

(i) 1 crore =  100 lakh
(ii) 1 crore = 10 million
(iii) 564 when estimated to the nearest hundred is 600.
(iv) The smallest 4-digit number with four different digits is 1023.

#### Page No 22:

F

Place value of 5 in 85419 = 5000
Face value of 5 in 85419 = 5
∴ Their difference = 5000 − 5 = 4995

#### Page No 22:

T

In Roman numerals, V, L and D are never repeated and never subtracted.

#### Page No 22:

T
Greatest five-digit number = 99999
Successor of 99999 = 99999 + 1 = 100000

#### Page No 22:

T

The number is 46,530. Its digit at the tens place is 3 < 5.
So, the number 46,530 is rounded off to the nearest hundred as 46,500.