RS Aggrawal 2020 2021 Solutions for Class 6 Maths Chapter 1 - Number System

Question 1:

Question 2:

(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

Question 3:

(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

Question 4:

(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)

Question 5:

(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

Question 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 â€’ 900) = 8999100

Question 7:

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

Question 8:

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

Question 9:

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

Question 10:

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

Question 11:

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

Question 12:

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

Question 13:

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

Question 14:

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

Question 15:

The number just before 10000000 is 10000000 â€’ 1 = 9999999.

Question 16:

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

Question 17:

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

Question 18:

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

Question 19:

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

Question 1:

Question 2:

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.

Question 3:

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.

Question 4:

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

Question 5:

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

Question 6:

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.

Question 7:

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

Question 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

Question 9:

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

Question 10:

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

Question 11:

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

Question 12:

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

Question 13:

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

Question 14:

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

Question 1:

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

Question 2:

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

Question 3:

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

Question 4:

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

Question 5:

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

Question 6:

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

Question 7:

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

Question 8:

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

Question 9:

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

Question 10:

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

Question 11:

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

∴ 5643879 is smaller than one crore by 4356121.

Question 12:

11010101 ‒ required number = 2635967

Thus, required number = 11010101 ‒ 2635967
                                        = 8374134

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

Question 13:

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

∴ The other number = 10750308 ‒ 8967519
                                    = 1782789

Question 14:

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

Question 15:

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

Question 16:

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

Question 17:

Cost of one chair = Rs 1485
Cost of 469 chairs = Rs 1485 $\times$ 469
                              = Rs 696465

∴ Cost of 469 chairs is Rs 696465.

Question 18:

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.

Question 19:

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.

Question 20:

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 $\times$ 156 = Rs 1365936

∴ Mr Bhaskar will save Rs 1365936 in 13 years.

Question 21:

Cost of one scooter = Rs 36725
Cost of 487 scooter = Rs 36725 $\times$ 487
                                = Rs 17885075

∴ The cost of 487 scooters will be Rs 17885075.

Question 22:

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

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

Question 23:

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

∴ The other number = 13421408 ÷ 364                                   
                                   = 36872

Question 24:

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

∴ Each flat costs Rs 1895875.

Question 25:

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

Question 26:

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
                                                               

Question 27:

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 $\times$ 16
                                                                             = 2.75 m $\times$ 16
                                                                             = 44 m

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

Question 28:

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.

Question 29:

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 $\times$ 14
                                              = 2.75 kg $\times$ 14
                                                = 38.5 kg = 38 kg 500 g

Question 30:

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

Question 1:

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]

Question 2:

(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

Question 3:

(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

Question 4:

(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

Question 5:

(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  

Question 6:

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

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

Question 7:

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

Question 8:

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

Question 9:

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

Question 10:

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

Question 11:

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

Question 12:

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

Question 13:

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

Question 14:

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

Question 15:

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

Question 16:

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

Question 17:

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

Question 18:

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

Question 19:

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

Question 20:

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

Question 21:

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

Question 22:

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

Question 23:

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

Question 24:

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

Question 25:

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

Question 26:

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

Question 27:

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

Question 28:

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

Question 29:

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

Question 30:

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

Question 31:

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

Question 1:

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

Question 2:

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

Question 3:

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

Question 4:

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

Question 5:

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

Question 6:

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

Question 7:

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

Question 8:

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

Question 9:

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

Question 10:

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

Question 11:

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

Question 12:

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

Question 13:

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

Question 14:

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

Question 15:

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

Question 16:

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

Question 17:

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

Question 18:

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

Question 19:

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

Question 20:

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

Question 21:

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

Question 1:

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

Question 2:

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

Question 3:

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

Question 4:

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

Question 5:

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

Question 6:

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

Question 7:

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

Question 8:

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

Question 9:

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

Question 10:

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

Question 1:

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

Question 2:

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

Question 3:

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

Question 4:

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

Question 1:

(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).

Question 2:

Option c is correct.

Place value of 6 = 6 lakhs = (6 $\times$ 100000) = 600000

Question 3:

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.

Question 4:

Option c is correct.

Place value of 5 = 5 $\times$ 10000 = 50000
Face value of 5 = 5

∴ Required difference = 50000 − 5 = 49995

Question 5:

Option b is correct.

The smallest counting number is 1.

Question 6:

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
                                           

Question 7:

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
                                        

Question 8:

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
                                               

Question 9:

Option b is correct.

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

Question 10:

Option a is correct.

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

Question 11:

Option c is correct.

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

Question 1:

Option b is correct.

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

Question 2:

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

Question 3:

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

Question 4:

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

Question 5:

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

Question 6:

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.

Question 7:

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

Question 8:

(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

Question 9:

(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

Question 10:

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

Question 11:

(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.

Question 12:

Option (a) is correct.

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

Question 13:

Option (b) is correct.

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

Question 14:

Option (b) is correct.

No Roman numeral can be repeated more than three times.

Question 15:

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.

Question 16:

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

Question 17:

Option (b) is correct.

1 million (1,000,000) = 10 lakh (10 $\times$ 1,00,000)

Question 18:

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.

Question 19:

Option (c) is correct.

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

Question 20:

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.

Question 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.

Question 22:

F

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

Question 23:

T

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

Question 24:

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

Question 25:

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.