NCERT Textbook - Decimals Class 6 Notes | EduRev

 Page 1


Savita and Shama were going to market to buy some stationary items.
Savita said, “I have 5 rupees and 75 paise”. Shama said, “I have 7 rupees
and 50 paise”.
They knew how to write rupees and paise using decimals.
So Savita said, I have Rs 5.75 and Shama said,
“I have Rs 7.50”.
Have they written correctly?
We know that the dot represents a decimal point.
In this chapter, we will learn more about working
with decimals.
8.2 Tenths
Ravi and Raju measured the lengths of their pencils. Ravi’s pencil was
7 cm 5mm long and Raju’s pencil was 8 cm 3 mm long. Can you express
these lengths in centimetre using decimals?
We know that 10 mm = 1 cm
Therefore,        1 mm = 
1
10
 cm or one-tenth cm = 0.1 cm
Now, length of Ravi’s pencil
=
 7cm 5mm
= 7
5
10
cm i.e. 7cm and 5 tenths of a cm
= 7.5cm
The length of Raju’s pencil = 8 cm 3 mm
= 8
3
10
 cm  i.e. 8 cm and 3 tenths of a cm
= 8.3 cm
8.1 Introduction
Chapter 8
D D De e ec c ci i im m ma a al l ls s s
Page 2


Savita and Shama were going to market to buy some stationary items.
Savita said, “I have 5 rupees and 75 paise”. Shama said, “I have 7 rupees
and 50 paise”.
They knew how to write rupees and paise using decimals.
So Savita said, I have Rs 5.75 and Shama said,
“I have Rs 7.50”.
Have they written correctly?
We know that the dot represents a decimal point.
In this chapter, we will learn more about working
with decimals.
8.2 Tenths
Ravi and Raju measured the lengths of their pencils. Ravi’s pencil was
7 cm 5mm long and Raju’s pencil was 8 cm 3 mm long. Can you express
these lengths in centimetre using decimals?
We know that 10 mm = 1 cm
Therefore,        1 mm = 
1
10
 cm or one-tenth cm = 0.1 cm
Now, length of Ravi’s pencil
=
 7cm 5mm
= 7
5
10
cm i.e. 7cm and 5 tenths of a cm
= 7.5cm
The length of Raju’s pencil = 8 cm 3 mm
= 8
3
10
 cm  i.e. 8 cm and 3 tenths of a cm
= 8.3 cm
8.1 Introduction
Chapter 8
D D De e ec c ci i im m ma a al l ls s s
DECIMALS
165
Let us recall what we have learnt earlier.
If we show units by blocks then one unit is
one block, two units are two blocks and so
on. One block divided into 10 equal parts
means each part is 
1
10
 (one-tenth) of a unit, 2 parts show 2 tenths and 5
parts show 5 tenths and so on. A combination of 2 blocks and 3 parts
(tenths) will be recorded as :
Ones Tenths
(1) (
1
10
)
2 3
It can be written as 2.3 and read as two point three.
Let us look at another example where we have more than ‘ones’. Each
tower represents 10 units. So, the number shown here is :
i.e. 20 + 3 + 
5
10
 = 23.5
This is read as ‘twenty three point five’.
1. Can you now write the following as decimals?
Hundreds Tens Ones Tenths
(100) (10) (1) (
1
10
)
5 3 8 1
2 7 3 4
3 5 4 6
2. Write the lengths of Ravi’s and Raju’s pencils in ‘cm’ using decimals.
3. Make three more examples similar to the one given in question 1 and
solve them.
     Tens Ones      Tenths
    (10)  (1)      (
1
10
)
      2   3         5
Page 3


Savita and Shama were going to market to buy some stationary items.
Savita said, “I have 5 rupees and 75 paise”. Shama said, “I have 7 rupees
and 50 paise”.
They knew how to write rupees and paise using decimals.
So Savita said, I have Rs 5.75 and Shama said,
“I have Rs 7.50”.
Have they written correctly?
We know that the dot represents a decimal point.
In this chapter, we will learn more about working
with decimals.
8.2 Tenths
Ravi and Raju measured the lengths of their pencils. Ravi’s pencil was
7 cm 5mm long and Raju’s pencil was 8 cm 3 mm long. Can you express
these lengths in centimetre using decimals?
We know that 10 mm = 1 cm
Therefore,        1 mm = 
1
10
 cm or one-tenth cm = 0.1 cm
Now, length of Ravi’s pencil
=
 7cm 5mm
= 7
5
10
cm i.e. 7cm and 5 tenths of a cm
= 7.5cm
The length of Raju’s pencil = 8 cm 3 mm
= 8
3
10
 cm  i.e. 8 cm and 3 tenths of a cm
= 8.3 cm
8.1 Introduction
Chapter 8
D D De e ec c ci i im m ma a al l ls s s
DECIMALS
165
Let us recall what we have learnt earlier.
If we show units by blocks then one unit is
one block, two units are two blocks and so
on. One block divided into 10 equal parts
means each part is 
1
10
 (one-tenth) of a unit, 2 parts show 2 tenths and 5
parts show 5 tenths and so on. A combination of 2 blocks and 3 parts
(tenths) will be recorded as :
Ones Tenths
(1) (
1
10
)
2 3
It can be written as 2.3 and read as two point three.
Let us look at another example where we have more than ‘ones’. Each
tower represents 10 units. So, the number shown here is :
i.e. 20 + 3 + 
5
10
 = 23.5
This is read as ‘twenty three point five’.
1. Can you now write the following as decimals?
Hundreds Tens Ones Tenths
(100) (10) (1) (
1
10
)
5 3 8 1
2 7 3 4
3 5 4 6
2. Write the lengths of Ravi’s and Raju’s pencils in ‘cm’ using decimals.
3. Make three more examples similar to the one given in question 1 and
solve them.
     Tens Ones      Tenths
    (10)  (1)      (
1
10
)
      2   3         5
MATHEMATICS
166
Representing Decimals on number line
We represented fractions on a number line. Let us now represent decimals
too on a number line. Let us represent 0.6 on a number line.
We know that 0.6 is more than zero but less than one. There are 6 tenths in
it. Divide the unit length between 0 and 1 into 10 equal parts and take 6 parts
as shown below :
Write five numbers between 0 and 1 and show them on the number line.
Can you now represent 2.3 on a number line? Check, how many ones and
tenths are there in 2.3. Where will it lie on the number line?
Show 1.4 on the number line.
Example 1 : Write the following numbers in the place value table : (a) 20.5
(b) 4.2
Solution : Let us make a common place value table, assigning appropriate place
value to the digits in the given numbers. We have,
Tens (10) Ones (1) Tenths (
1
10
)
20.5 2 0 5
4.2 0 4 2
Example 2 : Write each of the following as decimals : (a) Two ones and
five-tenths (b) Thirty and one-tenth
Solution : (a) Two ones and five-tenths = 2 + 
5
10
 = 2.5
(b) Thirty and one-tenth = 30 + 
1
10
 = 30.1
Example 3 : Write each of the following as decimals :
(a) 30 + 6 + 
2
10
(b) 600 + 2 + 
8
10
Solution : (a) 30 + 6 + 
2
10
How many tens, ones and tenths are there in this number? We have
3 tens, 6 ones and 2 tenths.
Therefore, the decimal representation is 36.2.
(b)  600 + 2 + 
8
10
Note that it has 6 hundreds, no tens, 2 ones and 8 tenths.
Therefore, the decimal representation is 602.8
Page 4


Savita and Shama were going to market to buy some stationary items.
Savita said, “I have 5 rupees and 75 paise”. Shama said, “I have 7 rupees
and 50 paise”.
They knew how to write rupees and paise using decimals.
So Savita said, I have Rs 5.75 and Shama said,
“I have Rs 7.50”.
Have they written correctly?
We know that the dot represents a decimal point.
In this chapter, we will learn more about working
with decimals.
8.2 Tenths
Ravi and Raju measured the lengths of their pencils. Ravi’s pencil was
7 cm 5mm long and Raju’s pencil was 8 cm 3 mm long. Can you express
these lengths in centimetre using decimals?
We know that 10 mm = 1 cm
Therefore,        1 mm = 
1
10
 cm or one-tenth cm = 0.1 cm
Now, length of Ravi’s pencil
=
 7cm 5mm
= 7
5
10
cm i.e. 7cm and 5 tenths of a cm
= 7.5cm
The length of Raju’s pencil = 8 cm 3 mm
= 8
3
10
 cm  i.e. 8 cm and 3 tenths of a cm
= 8.3 cm
8.1 Introduction
Chapter 8
D D De e ec c ci i im m ma a al l ls s s
DECIMALS
165
Let us recall what we have learnt earlier.
If we show units by blocks then one unit is
one block, two units are two blocks and so
on. One block divided into 10 equal parts
means each part is 
1
10
 (one-tenth) of a unit, 2 parts show 2 tenths and 5
parts show 5 tenths and so on. A combination of 2 blocks and 3 parts
(tenths) will be recorded as :
Ones Tenths
(1) (
1
10
)
2 3
It can be written as 2.3 and read as two point three.
Let us look at another example where we have more than ‘ones’. Each
tower represents 10 units. So, the number shown here is :
i.e. 20 + 3 + 
5
10
 = 23.5
This is read as ‘twenty three point five’.
1. Can you now write the following as decimals?
Hundreds Tens Ones Tenths
(100) (10) (1) (
1
10
)
5 3 8 1
2 7 3 4
3 5 4 6
2. Write the lengths of Ravi’s and Raju’s pencils in ‘cm’ using decimals.
3. Make three more examples similar to the one given in question 1 and
solve them.
     Tens Ones      Tenths
    (10)  (1)      (
1
10
)
      2   3         5
MATHEMATICS
166
Representing Decimals on number line
We represented fractions on a number line. Let us now represent decimals
too on a number line. Let us represent 0.6 on a number line.
We know that 0.6 is more than zero but less than one. There are 6 tenths in
it. Divide the unit length between 0 and 1 into 10 equal parts and take 6 parts
as shown below :
Write five numbers between 0 and 1 and show them on the number line.
Can you now represent 2.3 on a number line? Check, how many ones and
tenths are there in 2.3. Where will it lie on the number line?
Show 1.4 on the number line.
Example 1 : Write the following numbers in the place value table : (a) 20.5
(b) 4.2
Solution : Let us make a common place value table, assigning appropriate place
value to the digits in the given numbers. We have,
Tens (10) Ones (1) Tenths (
1
10
)
20.5 2 0 5
4.2 0 4 2
Example 2 : Write each of the following as decimals : (a) Two ones and
five-tenths (b) Thirty and one-tenth
Solution : (a) Two ones and five-tenths = 2 + 
5
10
 = 2.5
(b) Thirty and one-tenth = 30 + 
1
10
 = 30.1
Example 3 : Write each of the following as decimals :
(a) 30 + 6 + 
2
10
(b) 600 + 2 + 
8
10
Solution : (a) 30 + 6 + 
2
10
How many tens, ones and tenths are there in this number? We have
3 tens, 6 ones and 2 tenths.
Therefore, the decimal representation is 36.2.
(b)  600 + 2 + 
8
10
Note that it has 6 hundreds, no tens, 2 ones and 8 tenths.
Therefore, the decimal representation is 602.8
DECIMALS
167
Write 
3
2
4
5
8
5
, ,
 in decimal
notation.
Fractions as decimals
We have already seen how a fraction with denominator 10 can be represented
using decimals.
Let us now try to find decimal representation of (a) 
11
5
 (b) 
1
2
(a) We know that 
11
5
 = 
22
10
 = 
20 2
10
+
  = 
20
10
 + 
2
10
 = 2 + 
2
10
 = 2.2
Therefore, 
22
10
 = 2.2  (in decimal notation.)
(b) In 
1
2
, the denominator is 2. For writing in decimal notation, the
denominator should be 10. We already know
how to make an equivalent fraction. So,
1
2
 = 
1 5
2 5
5
10
×
×
=
 = 0.5
Therefore, 
1
2
 is 0.5 in decimal notation.
Decimals as fractions
Till now we have learnt how to write fractions with denominators 10, 2 or 5 as
decimals. Can we write a decimal number like 1.2 as a fraction?
Let us see 
12 1
2
10
. = + =
10
10
+ =
2
10
12
10
EXERCISE 8.1
1. Write the following as numbers in the given table.
(a) (b)
T ens Ones T enths Hundreds T ens T enths
Hundreds Tens Ones Tenths
(100) (10) (1) (
1
10
)
Page 5


Savita and Shama were going to market to buy some stationary items.
Savita said, “I have 5 rupees and 75 paise”. Shama said, “I have 7 rupees
and 50 paise”.
They knew how to write rupees and paise using decimals.
So Savita said, I have Rs 5.75 and Shama said,
“I have Rs 7.50”.
Have they written correctly?
We know that the dot represents a decimal point.
In this chapter, we will learn more about working
with decimals.
8.2 Tenths
Ravi and Raju measured the lengths of their pencils. Ravi’s pencil was
7 cm 5mm long and Raju’s pencil was 8 cm 3 mm long. Can you express
these lengths in centimetre using decimals?
We know that 10 mm = 1 cm
Therefore,        1 mm = 
1
10
 cm or one-tenth cm = 0.1 cm
Now, length of Ravi’s pencil
=
 7cm 5mm
= 7
5
10
cm i.e. 7cm and 5 tenths of a cm
= 7.5cm
The length of Raju’s pencil = 8 cm 3 mm
= 8
3
10
 cm  i.e. 8 cm and 3 tenths of a cm
= 8.3 cm
8.1 Introduction
Chapter 8
D D De e ec c ci i im m ma a al l ls s s
DECIMALS
165
Let us recall what we have learnt earlier.
If we show units by blocks then one unit is
one block, two units are two blocks and so
on. One block divided into 10 equal parts
means each part is 
1
10
 (one-tenth) of a unit, 2 parts show 2 tenths and 5
parts show 5 tenths and so on. A combination of 2 blocks and 3 parts
(tenths) will be recorded as :
Ones Tenths
(1) (
1
10
)
2 3
It can be written as 2.3 and read as two point three.
Let us look at another example where we have more than ‘ones’. Each
tower represents 10 units. So, the number shown here is :
i.e. 20 + 3 + 
5
10
 = 23.5
This is read as ‘twenty three point five’.
1. Can you now write the following as decimals?
Hundreds Tens Ones Tenths
(100) (10) (1) (
1
10
)
5 3 8 1
2 7 3 4
3 5 4 6
2. Write the lengths of Ravi’s and Raju’s pencils in ‘cm’ using decimals.
3. Make three more examples similar to the one given in question 1 and
solve them.
     Tens Ones      Tenths
    (10)  (1)      (
1
10
)
      2   3         5
MATHEMATICS
166
Representing Decimals on number line
We represented fractions on a number line. Let us now represent decimals
too on a number line. Let us represent 0.6 on a number line.
We know that 0.6 is more than zero but less than one. There are 6 tenths in
it. Divide the unit length between 0 and 1 into 10 equal parts and take 6 parts
as shown below :
Write five numbers between 0 and 1 and show them on the number line.
Can you now represent 2.3 on a number line? Check, how many ones and
tenths are there in 2.3. Where will it lie on the number line?
Show 1.4 on the number line.
Example 1 : Write the following numbers in the place value table : (a) 20.5
(b) 4.2
Solution : Let us make a common place value table, assigning appropriate place
value to the digits in the given numbers. We have,
Tens (10) Ones (1) Tenths (
1
10
)
20.5 2 0 5
4.2 0 4 2
Example 2 : Write each of the following as decimals : (a) Two ones and
five-tenths (b) Thirty and one-tenth
Solution : (a) Two ones and five-tenths = 2 + 
5
10
 = 2.5
(b) Thirty and one-tenth = 30 + 
1
10
 = 30.1
Example 3 : Write each of the following as decimals :
(a) 30 + 6 + 
2
10
(b) 600 + 2 + 
8
10
Solution : (a) 30 + 6 + 
2
10
How many tens, ones and tenths are there in this number? We have
3 tens, 6 ones and 2 tenths.
Therefore, the decimal representation is 36.2.
(b)  600 + 2 + 
8
10
Note that it has 6 hundreds, no tens, 2 ones and 8 tenths.
Therefore, the decimal representation is 602.8
DECIMALS
167
Write 
3
2
4
5
8
5
, ,
 in decimal
notation.
Fractions as decimals
We have already seen how a fraction with denominator 10 can be represented
using decimals.
Let us now try to find decimal representation of (a) 
11
5
 (b) 
1
2
(a) We know that 
11
5
 = 
22
10
 = 
20 2
10
+
  = 
20
10
 + 
2
10
 = 2 + 
2
10
 = 2.2
Therefore, 
22
10
 = 2.2  (in decimal notation.)
(b) In 
1
2
, the denominator is 2. For writing in decimal notation, the
denominator should be 10. We already know
how to make an equivalent fraction. So,
1
2
 = 
1 5
2 5
5
10
×
×
=
 = 0.5
Therefore, 
1
2
 is 0.5 in decimal notation.
Decimals as fractions
Till now we have learnt how to write fractions with denominators 10, 2 or 5 as
decimals. Can we write a decimal number like 1.2 as a fraction?
Let us see 
12 1
2
10
. = + =
10
10
+ =
2
10
12
10
EXERCISE 8.1
1. Write the following as numbers in the given table.
(a) (b)
T ens Ones T enths Hundreds T ens T enths
Hundreds Tens Ones Tenths
(100) (10) (1) (
1
10
)
MATHEMATICS
168
2. Write the following decimals in the place value table.
(a) 19.4 (b) 0.3 (c) 10.6 (d) 205.9
3. Write each of the following as decimals :
(a) Seven-tenths (b) Two tens and nine-tenths
(c) Fourteen point six (d) One hundred and two ones
(e) Six hundred point eight
4. Write each of the following as decimals:
(a)
5
10
(b) 3 + 
7
10
(c) 200 + 60 + 5 + 
1
10
(d) 70 + 
8
10
(e)
88
10
(f)
4
2
10
(g)
3
2
(h)
2
5
(i)
12
5
(j)
3
3
5
(k)
4
1
2
5. Write the following decimals as fractions. Reduce the fractions to lowest form.
(a) 0.6 (b) 2.5 (c) 1.0 (d) 3.8 (e) 13.7 (f) 21.2 (g) 6.4
6. Express the following as cm using decimals.
(a) 2 mm (b) 30 mm (c) 116 mm (d) 4 cm 2 mm (e) 162 mm
(f) 83 mm
7. Between which two whole numbers on the number line are the given numbers lie?
Which of these whole numbers is nearer the number?
(a) 0.8 (b) 5.1 (c) 2.6 (d) 6.4 (e) 9.1 (f) 4.9
8. Show the following numbers on the number line.
(a) 0.2 (b) 1.9 (c) 1.1 (d) 2.5
9. Write the decimal number represented by the points A, B, C, D on the given
number line.
10. (a) The length of Ramesh’s notebook is 9 cm 5 mm. What will be its length in cm?
(b) The length of a young gram plant is 65 mm. Express its length in cm.
8.3 Hundredths
David was measuring the length of his room.
He found that the length of his room is 4 m
and 25 cm.
He wanted to write the length in metres.
Can you help him? What part of a metre will
be one centimetre?