RS Aggarwal Solutions: Decimals | Mathematics (Maths) Class 6 PDF Download

 Page 1


Points to Remember :
1. Decimal Fractions. A fraction whose
denominator is 10 or 100 or 1000 etc. is
known as decimal fraction : e.g.
9
10
11
100
33
1000
, ,
 etc.
2. Decimals. The numbers expressed in
decimal form are called decimal numbers
on simply decimals. e.g. 0·3, 2·8, 21·78,
123·432 etc.
A decimal has two parts :
(i) whole number (ii) decimal part.
In the above examples,
0, 2, 21, 123 .... are whole numbers and
·3, ·8, ·78, ·432 ... are decimal parts.
The whole number and decimal parts are
separated by a point (·) which is called
decimal point.
3. Decimal places. The number of digits
contained in the decimal part of a decimal,
are the decimal places.
4. Like and unlike decimals :
(i) Like decimals. Decimals having the
same number of decimal places are called
like decimals.
(ii) Unlike decimals. Decimals having
different places of decimals are called
unlike decimals.
Note. (i) Putting any numebr of zeros to
the extreme right of a decimal part of a
decimal number, does not change its value.
e.g. 2·34, 2·340, 2·3400, 2·34000 ... are
all of the same value.
(ii) We put zeros to the extreme right of
a decimal part in order to make like
decimals.
5. Comparison of decimals :
Suppose we have to compare two
decimals. Then, we proceed according
to the following steps.
Step 1. Convert the given decimals into like
decimals.
Step 2. First compare the whole-number part.
The decimal with the greater whole-
number part is greater.
Step 3. If the whole-number parts are equal,
compare the tenths digits.
The decimal with the bigger digit in the
tenths place is greater.
Step 4. If the tenths digits are also equal, compare
the hundredths digit, and so on.
6. Converting of a decimal into a
fractions :
Method : Step 1. Write the given decimal without
the decimal point as the numerator of the
fraction.
Step 2. In the denominator, write 1 followed by
as many zeros as there are decimal places
in the given decimal.
Step 3. Convert the above fraction to the simplest
form.
7. Converting of a fraction into a
decimal.
General Method of Converting a
Fraction into a Decimal
Step 1. Divide the numerator by the denominator
till a nonzero remainder is obtained.
Step 2. Put a decimal point in the dividend as
well as in the quotient.
Step 3. Put a zero on the right of the decimal
point in the dividend as well as on the
right of the remainder.
Step 4. Divide again just as we do in whole
numbers.
Page 2


Points to Remember :
1. Decimal Fractions. A fraction whose
denominator is 10 or 100 or 1000 etc. is
known as decimal fraction : e.g.
9
10
11
100
33
1000
, ,
 etc.
2. Decimals. The numbers expressed in
decimal form are called decimal numbers
on simply decimals. e.g. 0·3, 2·8, 21·78,
123·432 etc.
A decimal has two parts :
(i) whole number (ii) decimal part.
In the above examples,
0, 2, 21, 123 .... are whole numbers and
·3, ·8, ·78, ·432 ... are decimal parts.
The whole number and decimal parts are
separated by a point (·) which is called
decimal point.
3. Decimal places. The number of digits
contained in the decimal part of a decimal,
are the decimal places.
4. Like and unlike decimals :
(i) Like decimals. Decimals having the
same number of decimal places are called
like decimals.
(ii) Unlike decimals. Decimals having
different places of decimals are called
unlike decimals.
Note. (i) Putting any numebr of zeros to
the extreme right of a decimal part of a
decimal number, does not change its value.
e.g. 2·34, 2·340, 2·3400, 2·34000 ... are
all of the same value.
(ii) We put zeros to the extreme right of
a decimal part in order to make like
decimals.
5. Comparison of decimals :
Suppose we have to compare two
decimals. Then, we proceed according
to the following steps.
Step 1. Convert the given decimals into like
decimals.
Step 2. First compare the whole-number part.
The decimal with the greater whole-
number part is greater.
Step 3. If the whole-number parts are equal,
compare the tenths digits.
The decimal with the bigger digit in the
tenths place is greater.
Step 4. If the tenths digits are also equal, compare
the hundredths digit, and so on.
6. Converting of a decimal into a
fractions :
Method : Step 1. Write the given decimal without
the decimal point as the numerator of the
fraction.
Step 2. In the denominator, write 1 followed by
as many zeros as there are decimal places
in the given decimal.
Step 3. Convert the above fraction to the simplest
form.
7. Converting of a fraction into a
decimal.
General Method of Converting a
Fraction into a Decimal
Step 1. Divide the numerator by the denominator
till a nonzero remainder is obtained.
Step 2. Put a decimal point in the dividend as
well as in the quotient.
Step 3. Put a zero on the right of the decimal
point in the dividend as well as on the
right of the remainder.
Step 4. Divide again just as we do in whole
numbers.
Step 5. Repeat step 4 till the remainder is zero.
8. Addition of Decimals
Method :
Step 1. Convert the given decimals into like
decimals.
Step 2. Write the addends one under the other in
column form, keeping the decimal points
of all the addends in the same column
and the digits of the same place in the
same column.
Step 3. Add as in the case of whole numbers.
Step 4. In the sum, put the decimal point directly
under decimal points in the addonds.
9. Subtraction of Decimals
Method : Step 1. Convert the given
decimals into like decimals.
Step 2. Write the smaller number under the larger
one in column form in such a way that
the decimal points of both the numbers
are in the same column and the digits of
the same place lie in the same column.
Step 3. Subtract as we do in case of whole
numbers.
Step 4. In the difference, put the decimal point
directly under the decimal points of the
given numbers.
10. 1000 g = 1 kg, 100 cm = 1 metre
100 paise = 1 rupee
          Q. 1. Write each of the following in figures :
(i) Fifty-eight point six three
(ii) One hundred twenty-four point four two
five
(iii) Seven point seven six
(iv) Nineteen point eight
(v) Four hundred four point zero four four
(vi) Point one seven three
(vii) Point zero one five
Sol. (i) Fifty-eight point six three = 58·63
(ii) One hundred twenty-four point four two
five = 124·425
(iii) Seven point seven six = 7·76
(iv) Nineteen point eight = 19·8
(v) Four hundred four point zero four four
= 404·044
(vi) Point one seven three = ·173
(vii) Point zero one five = ·015 Ans.
Q. 2. Write the place value of each digit in each
of the following decimals :
(i) 14·83 (ii) 275·269 (iii) 46·075
(iv) 302·459 (v) 5370·34 (vi) 186·209
Sol. (i) 14·83
Place value of 1 = 10,
Place value of 4 = 4,
Place value of 8 = 
8
10
,
Place value of 3 = 
3
100
(ii) 275·269
Place value of 2 = 200,
Place value of 7 = 70,
Place value of 5 = 5,
Place value of 2 = 
2
10
,
Place value of 6 = 
6
100
,
Place value of 
9
9
1000
(iii) 46·075
Place value of 4 = 40,
Place value of 6 = 6,
Place value of 0 = 0,
Place value of 7 = 
7
100
,
Place value of 5 = 
5
1000
(iv) 302·459
Place value of 3 = 300,
Place value of 0 = 0,
Place value of 2 = 2,
Place value of 4
4
10
Place value of 5
5
100
Place value of 9 
9
1000
Page 3


Points to Remember :
1. Decimal Fractions. A fraction whose
denominator is 10 or 100 or 1000 etc. is
known as decimal fraction : e.g.
9
10
11
100
33
1000
, ,
 etc.
2. Decimals. The numbers expressed in
decimal form are called decimal numbers
on simply decimals. e.g. 0·3, 2·8, 21·78,
123·432 etc.
A decimal has two parts :
(i) whole number (ii) decimal part.
In the above examples,
0, 2, 21, 123 .... are whole numbers and
·3, ·8, ·78, ·432 ... are decimal parts.
The whole number and decimal parts are
separated by a point (·) which is called
decimal point.
3. Decimal places. The number of digits
contained in the decimal part of a decimal,
are the decimal places.
4. Like and unlike decimals :
(i) Like decimals. Decimals having the
same number of decimal places are called
like decimals.
(ii) Unlike decimals. Decimals having
different places of decimals are called
unlike decimals.
Note. (i) Putting any numebr of zeros to
the extreme right of a decimal part of a
decimal number, does not change its value.
e.g. 2·34, 2·340, 2·3400, 2·34000 ... are
all of the same value.
(ii) We put zeros to the extreme right of
a decimal part in order to make like
decimals.
5. Comparison of decimals :
Suppose we have to compare two
decimals. Then, we proceed according
to the following steps.
Step 1. Convert the given decimals into like
decimals.
Step 2. First compare the whole-number part.
The decimal with the greater whole-
number part is greater.
Step 3. If the whole-number parts are equal,
compare the tenths digits.
The decimal with the bigger digit in the
tenths place is greater.
Step 4. If the tenths digits are also equal, compare
the hundredths digit, and so on.
6. Converting of a decimal into a
fractions :
Method : Step 1. Write the given decimal without
the decimal point as the numerator of the
fraction.
Step 2. In the denominator, write 1 followed by
as many zeros as there are decimal places
in the given decimal.
Step 3. Convert the above fraction to the simplest
form.
7. Converting of a fraction into a
decimal.
General Method of Converting a
Fraction into a Decimal
Step 1. Divide the numerator by the denominator
till a nonzero remainder is obtained.
Step 2. Put a decimal point in the dividend as
well as in the quotient.
Step 3. Put a zero on the right of the decimal
point in the dividend as well as on the
right of the remainder.
Step 4. Divide again just as we do in whole
numbers.
Step 5. Repeat step 4 till the remainder is zero.
8. Addition of Decimals
Method :
Step 1. Convert the given decimals into like
decimals.
Step 2. Write the addends one under the other in
column form, keeping the decimal points
of all the addends in the same column
and the digits of the same place in the
same column.
Step 3. Add as in the case of whole numbers.
Step 4. In the sum, put the decimal point directly
under decimal points in the addonds.
9. Subtraction of Decimals
Method : Step 1. Convert the given
decimals into like decimals.
Step 2. Write the smaller number under the larger
one in column form in such a way that
the decimal points of both the numbers
are in the same column and the digits of
the same place lie in the same column.
Step 3. Subtract as we do in case of whole
numbers.
Step 4. In the difference, put the decimal point
directly under the decimal points of the
given numbers.
10. 1000 g = 1 kg, 100 cm = 1 metre
100 paise = 1 rupee
          Q. 1. Write each of the following in figures :
(i) Fifty-eight point six three
(ii) One hundred twenty-four point four two
five
(iii) Seven point seven six
(iv) Nineteen point eight
(v) Four hundred four point zero four four
(vi) Point one seven three
(vii) Point zero one five
Sol. (i) Fifty-eight point six three = 58·63
(ii) One hundred twenty-four point four two
five = 124·425
(iii) Seven point seven six = 7·76
(iv) Nineteen point eight = 19·8
(v) Four hundred four point zero four four
= 404·044
(vi) Point one seven three = ·173
(vii) Point zero one five = ·015 Ans.
Q. 2. Write the place value of each digit in each
of the following decimals :
(i) 14·83 (ii) 275·269 (iii) 46·075
(iv) 302·459 (v) 5370·34 (vi) 186·209
Sol. (i) 14·83
Place value of 1 = 10,
Place value of 4 = 4,
Place value of 8 = 
8
10
,
Place value of 3 = 
3
100
(ii) 275·269
Place value of 2 = 200,
Place value of 7 = 70,
Place value of 5 = 5,
Place value of 2 = 
2
10
,
Place value of 6 = 
6
100
,
Place value of 
9
9
1000
(iii) 46·075
Place value of 4 = 40,
Place value of 6 = 6,
Place value of 0 = 0,
Place value of 7 = 
7
100
,
Place value of 5 = 
5
1000
(iv) 302·459
Place value of 3 = 300,
Place value of 0 = 0,
Place value of 2 = 2,
Place value of 4
4
10
Place value of 5
5
100
Place value of 9 
9
1000
(v) 5370·34
Place value of 5 = 5000,
Place value of 3 = 300,
Place value of 7 = 70,
Place value of 0 = 0,
Place value of 
3
3
10
,
Place value of 
4
4
100
(vi) 186·209
Place value of 1 = 100,
Place value of 8 = 80,
Place value of 6 = 6,
Place value of 2 = 
2
10
,
Place value of 0 = 0,
Place value of 9 = 
9
1000
Q. 3. Write each of the following decimals in
the expanded form :
(i) 67·83 (ii) 283·61
(iii) 24·675 (iv) 0·294
(v) 8·006 (vi) 4615·72
Sol. (i) 67·83 = (6 × 10) + (7 × 1)
+ 
8
1
10
3
1
100
F
H
G
I
K
J
F
H
G
I
K
J
(ii) 283·61 = (2 × 100) + (8 × 10) + (3 × 1)
+ 
6
1
10
1
1
100
F
H
G
I
K
J
F
H
G
I
K
J
(iii) 24·675 = (2 × 10) + (4 × 1)
+ 
6
1
10
7
1
100
5
1
1000
F
H
G
I
K
J
F
H
G
I
K
J
F
H
G
I
K
J
(iv) 0·294 =
2
1
10
9
1
100
4
1
1000
F
H
G
I
K
J
F
H
G
I
K
J
F
H
G
I
K
J
(v) 8·006 = (8 × 1) + 
6
1
1000
F
H
G
I
K
J
(vi) 4615·72 = (4 × 1000) + (6 × 100)
+ (1 × 10) + (5 × 1)
+ 7
1
10
2
1
100
F
H
G
I
K
J
F
H
G
I
K
J Ans.
Q. 4. Write each of the following in the decimal
form :
(i)
40 6
7
10
9
100
(ii)
500 70 8
3
10
1
100
6
1000
(iii)
700 30 1
8
10
4
100
(iv)
600 5
7
100
9
1000
(v)
800 5
8
10
6
1000
(vi)
30 9
4
100
8
1000
Sol. (i) 
40 6
7
10
9
100
 = 46·79
(ii)
500 70 8
3
10
1
100
6
1000
= 578·316
(iii)
700 30 1
8
10
4
100
 = 731·84
(iv)
600 5
7
100
9
1000
 = 605·079
(v)
800 5
8
10
6
1000
 = 805·806
(vi)
30 9
4
100
8
1000
 = 39·048 Ans.
Q. 5. Convert each of the following into like
decimals :
(i) 7·5, 64·23, 0·074
(ii) 0·6, 5·937, 2·36, 4·2
(iii) 1·6, 0·07, 3·58, 2·9
(iv) 2·5, 0·63, 14·08, 1·637
Sol. (i) 7·5, 64·23, 0·074 = 7·500, 64·230,
0·074
(Here, at the most 0·074 has 3 places)
(ii) 0·6, 5·937, 2·36, 4·2 = 0·600, 5·937,
2·360, 4·200
(Here, 5·937 has at most 3 places)
(iii) 1·6, 0·07, 3·58, 2·9 = 1·60, 0·07, 3·58,
2·90
Page 4


Points to Remember :
1. Decimal Fractions. A fraction whose
denominator is 10 or 100 or 1000 etc. is
known as decimal fraction : e.g.
9
10
11
100
33
1000
, ,
 etc.
2. Decimals. The numbers expressed in
decimal form are called decimal numbers
on simply decimals. e.g. 0·3, 2·8, 21·78,
123·432 etc.
A decimal has two parts :
(i) whole number (ii) decimal part.
In the above examples,
0, 2, 21, 123 .... are whole numbers and
·3, ·8, ·78, ·432 ... are decimal parts.
The whole number and decimal parts are
separated by a point (·) which is called
decimal point.
3. Decimal places. The number of digits
contained in the decimal part of a decimal,
are the decimal places.
4. Like and unlike decimals :
(i) Like decimals. Decimals having the
same number of decimal places are called
like decimals.
(ii) Unlike decimals. Decimals having
different places of decimals are called
unlike decimals.
Note. (i) Putting any numebr of zeros to
the extreme right of a decimal part of a
decimal number, does not change its value.
e.g. 2·34, 2·340, 2·3400, 2·34000 ... are
all of the same value.
(ii) We put zeros to the extreme right of
a decimal part in order to make like
decimals.
5. Comparison of decimals :
Suppose we have to compare two
decimals. Then, we proceed according
to the following steps.
Step 1. Convert the given decimals into like
decimals.
Step 2. First compare the whole-number part.
The decimal with the greater whole-
number part is greater.
Step 3. If the whole-number parts are equal,
compare the tenths digits.
The decimal with the bigger digit in the
tenths place is greater.
Step 4. If the tenths digits are also equal, compare
the hundredths digit, and so on.
6. Converting of a decimal into a
fractions :
Method : Step 1. Write the given decimal without
the decimal point as the numerator of the
fraction.
Step 2. In the denominator, write 1 followed by
as many zeros as there are decimal places
in the given decimal.
Step 3. Convert the above fraction to the simplest
form.
7. Converting of a fraction into a
decimal.
General Method of Converting a
Fraction into a Decimal
Step 1. Divide the numerator by the denominator
till a nonzero remainder is obtained.
Step 2. Put a decimal point in the dividend as
well as in the quotient.
Step 3. Put a zero on the right of the decimal
point in the dividend as well as on the
right of the remainder.
Step 4. Divide again just as we do in whole
numbers.
Step 5. Repeat step 4 till the remainder is zero.
8. Addition of Decimals
Method :
Step 1. Convert the given decimals into like
decimals.
Step 2. Write the addends one under the other in
column form, keeping the decimal points
of all the addends in the same column
and the digits of the same place in the
same column.
Step 3. Add as in the case of whole numbers.
Step 4. In the sum, put the decimal point directly
under decimal points in the addonds.
9. Subtraction of Decimals
Method : Step 1. Convert the given
decimals into like decimals.
Step 2. Write the smaller number under the larger
one in column form in such a way that
the decimal points of both the numbers
are in the same column and the digits of
the same place lie in the same column.
Step 3. Subtract as we do in case of whole
numbers.
Step 4. In the difference, put the decimal point
directly under the decimal points of the
given numbers.
10. 1000 g = 1 kg, 100 cm = 1 metre
100 paise = 1 rupee
          Q. 1. Write each of the following in figures :
(i) Fifty-eight point six three
(ii) One hundred twenty-four point four two
five
(iii) Seven point seven six
(iv) Nineteen point eight
(v) Four hundred four point zero four four
(vi) Point one seven three
(vii) Point zero one five
Sol. (i) Fifty-eight point six three = 58·63
(ii) One hundred twenty-four point four two
five = 124·425
(iii) Seven point seven six = 7·76
(iv) Nineteen point eight = 19·8
(v) Four hundred four point zero four four
= 404·044
(vi) Point one seven three = ·173
(vii) Point zero one five = ·015 Ans.
Q. 2. Write the place value of each digit in each
of the following decimals :
(i) 14·83 (ii) 275·269 (iii) 46·075
(iv) 302·459 (v) 5370·34 (vi) 186·209
Sol. (i) 14·83
Place value of 1 = 10,
Place value of 4 = 4,
Place value of 8 = 
8
10
,
Place value of 3 = 
3
100
(ii) 275·269
Place value of 2 = 200,
Place value of 7 = 70,
Place value of 5 = 5,
Place value of 2 = 
2
10
,
Place value of 6 = 
6
100
,
Place value of 
9
9
1000
(iii) 46·075
Place value of 4 = 40,
Place value of 6 = 6,
Place value of 0 = 0,
Place value of 7 = 
7
100
,
Place value of 5 = 
5
1000
(iv) 302·459
Place value of 3 = 300,
Place value of 0 = 0,
Place value of 2 = 2,
Place value of 4
4
10
Place value of 5
5
100
Place value of 9 
9
1000
(v) 5370·34
Place value of 5 = 5000,
Place value of 3 = 300,
Place value of 7 = 70,
Place value of 0 = 0,
Place value of 
3
3
10
,
Place value of 
4
4
100
(vi) 186·209
Place value of 1 = 100,
Place value of 8 = 80,
Place value of 6 = 6,
Place value of 2 = 
2
10
,
Place value of 0 = 0,
Place value of 9 = 
9
1000
Q. 3. Write each of the following decimals in
the expanded form :
(i) 67·83 (ii) 283·61
(iii) 24·675 (iv) 0·294
(v) 8·006 (vi) 4615·72
Sol. (i) 67·83 = (6 × 10) + (7 × 1)
+ 
8
1
10
3
1
100
F
H
G
I
K
J
F
H
G
I
K
J
(ii) 283·61 = (2 × 100) + (8 × 10) + (3 × 1)
+ 
6
1
10
1
1
100
F
H
G
I
K
J
F
H
G
I
K
J
(iii) 24·675 = (2 × 10) + (4 × 1)
+ 
6
1
10
7
1
100
5
1
1000
F
H
G
I
K
J
F
H
G
I
K
J
F
H
G
I
K
J
(iv) 0·294 =
2
1
10
9
1
100
4
1
1000
F
H
G
I
K
J
F
H
G
I
K
J
F
H
G
I
K
J
(v) 8·006 = (8 × 1) + 
6
1
1000
F
H
G
I
K
J
(vi) 4615·72 = (4 × 1000) + (6 × 100)
+ (1 × 10) + (5 × 1)
+ 7
1
10
2
1
100
F
H
G
I
K
J
F
H
G
I
K
J Ans.
Q. 4. Write each of the following in the decimal
form :
(i)
40 6
7
10
9
100
(ii)
500 70 8
3
10
1
100
6
1000
(iii)
700 30 1
8
10
4
100
(iv)
600 5
7
100
9
1000
(v)
800 5
8
10
6
1000
(vi)
30 9
4
100
8
1000
Sol. (i) 
40 6
7
10
9
100
 = 46·79
(ii)
500 70 8
3
10
1
100
6
1000
= 578·316
(iii)
700 30 1
8
10
4
100
 = 731·84
(iv)
600 5
7
100
9
1000
 = 605·079
(v)
800 5
8
10
6
1000
 = 805·806
(vi)
30 9
4
100
8
1000
 = 39·048 Ans.
Q. 5. Convert each of the following into like
decimals :
(i) 7·5, 64·23, 0·074
(ii) 0·6, 5·937, 2·36, 4·2
(iii) 1·6, 0·07, 3·58, 2·9
(iv) 2·5, 0·63, 14·08, 1·637
Sol. (i) 7·5, 64·23, 0·074 = 7·500, 64·230,
0·074
(Here, at the most 0·074 has 3 places)
(ii) 0·6, 5·937, 2·36, 4·2 = 0·600, 5·937,
2·360, 4·200
(Here, 5·937 has at most 3 places)
(iii) 1·6, 0·07, 3·58, 2·9 = 1·60, 0·07, 3·58,
2·90
(Here, at the most are two places)
(iv) 2·5, 0·63, 14·08, 1·637 = 2·500, 0·630,
14·080, 1·637 Ans.
(Here, at the most are three places)
Q. 6. Fill in each of the place holders with the
correct symbol > or < :
(i) 84·23  76·35
(ii) 7·608  7·68
(iii) 8·34  8·43
(iv) 12·06  12·006
(v) 3·85  3·805
(vi) 0·97  1·07
Sol. Making like decimals where ever it is
necessary,
(i) 84·23  76·35  84·23 > 76·35
(ii) 7·608  7·68   7·608  7·680
 7·608 < 7·680
(iii) 8·34  8·43  8·34 < 8·43
(iv) 12·06  12·006
 12·060  12·006
 12·06 > 12·006
(v) 3·85  3·805  3·850  3·805
 3·850 > 3·805
(vi) 0·97  1·07  0·97 < 1·07 Ans.
Q. 7. Arrange the following decimals in an
ascending order :
(i) 5·8, 7·2, 5·69, 7·14, 5·06
(ii) 0·6, 6·6, 6·06, 66·6, 0·06
(iii) 6·54, 6·45, 6·4, 6·5, 6·05
(iv) 3·3, 3·303, 3·033, 0·33, 3·003
Sol. First of all making them in like decimals,
(i) 5·8, 7·2, 5·69, 7·14, 5·06
 5·80, 7·20, 5·69, 7·14, 5·06
Arranging in ascending order,
5·06 < 5·69 < 5·80 < 7·14 < 7·20
 5·06 < 5·69 < 5·8 < 7·14 < 7·2 Ans.
(ii) 0·6, 6·6, 6·06, 66·6, 0·06
 0·60, 6·60, 6·06, 66·60, 0·06
Arranging in ascending order,
0·06 < 0·60 < 6·06 < 6·60 < 66·60
 0·06 < 0·6 < 6·06 < 6·6 < 66·6 Ans.
(iii) 6·54, 6·45, 6·4, 6·5, 6·05
 6·54, 6·45, 6·4, 6·5, 6·05
Arranging in ascending order,
6·05 < 6·40 < 6·45 < 6·50 < 6·54
 6·05 < 6·4 < 6·45 < 6·5 < 6·54 Ans.
(iv) 3·3, 3·303, 3·033, 0·33, 3·003
 3·300, 3·303, 3·033, 0·330, 3·003
Arranging in descending order,
0·330 < 3·003 < 3·033 < 3·300 < 3·303
 0·33 < 3·003 < 3·033 < 3·3 < 3·303
Ans.
Q. 8. Arrange the following decimals in a
descending order :
(i) 7·3, 8·73, 73·03, 7·33, 8·073
(ii) 3·3, 3·03, 30·3, 30·03, 3·003
(iii) 2·7, 7·2, 2·27, 2·72, 2·02, 2·007
(iv) 8·88, 8·088, 88·8, 88·08, 8·008
Sol. Making them in like decimals and them
comparing
(i) 7·3, 8·73, 73·03, 7·33, 8·073
 7·300, 8·730, 73·030, 7·330, 8·073
Arranging in decending order
73·030 > 8·730 > 8·073 > 7·330 > 7·300
 73·03 > 8·73 > 8·073 > 7·33 > 7·3
Ans.
(ii) 3·3, 3·03, 30·3, 30·03, 3·003
 3·300, 3·030, 30·300, 30·030, 3·003
Arranging in descending order
30·300 > 30·030 > 3·300 > 3·030 > 3·003
 30·3 > 30·03 > 3·3 > 3·03 > 3·003 Ans.
(iii) 2·7, 7·2, 2·27, 2·72, 2·02, 2·007
 2·700, 7·200, 2·270, 2·720, 2·020,
2·007
Arranging in descending order
7·200 > 2·720 > 2·700 > 2·270 > 2·020>
2·007
 7·2 > 2·72 > 2·7 > 2·27 > 2·02 > 2·007
Ans.
(iv) 8·88, 8·088, 88·8, 88·08, 8·008
 8·880, 8·088, 88·800, 88·080, 8·008
Arranging in decending order,
88·800 > 88·080 > 8·880 > 8·088 > 8·008
 88·8 > 88·08 > 8·88 > 8·088 > 8·008
Ans.
Page 5


Points to Remember :
1. Decimal Fractions. A fraction whose
denominator is 10 or 100 or 1000 etc. is
known as decimal fraction : e.g.
9
10
11
100
33
1000
, ,
 etc.
2. Decimals. The numbers expressed in
decimal form are called decimal numbers
on simply decimals. e.g. 0·3, 2·8, 21·78,
123·432 etc.
A decimal has two parts :
(i) whole number (ii) decimal part.
In the above examples,
0, 2, 21, 123 .... are whole numbers and
·3, ·8, ·78, ·432 ... are decimal parts.
The whole number and decimal parts are
separated by a point (·) which is called
decimal point.
3. Decimal places. The number of digits
contained in the decimal part of a decimal,
are the decimal places.
4. Like and unlike decimals :
(i) Like decimals. Decimals having the
same number of decimal places are called
like decimals.
(ii) Unlike decimals. Decimals having
different places of decimals are called
unlike decimals.
Note. (i) Putting any numebr of zeros to
the extreme right of a decimal part of a
decimal number, does not change its value.
e.g. 2·34, 2·340, 2·3400, 2·34000 ... are
all of the same value.
(ii) We put zeros to the extreme right of
a decimal part in order to make like
decimals.
5. Comparison of decimals :
Suppose we have to compare two
decimals. Then, we proceed according
to the following steps.
Step 1. Convert the given decimals into like
decimals.
Step 2. First compare the whole-number part.
The decimal with the greater whole-
number part is greater.
Step 3. If the whole-number parts are equal,
compare the tenths digits.
The decimal with the bigger digit in the
tenths place is greater.
Step 4. If the tenths digits are also equal, compare
the hundredths digit, and so on.
6. Converting of a decimal into a
fractions :
Method : Step 1. Write the given decimal without
the decimal point as the numerator of the
fraction.
Step 2. In the denominator, write 1 followed by
as many zeros as there are decimal places
in the given decimal.
Step 3. Convert the above fraction to the simplest
form.
7. Converting of a fraction into a
decimal.
General Method of Converting a
Fraction into a Decimal
Step 1. Divide the numerator by the denominator
till a nonzero remainder is obtained.
Step 2. Put a decimal point in the dividend as
well as in the quotient.
Step 3. Put a zero on the right of the decimal
point in the dividend as well as on the
right of the remainder.
Step 4. Divide again just as we do in whole
numbers.
Step 5. Repeat step 4 till the remainder is zero.
8. Addition of Decimals
Method :
Step 1. Convert the given decimals into like
decimals.
Step 2. Write the addends one under the other in
column form, keeping the decimal points
of all the addends in the same column
and the digits of the same place in the
same column.
Step 3. Add as in the case of whole numbers.
Step 4. In the sum, put the decimal point directly
under decimal points in the addonds.
9. Subtraction of Decimals
Method : Step 1. Convert the given
decimals into like decimals.
Step 2. Write the smaller number under the larger
one in column form in such a way that
the decimal points of both the numbers
are in the same column and the digits of
the same place lie in the same column.
Step 3. Subtract as we do in case of whole
numbers.
Step 4. In the difference, put the decimal point
directly under the decimal points of the
given numbers.
10. 1000 g = 1 kg, 100 cm = 1 metre
100 paise = 1 rupee
          Q. 1. Write each of the following in figures :
(i) Fifty-eight point six three
(ii) One hundred twenty-four point four two
five
(iii) Seven point seven six
(iv) Nineteen point eight
(v) Four hundred four point zero four four
(vi) Point one seven three
(vii) Point zero one five
Sol. (i) Fifty-eight point six three = 58·63
(ii) One hundred twenty-four point four two
five = 124·425
(iii) Seven point seven six = 7·76
(iv) Nineteen point eight = 19·8
(v) Four hundred four point zero four four
= 404·044
(vi) Point one seven three = ·173
(vii) Point zero one five = ·015 Ans.
Q. 2. Write the place value of each digit in each
of the following decimals :
(i) 14·83 (ii) 275·269 (iii) 46·075
(iv) 302·459 (v) 5370·34 (vi) 186·209
Sol. (i) 14·83
Place value of 1 = 10,
Place value of 4 = 4,
Place value of 8 = 
8
10
,
Place value of 3 = 
3
100
(ii) 275·269
Place value of 2 = 200,
Place value of 7 = 70,
Place value of 5 = 5,
Place value of 2 = 
2
10
,
Place value of 6 = 
6
100
,
Place value of 
9
9
1000
(iii) 46·075
Place value of 4 = 40,
Place value of 6 = 6,
Place value of 0 = 0,
Place value of 7 = 
7
100
,
Place value of 5 = 
5
1000
(iv) 302·459
Place value of 3 = 300,
Place value of 0 = 0,
Place value of 2 = 2,
Place value of 4
4
10
Place value of 5
5
100
Place value of 9 
9
1000
(v) 5370·34
Place value of 5 = 5000,
Place value of 3 = 300,
Place value of 7 = 70,
Place value of 0 = 0,
Place value of 
3
3
10
,
Place value of 
4
4
100
(vi) 186·209
Place value of 1 = 100,
Place value of 8 = 80,
Place value of 6 = 6,
Place value of 2 = 
2
10
,
Place value of 0 = 0,
Place value of 9 = 
9
1000
Q. 3. Write each of the following decimals in
the expanded form :
(i) 67·83 (ii) 283·61
(iii) 24·675 (iv) 0·294
(v) 8·006 (vi) 4615·72
Sol. (i) 67·83 = (6 × 10) + (7 × 1)
+ 
8
1
10
3
1
100
F
H
G
I
K
J
F
H
G
I
K
J
(ii) 283·61 = (2 × 100) + (8 × 10) + (3 × 1)
+ 
6
1
10
1
1
100
F
H
G
I
K
J
F
H
G
I
K
J
(iii) 24·675 = (2 × 10) + (4 × 1)
+ 
6
1
10
7
1
100
5
1
1000
F
H
G
I
K
J
F
H
G
I
K
J
F
H
G
I
K
J
(iv) 0·294 =
2
1
10
9
1
100
4
1
1000
F
H
G
I
K
J
F
H
G
I
K
J
F
H
G
I
K
J
(v) 8·006 = (8 × 1) + 
6
1
1000
F
H
G
I
K
J
(vi) 4615·72 = (4 × 1000) + (6 × 100)
+ (1 × 10) + (5 × 1)
+ 7
1
10
2
1
100
F
H
G
I
K
J
F
H
G
I
K
J Ans.
Q. 4. Write each of the following in the decimal
form :
(i)
40 6
7
10
9
100
(ii)
500 70 8
3
10
1
100
6
1000
(iii)
700 30 1
8
10
4
100
(iv)
600 5
7
100
9
1000
(v)
800 5
8
10
6
1000
(vi)
30 9
4
100
8
1000
Sol. (i) 
40 6
7
10
9
100
 = 46·79
(ii)
500 70 8
3
10
1
100
6
1000
= 578·316
(iii)
700 30 1
8
10
4
100
 = 731·84
(iv)
600 5
7
100
9
1000
 = 605·079
(v)
800 5
8
10
6
1000
 = 805·806
(vi)
30 9
4
100
8
1000
 = 39·048 Ans.
Q. 5. Convert each of the following into like
decimals :
(i) 7·5, 64·23, 0·074
(ii) 0·6, 5·937, 2·36, 4·2
(iii) 1·6, 0·07, 3·58, 2·9
(iv) 2·5, 0·63, 14·08, 1·637
Sol. (i) 7·5, 64·23, 0·074 = 7·500, 64·230,
0·074
(Here, at the most 0·074 has 3 places)
(ii) 0·6, 5·937, 2·36, 4·2 = 0·600, 5·937,
2·360, 4·200
(Here, 5·937 has at most 3 places)
(iii) 1·6, 0·07, 3·58, 2·9 = 1·60, 0·07, 3·58,
2·90
(Here, at the most are two places)
(iv) 2·5, 0·63, 14·08, 1·637 = 2·500, 0·630,
14·080, 1·637 Ans.
(Here, at the most are three places)
Q. 6. Fill in each of the place holders with the
correct symbol > or < :
(i) 84·23  76·35
(ii) 7·608  7·68
(iii) 8·34  8·43
(iv) 12·06  12·006
(v) 3·85  3·805
(vi) 0·97  1·07
Sol. Making like decimals where ever it is
necessary,
(i) 84·23  76·35  84·23 > 76·35
(ii) 7·608  7·68   7·608  7·680
 7·608 < 7·680
(iii) 8·34  8·43  8·34 < 8·43
(iv) 12·06  12·006
 12·060  12·006
 12·06 > 12·006
(v) 3·85  3·805  3·850  3·805
 3·850 > 3·805
(vi) 0·97  1·07  0·97 < 1·07 Ans.
Q. 7. Arrange the following decimals in an
ascending order :
(i) 5·8, 7·2, 5·69, 7·14, 5·06
(ii) 0·6, 6·6, 6·06, 66·6, 0·06
(iii) 6·54, 6·45, 6·4, 6·5, 6·05
(iv) 3·3, 3·303, 3·033, 0·33, 3·003
Sol. First of all making them in like decimals,
(i) 5·8, 7·2, 5·69, 7·14, 5·06
 5·80, 7·20, 5·69, 7·14, 5·06
Arranging in ascending order,
5·06 < 5·69 < 5·80 < 7·14 < 7·20
 5·06 < 5·69 < 5·8 < 7·14 < 7·2 Ans.
(ii) 0·6, 6·6, 6·06, 66·6, 0·06
 0·60, 6·60, 6·06, 66·60, 0·06
Arranging in ascending order,
0·06 < 0·60 < 6·06 < 6·60 < 66·60
 0·06 < 0·6 < 6·06 < 6·6 < 66·6 Ans.
(iii) 6·54, 6·45, 6·4, 6·5, 6·05
 6·54, 6·45, 6·4, 6·5, 6·05
Arranging in ascending order,
6·05 < 6·40 < 6·45 < 6·50 < 6·54
 6·05 < 6·4 < 6·45 < 6·5 < 6·54 Ans.
(iv) 3·3, 3·303, 3·033, 0·33, 3·003
 3·300, 3·303, 3·033, 0·330, 3·003
Arranging in descending order,
0·330 < 3·003 < 3·033 < 3·300 < 3·303
 0·33 < 3·003 < 3·033 < 3·3 < 3·303
Ans.
Q. 8. Arrange the following decimals in a
descending order :
(i) 7·3, 8·73, 73·03, 7·33, 8·073
(ii) 3·3, 3·03, 30·3, 30·03, 3·003
(iii) 2·7, 7·2, 2·27, 2·72, 2·02, 2·007
(iv) 8·88, 8·088, 88·8, 88·08, 8·008
Sol. Making them in like decimals and them
comparing
(i) 7·3, 8·73, 73·03, 7·33, 8·073
 7·300, 8·730, 73·030, 7·330, 8·073
Arranging in decending order
73·030 > 8·730 > 8·073 > 7·330 > 7·300
 73·03 > 8·73 > 8·073 > 7·33 > 7·3
Ans.
(ii) 3·3, 3·03, 30·3, 30·03, 3·003
 3·300, 3·030, 30·300, 30·030, 3·003
Arranging in descending order
30·300 > 30·030 > 3·300 > 3·030 > 3·003
 30·3 > 30·03 > 3·3 > 3·03 > 3·003 Ans.
(iii) 2·7, 7·2, 2·27, 2·72, 2·02, 2·007
 2·700, 7·200, 2·270, 2·720, 2·020,
2·007
Arranging in descending order
7·200 > 2·720 > 2·700 > 2·270 > 2·020>
2·007
 7·2 > 2·72 > 2·7 > 2·27 > 2·02 > 2·007
Ans.
(iv) 8·88, 8·088, 88·8, 88·08, 8·008
 8·880, 8·088, 88·800, 88·080, 8·008
Arranging in decending order,
88·800 > 88·080 > 8·880 > 8·088 > 8·008
 88·8 > 88·08 > 8·88 > 8·088 > 8·008
Ans.
          Convert each of the following into a
fraction in its simplest form :
Q. 1. ·9 Q. 2. 0·6
Q. 3. ·08 Q. 4. 0·15
Q. 5. 0·48 Q. 6. ·053
Q. 7. 0·125 Q. 8. ·224
Sol. 1. ·9 = 
9
10
2. 0·6 = 
6
10
6 2
10 2
3
5
(Dividing by 2, the HCF of 6, 10)
3. ·08 = 
8
100
8 4
100 4
2
25
(Dividing by 4, the HCF of 8, 100)
4. 0·15 = 
15
100
15 5
100 5
3
20
(Dividing by 5, the HCF of 15, 100)
5. 0·48 = 
48
100
48 4
100 4
12
25
(Dividing by 4, the HCF of 48, 100)
6. ·053 = 
53
1000
7. 0·125 = 
125
1000
125 125
1000 125
1
8
(Dividing by 125, the HCF of 125, 1000)
8. ·224 = 
224
1000
224 8
1000 8
28
125
 Ans.
(Dividing by 8, the HCF of 224 and 1000)
Convert each of the following as a
mixed fraction :
Q. 9. 6·4 Q. 10. 16·5
Q. 11. 8·36 Q. 11. 4·275
Q. 13. 25·06 Q. 14. 7·004
Q. 15. 2·052 Q. 16. 3·108
Sol. 9. 6·4 = 
64
10
64 2
10 2
32
5
6
2
5
(Dividing by 2, the HCF of 64 and 10)
10. 16·5 = 
165
10
165 5
10 5
33
2
16
1
2
(Dividing by 5, the HCF of 165 and 10)
11. 8·36 = 
836
100
836 4
100 4
209
25
8
9
25
(Dividing by 4, the HCF of 836 and 100)
12. 4·275
= 
4275
1000
4275 25
1000 25
171
40
4
11
40
(Dividing by 25)
13. 25·06
= 
2506
100
2506 2
100 2
1253
50
25
3
50
(Dividing by 2)
14. 7·004
= 
7004
1000
7004 4
1000 4
1751
250
7
1
250
(Dividing by 4)
15. 2·052
= 
2052
1000
2052 4
1000 4
513
250
2
13
250
(Dividing by 4)
16. 3·108
= 
3108
1000
3108 4
1000 4
777
250
3
27
250
(Dividing by 4)    Ans.
Convert each of the following into a
decimal :
Q. 17.
23
10
Q. 18.
167
100
Q. 19.
1589
100
Q. 20.
5413
1000
Q. 21.
21415
1000
Q. 22.
25
4
Q. 23.
3
3
5
Q. 24.
1
4
25