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CHAPTER
1
Knowing Our Numbers
POINTS FOR QUICK REVISION
l
l
l
l
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Natural Numbers: To count and calculate we
use numbers 1, 2, 3, 4, ....... which are called
counting numbers or natural numbers.
There is no greatest natural number i.e. these
are uncountable.
Number 0 together with natural numbers gives
us 0, 1, 2, 3, ....... are called whole numbers and
denoted by W = {0, 1, 2, .......}
Ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 which are
used to represent any number are called digits
or figures.
While forming numbers from given digits we
should take care of condition(s) given.
Place value of a digit in a number = Its face
value × position value.
1 km
=
1000 m
1m
=
100 cm
1 cm
=
1000 mm
1 kg
=
1000 gm
1 litre
=
1000 ml
l
l
l
l
l
l
l
l
1 million
10 million
100 million
=
=
=
10 lakh
1 crore
100 crore or 1
Arab.
Estimation means approximating a quantity
to an accuracy required.
In calculation use of brackets allow us to carry
out more than one number operation.
Roman numerals
IVXLCDM
Hindu-Arabic numerals 1 5 10 50 100 500
1000
There is no zero in Roman system. Roman
system does not use place value.
If a bar is placed over a numeral it is multiplied
by 1000 i.e. V = 5000 and L = 50000.
Palindromic Number is a number that is same
whether read backwards or forward for
example:– 575.
In India we use two major numeration
systems–Indian system and International
system. Use of commas helps in reading and
writing large numbers.
2
l
MATHEMATICS—VI
In the Indian system of numeration we have
commas after 3 digits starting from the right
and thereafter after every 2 digits. The commas
after 3, 5 and 7 digits separate thousand, lakh
and crore respectively.
l
In the International system of numeration
commas are placed after every 3 digits
starting from the right. The commas after
3 and 6 digits separate thousand and
million respectively.
NCERT TEXT BOOK QUESTIONS
TRY THESE
(With Solutions)
(Page 2)
Can you instantly find the greatest and the
smallest numbers in each row?
Q. 1. 382, 4972, 18, 59785, 750
Sol. 59785 is the greatest and 18 is the smallest.
Q. 2. 1473, 89423, 100, 5000, 310
Sol. 89423 is the greatest and 100 is the
smallest.
Q. 3. 1834, 75284, 111, 2333, 450
Sol. 75284 is the greatest and 111 is the
smallest.
Q. 4. 2853, 7691, 9999, 12002, 124
Sol. 12002 is the greatest and 124 is the
smallest.
Q. 5. Was it easy? Why was it easy?
Sol. Yes, it was easy. We just looked at the
number of digits and obtained the answer. The number
in which the number of digits was greatest, was the
greatest number. If in two numbers the number of
digits was the same, then the number in which on
moving to the left the leftmost digit was greater, was
the greater number. If the leftmost digits were also the
same, than we saw the next digit. The number in which
the next digit was greater, was the greater number. If
requirement be, we proceeded next to next digits.
TRY THESE
(With Solutions)
(Page 2)
Make five more problems of this kind and give
to your friends to solve.
Sol. (a) The greatest number is 4892.
The smallest number is 4370.
(b) The greatest number is 15800.
Q. 1. Find the greatest and the smallest numbers.
(a) 4536, 4892, 4370, 4452
(b) 15623, 15073, 15189, 15800
(c) 25286, 25245, 25270, 25210
(d) 6895, 23787, 24569, 24659
1
3
KNOWING OUR NUMBERS
3. Find the greatest and the smallest numbers.
42375, 42367, 42329, 42338
Greatest : 42375
Smallest : 42329
4. Find the greatest and the smallest numbers.
31896, 31897, 31899, 31890
Greatest : 31899
Smallest : 31890
5. Find the greatest and the smallest numbers.
25279, 20791, 3987, 3978
Greatest : 25279
Smallest : 3978
Smallest : 31545
TRY THESE
(With Solutions)
(Pages 3–4)
Q. 1. Use the given digits without repetition and
make the greatest and smallest four digit numbers ?
(a) 2, 8, 7, 4
(b) 9, 7, 4, 1
(c) 4, 7, 5, 0
(d) 1, 7, 6, 2
(e) 5, 4, 0, 3
(Hint. 0754 is a 3-digit number.)
Sol. (a) Greatest four digit number = 8742 ;
Smallest four digit number = 2478
(b) Greatest four digit number = 9741 ;
Smallest four digit number = 1479
(c) Greatest four digit number = 7540 ;
Smallest four digit number = 4057
(d) Greatest four digit number = 7621 ;
Smallest four digit number = 1267
(e) Greatest four digit number = 5430 ;
Smallest four digit number = 3045
Q. 2. Now make the greatest and the smallest
four digit numbers by using any one digit twice.
(a) 3, 8, 7
(b) 9, 0, 5
(c) 0, 4, 9
(d) 8, 5, 1
(Hint. Think in each case which digit will you
use twice.)
Sol. (a) Greatest four digit number = 8873 ;
Smallest four digit number = 3378
(b) Greatest four digit number = 9950 ;
Smallest four digit number = 5009
(c) Greatest four digit number = 9940 ;
Smallest four digit number = 4009
(d) Greatest four digit number = 8851 ;
Smallest four digit number = 1158
In the greatest number, the greatest digit is used
twice while in the smaller number, the smallest digit
is used twice.
Q. 3. Make the greatest and the smallest 4- digit
numbers using any four different digits, with conditions as given.
(a) Digit 7 is always at ones place
Greatest
9 8 6 7
Smallest
1 0 2 7
(Note, the number cannot begin with the digit
0. Why ?)
(b) Digit 4 is always at tens place
Greatest
4
Smallest
4
CHAPTER 1
The smallest number is 15073.
(c) The greatest number is 25286.
The smallest number is 25210.
(d) The greatest number is 24659.
The smallest number is 6895.
Five more problems of this kind are given below.
1. Find the greatest and the smallest numbers.
5936, 5079, 5788, 5672
Greatest : 5936
Smallest : 5079
2. Find the greatest and the smallest numbers.
31972, 31729, 31649, 31545
Greatest : 31972
4
MATHEMATICS—VI
(c) Digit 9 is always at hundreds place
Greatest
9
Smallest
9
(d) Digit 1 is always at thousands place
Greatest
1
Smallest
1
Sol. (b) Greatest
9
8
4
7
Smallest
1
0
4
2
(c) Greatest
8
9
7
6
Smallest
1
9
0
2
(d) Greatest
1
9
8
7
Smallest
1
0
2
3
The number cannot begin with the digit 0 as
then number will actually be a 3-digit number.
Q. 4. Take two digits, say 2 and 3. Make 4-digit
numbers, using both the digits equal number of times.
Which is the greatest number ?
Which is the smallest number ?
How many different numbers can you make in
all ?
Sol. Four digit numbers formed by using two
digits 2 and 3 equal number of times are :
2233, 2323, 2332, 3223, 3232, 3322
The greatest number is 3322
The smallest number is 2233
We can make 6 different numbers in all.
TRY THESE
(With Solutions)
(Page 4)
Q. Think of five more situations where you
compare three or more quantities.
Sol. Five more situations of this type are as
follows:
1. The weights of 3 students Manish, Krishna
and Yash of section A of class VI are respectively
25 kg, 20 kg and 18 kg.
2. The number of runs scored in an inning by
four players Ram, Rahim, Mohan and Shubham is
respectively 30, 26, 46 and 15.
3. The heights of five girls are respectively
130 cm, 150 cm, 145 cm, 167 cm and 158 cm.
4. The monthly incomes of four persons Mukut,
Girraj, Dandayal and Vishnu are respectively ` 10000,
` 8000, ` 9000 and ` 7500.
5. The total number of students in three schools
A, B and C are 1900, 2000 and 1640 respectively.
TRY THESE
(With Solutions)
(Page 5)
Q. 1. Arrange the following numbers in
ascending order.
(a) 847, 9754, 8320, 571
(b) 9801, 25751, 36501, 38802
Sol. (a) The given numbers in ascending order
are :
571, 847, 8320, 9754
(b) The given numbers in ascending order are :
9801, 25751, 36501, 38802
Q. 2. Arrange the following numbers in
descending order.
(a) 5000, 7500, 85400, 7861
(b) 1971, 45321, 88715, 92547
Sol. (a) The given numbers in descending order
are :
85400, 7861, 7500, 5000
(b) The given numbers in descending order are :
92547, 88715, 45321, 1971
Q. 3. Make ten such examples of ascending/
descending order and solve them.
Sol. Ten such examples of ascending/
descending order are as follows:
Arrange the following numbers in ascending
order:
Ex. 1. 958, 9865, 9431, 682
Ex. 2. 736, 8643, 7210, 460
Ex. 3. 625, 7532, 6101, 350
Ex. 4. 514, 6421, 5012, 240
Ex. 5. 403, 5310, 4001, 130
Sol. The numbers arranged in ascending order
are as follows:
1. 682, 958, 9431, 9865
2. 460, 736, 7210, 8643
3. 350, 625, 6101, 7532
4. 240, 514, 5012, 6421
5. 130, 403, 4001, 5310
Arrange the following numbers in descending
order:
Ex. 1. 6000, 8600, 86500, 8972
Ex. 2. 7000, 6400, 74300, 6750
Ex. 3. 2892, 56432, 99826, 93658
Ex. 4. 2760, 34210, 66604, 81436
Ex. 5. 1650, 23101, 55513, 70328
Sol. The numbers arranged in descending order
are as follows:
1. 86500, 8972, 8600, 6000
2. 74300, 7000, 6750, 6400
3. 99826, 93658, 56432, 2892
4. 81436, 66604, 34210, 2760
5. 70328, 55513, 23101, 1650
TRY THESE
(With Solutions)
(Page 6)
Q. 1. Read and expand the numbers wherever there are blanks.
Number Number Name
Expansion
20000
26000
38400
65740
2 × 10000
2 × 10000 + 6 × 1000
3 × 10000 + 8 × 1000 + 4 × 100
6 × 10000 + 5 × 1000 + 7 × 100 + 4 × 10
89324
50000
41000
47300
57630
29485
29085
20085
20005
Twenty thousand
Twenty six thousand
Thirty eight thousand four hundred
Sixty five thousand seven hundred
forty
Eighty nine thousand, three hundred
twenty four
_________________________
_________________________
_________________________
_________________________
_________________________
_________________________
_________________________
_________________________
8 × 10000 + 9 × 1000 + 3 × 100 + 2 × 10 + 4 × 1
_________________________
_________________________
_________________________
_________________________
_________________________
_________________________
_________________________
_________________________
CHAPTER 1
5
KNOWING OUR NUMBERS
6
MATHEMATICS—VI
Sol.
50000
41000
47300
57630
29485
Fifty thousand
Forty one thousand
Forty seven thousand three hundred
Fifty seven thousand six hundred thirty
Twenty nine thousand four hundred
eighty five
29085 Twenty nine thousand eighty five
20085 Twenty thousand eighty five
20005 Twenty thousand five
5 × 10000
4 × 10000 + 1 × 1000
4 × 10000 + 7 × 1000 + 3 × 100
5 × 10000 + 7 × 1000 + 6 × 100 + 3 × 10
2 × 10000 + 9 × 1000 + 4 × 100 + 8 × 10 + 5 × 1
2 × 10000 + 9 × 1000 + 8 × 10 + 5 × 1
2 × 10000 + 8 × 10 + 5 × 1
2 × 10000 + 5 × 1
Q. 2. Write five more 5-digit numbers, read them and expand them.
Sol.
Number
Number Name
Expansion
30000
Thirty thousand
3 × 10,000
52000
Fifty two thousand
5 × 10,000 + 2 × 1,000
36400
Thirty six thousand
four hundred
3 × 10,000 + 6 × 1,000
+ 4 × 100
68940
Sixty eight thousand
nine hundred forty
6 × 10,000 + 8 × 1,000
+ 9 × 100 + 4 × 10
24579
Twenty four thousand
five hundred seventy nine
2 × 10,000 + 4 × 1,000
+ 5 × 100 + 7 × 10 + 9 × 1
TRY THESE
(With Solutions)
(Page 7)
Q. Read and expand the numbers wherever there are blanks.
Number
3,00,000
3,50,000
3,53, 500
Number Name
Expansion
Three lakh
3 × 1,00,000
Three lakh fifty thousand
3 × 1, 00,000 + 5 × 10,000
Three lakh fifty three thousand
3 × 1,00,000 + 5 × 10,000 + 3 × 1000 + 5 × 100
five hundred
_________________________
_________________________
4,57,928
4,07,928
_________________________
_________________________
4,00,829
_________________________
_________________________
4,00,029
_________________________
_________________________
Sol. 4,57,928
Four lakh fifty seven thousand
4 × 1,00,000 + 5 × 10,000 + 7 × 1000
nine hundred twenty eight
+ 9 × 100 + 2 × 10 + 8 × 1
4,07,928 Four lakh seven thousand nine
4 × 1,00,000 + 7 × 1000 + 9 × 100
hundred twenty eight
+ 2 × 10 + 8 × 1
4,00,829 Four lakh eight hundred twenty nine
4 × 1,00,000 + 8 × 100 + 2 × 10 + 9
4,00,029
Four lakh twenty nine
4 × 1,00,000 + 2 × 10 + 9 × 1
7
KNOWING OUR NUMBERS
CHAPTER 1
TRY THESE
(With Solutions)
(Page 8)
Q. 1. What is 10 – 1 = ?
2. What is 100 – 1 = ?
3. What is 10,000 – 1 = ?
4. What is 1,00,000 – 1 = ?
5. What is 1,00,00,000 – 1 = ?
(Hint: Use the said pattern.)
Sol. 1.
2.
3.
4.
5.
10 – 1 =
100 – 1 =
10,000 – 1 =
1,00,000 – 1 =
1,00,00,000 – 1 =
9
99
9,999
99,999
99,99,999
TRY THESE
(With Solutions)
(Page 8)
Q. 1. Give five examples where the number of
9, 99, 998 ; 9, 99, 997 ; 9, 99, 996 ; 9, 99,
things counted would be more than 6-digit number. 995 ; 9, 99, 994.
Sol. 1. Number of people in U.P. state.
Q. 3. Starting from the smallest 8-digit number
2. Number of copies of Ram Charit Manas write the next five numbers in ascending order and
published each year.
read them.
3. Number of grains in a sack full of wheat.
Sol. The smallest 8-digit number is 1,00,00,000.
The
next
5 numbers in ascending order are :
4. Number of tickets sold in 2002–2003 in U.P.
1,00,00,001
One crore one
5. Number of saving account holders in
nationalised banks in U.P.
1,00,00,002
One crore two
Q. 2. Starting from the greatest 6-digit number,
1,00,00,003
One crore three
write the previous five numbers in descending order.
1,00,00,004
One crore four
Sol. The largest 6-digit number is 9,99,999.
1,00,00,005
One crore five
The previous five numbers in descending order are
TRY THESE
(With Solutions)
(Page 11)
Q. 1. Read these numbers. Write them using placement boxes and then write their expanded forms.
(i) 475320
(ii) 9847215
(iii) 97645310
(iv) 30458094
(a) Which is the smallest number?
(b) Which is the greatest number?
(c) Arrange these numbers in ascending and descending order.
Sol.
S.No. Number T Cr
(i) 475320
—
Cr
T La
La
T Th
Th
H
T
O
—
—
4
7
5
3
2
0
Number name Four lakh, seventy five thousand, three hundred twenty
Expansion
4 × 1,00,000 + 7 × 10,000 + 5 × 1000 + 3 × 100 + 2 × 10 + 0
(Contd.)
8
MATHEMATICS—VI
(ii) 9847215 —
—
9
8
4
7
2
1
5
Ninety eight lakh, forty seven thousand, two hundred fifteen
9 × 10,00,000 + 8 × 1,00,000 + 4 × 10,000 + 7 × 1000 + 2 × 100 + 1 × 10 + 5
(iii) 97645310 —
9
7
6
4
5
3
1
0
Nine crore, seventy six lakh, forty-five thousand, three hundred ten
9 × 1,00,00,000 + 7 × 10,00,000 + 6 × 1,00,000 + 4 × 10,000 + 5 × 1,000 + 3 × 100 + 1 × 10 + 0
(iv) 30458094—
3
0
4
5
8
0
9
4
Three crore, four lakh, fifty eight thousand ninety four
3 × 1,00,00,000 + 4 × 1,00,000 + 5 × 10,000 + 8 × 1,000 + 9 × 10 + 4
(a) The smallest number is 475320.
(b) The greatest number is 97645310.
(c) The numbers arranged in ascending order are as follows :
475320, 9847215, 30458094, 97645310
The numbers arranged in descending order are as follows :
97645310, 30458094, 9847215, 475320.
Q. 2. Read these numbers
(i) 527864
(ii) 95432
(iii) 18950049 (iv) 70002509.
(a) Write these numbers using placement boxes and then using commas in Indian as well as International
System of Numeration.
(b) Arrange these in ascending and descending order.
Sol. (a) (I) Using Placement Boxes
S.No. Number T Cr
Cr
T La
La
T Th
Th
H
T
O
Number name
(i)
527864
—
—
—
5
2
7
8
6
4
Five lakh twenty seven thousand
eight hundred sixty four
(ii)
95432
—
—
—
—
9
5
4
3
2
Ninety five thousand four
hundred thirty two
(iii)
18950049 —
1
8
9
5
0
0
4
9
One crore eighty nine lakh fifty
thousand forty nine
(iv)
70002509 —
7
0
0
0
2
5
0
9
Seven crore two thousand five
hundred nine
9
KNOWING OUR NUMBERS
Number
Number using commas in
Indian system of numeration
Number using commas
in International system of
numeration
(i) 527864
5,27,864
527,864
(ii) 95432
95,432
95,432
(iii) 18950049
1,89,50,049
18,950,049
(iv) 70002509
7,00,02,509
70,002,509
(b) Arranged in ascending order, these numbers are as follows :
95432, 527864, 18950049, 70002509
Arranged in descending order, these numbers are as follows :
70002509, 18950049, 527864, 95432.
Q. 3. Take three more groups of large numbers and do the exercise given above.
Sol. Three more groups of large numbers are as follows:
I. 637534, 86543, 27860038
II. 748645, 97654, 38970049
III. 526423, 75432, 16750027
(I) (a) (1) Using Placement Boxes
S. No.
Number
T Cr
Cr
T La
La
T Th
Th
H
T
O
(i)
637534
—
—
—
6
3
7
5
3
4
Number name Six lakh thirty seven thousand five hundred thirty four
(ii)
86543
—
—
—
—
8
6
5
4
3
6
0
0
3
8
Eighty six thousand five hundred forty three
(iii)
27860038
—
2
7
8
Two crore seventy eight lakh sixty thousand thirty eight
(2) Using Commas
Number
Number using commas in
Indian system of numeration
Number using commas in
International system of numeration
(i)
637534
6,37,534
637,534
(ii)
86543
86,543
86,543
(iii)
27860038
2,78,60,038
27,860,038
(b) Arranged in ascending order, these numbers are as follows:
86543, 637534, 27860038
Arranged in descending order, these numbers are as follows:
27860038, 637534, 86543
CHAPTER 1
(II) Using Commas
10
MATHEMATICS—VI
(II) (a) (1) Using Placement Boxes
S. No.
(i)
Number
T Cr
Cr
T La
La
T Th
Th
H
T
O
748645
—
—
—
7
4
8
6
4
5
Number name
(ii)
Seven lakh forty eight thousand six hundred forty five
97654
—
—
—
—
9
7
6
5
4
7
0
0
4
9
Ninety seven thousand six hundred fifty four
(iii)
38970049
—
3
8
9
Three crore, eighty nine lakh seventy thousand forty nine
(2) Using Commas
Number
Number using commas in Indian
system of numeration
Number using commas in
International system of numeration
(i)
748645
7,48,645
748,645
(ii)
97654
97,654
97,654
(iii)
38970049
3,89,70,049
38,970,049
(b) Arranged in ascending order, these numbers are as follows:
97654, 748645, 38970049
Arranged in descending order, these numbers are as follows:
38970049, 748645, 97654
(III) (a) (1) Using Placement Boxes
S. No.
(i)
Number
T Cr
Cr
T La
La
T Th
Th
H
T
O
526423
—
—
—
5
2
6
4
2
3
4
3
2
16750027
—
1
6
7
5
0
One crore sixty seven lakh fifty thousand twenty seven
0
2
Number name
(ii)
Five lakh twenty six thousand four hundred twenty three
75432
—
—
—
7
5
Seventy five thousand four hundred thirty two
(iii)
7
(2) Using Commas
Number
Number using commas in
Indian system of numeration
Number using commas in
International system of numeration
(i)
526423
5,26,423
526,423
(ii)
75432
75,432
75,432
(iii)
16750027
1,67,50,027
16,750,027