Class 7 Maths Chapter 3 NCERT Book - A Peek Beyond the Point

 Page 1


3.1 The Need for Smaller Units
Sonu’s mother was fixing a 
toy. She was trying to join 
two pieces with the help of 
a screw. Sonu was 
watching his mother with 
great curiosity. His mother 
was unable to join the 
pieces. Sonu asked why. 
His mother said that the 
screw was not of the 
right size.
She brought another screw from the box and was able to fix the toy. 
The two screws looked the same to Sonu. But when he observed them 
closely, he saw they were of slightly different lengths.
Sonu was fascinated 
by how such a small 
difference in lengths 
could matter so much. 
He was curious to 
know the difference 
in lengths. He was also 
curious to know how 
little the difference 
was because the 
screws looked nearly 
the same.
In the following figure, screws are placed above a scale. Measure 
them and write their length in the space provided.
A PEEK BEYOND 
THE POINT
3
Chapter-3.indd   46 Chapter-3.indd   46 4/12/2025   11:39:13 AM 4/12/2025   11:39:13 AM
Page 2


3.1 The Need for Smaller Units
Sonu’s mother was fixing a 
toy. She was trying to join 
two pieces with the help of 
a screw. Sonu was 
watching his mother with 
great curiosity. His mother 
was unable to join the 
pieces. Sonu asked why. 
His mother said that the 
screw was not of the 
right size.
She brought another screw from the box and was able to fix the toy. 
The two screws looked the same to Sonu. But when he observed them 
closely, he saw they were of slightly different lengths.
Sonu was fascinated 
by how such a small 
difference in lengths 
could matter so much. 
He was curious to 
know the difference 
in lengths. He was also 
curious to know how 
little the difference 
was because the 
screws looked nearly 
the same.
In the following figure, screws are placed above a scale. Measure 
them and write their length in the space provided.
A PEEK BEYOND 
THE POINT
3
Chapter-3.indd   46 Chapter-3.indd   46 4/12/2025   11:39:13 AM 4/12/2025   11:39:13 AM
A Peek Beyond the Point
Which scale helped you measure the length of the screws accurately? 
Why? 
What is the meaning of 2 
7
10
  cm (the length of the first screw)?
As seen on the ruler, the unit length between two consecutive 
numbers is divided into 10 equal parts. To get the length 2 
7
10
 cm, we 
go from 0 to 2 and then take seven parts of 
1
10
. The length of the screw 
is 2 cm and 
7
10
 cm. Similarly, we can make sense of the length 3 
2
10
 cm.
We read 2 
7
10
 cm as two and seven-tenth centimeters, and 3 
2
10
 cm 
as three and two-tenth centimeters.
Can you explain why the unit was divided into smaller parts to measure 
the screws?
Measure the following objects using a scale and write their 
measurements in centimeters (as shown earlier for the lengths of the 
screws): pen, sharpener, and any other object of your choice.
Write the measurements of the objects shown in the picture:
Between 2 cm
 and 3 cm
_______________
_______________
_______________
More than 2 
1
2
 
cm but less than
3 cm
2 
7
10
 cm
47
Chapter-3.indd   47 Chapter-3.indd   47 4/12/2025   11:39:14 AM 4/12/2025   11:39:14 AM
Page 3


3.1 The Need for Smaller Units
Sonu’s mother was fixing a 
toy. She was trying to join 
two pieces with the help of 
a screw. Sonu was 
watching his mother with 
great curiosity. His mother 
was unable to join the 
pieces. Sonu asked why. 
His mother said that the 
screw was not of the 
right size.
She brought another screw from the box and was able to fix the toy. 
The two screws looked the same to Sonu. But when he observed them 
closely, he saw they were of slightly different lengths.
Sonu was fascinated 
by how such a small 
difference in lengths 
could matter so much. 
He was curious to 
know the difference 
in lengths. He was also 
curious to know how 
little the difference 
was because the 
screws looked nearly 
the same.
In the following figure, screws are placed above a scale. Measure 
them and write their length in the space provided.
A PEEK BEYOND 
THE POINT
3
Chapter-3.indd   46 Chapter-3.indd   46 4/12/2025   11:39:13 AM 4/12/2025   11:39:13 AM
A Peek Beyond the Point
Which scale helped you measure the length of the screws accurately? 
Why? 
What is the meaning of 2 
7
10
  cm (the length of the first screw)?
As seen on the ruler, the unit length between two consecutive 
numbers is divided into 10 equal parts. To get the length 2 
7
10
 cm, we 
go from 0 to 2 and then take seven parts of 
1
10
. The length of the screw 
is 2 cm and 
7
10
 cm. Similarly, we can make sense of the length 3 
2
10
 cm.
We read 2 
7
10
 cm as two and seven-tenth centimeters, and 3 
2
10
 cm 
as three and two-tenth centimeters.
Can you explain why the unit was divided into smaller parts to measure 
the screws?
Measure the following objects using a scale and write their 
measurements in centimeters (as shown earlier for the lengths of the 
screws): pen, sharpener, and any other object of your choice.
Write the measurements of the objects shown in the picture:
Between 2 cm
 and 3 cm
_______________
_______________
_______________
More than 2 
1
2
 
cm but less than
3 cm
2 
7
10
 cm
47
Chapter-3.indd   47 Chapter-3.indd   47 4/12/2025   11:39:14 AM 4/12/2025   11:39:14 AM
Ganita Prakash | Grade 7 
As seen here, when exact measures are required we can make use of 
smaller units of measurement.
3.2 A Tenth Part
The length of the pencil shown in the figure below is 3 
4
10
 units, which can 
also be read as 3 units and four one-tenths, i.e., (3 × 1) + 
(
4 × 
1
10
)
 units.
= 10 times 
1
10
 = 10 × 
1
10
 = 1 unit 
This length is the same as 34 one-tenths units because 10 one-tenths 
units make one unit.
1
10
 + 
1
10
 + 
1
10
 + 
1
10
 + 
1
10
 + 
1
10
 + 
1
10
 + 
1
10
 + 
1
10
 + 
1
10
48
Chapter-3.indd   48 Chapter-3.indd   48 4/12/2025   11:39:14 AM 4/12/2025   11:39:14 AM
Page 4


3.1 The Need for Smaller Units
Sonu’s mother was fixing a 
toy. She was trying to join 
two pieces with the help of 
a screw. Sonu was 
watching his mother with 
great curiosity. His mother 
was unable to join the 
pieces. Sonu asked why. 
His mother said that the 
screw was not of the 
right size.
She brought another screw from the box and was able to fix the toy. 
The two screws looked the same to Sonu. But when he observed them 
closely, he saw they were of slightly different lengths.
Sonu was fascinated 
by how such a small 
difference in lengths 
could matter so much. 
He was curious to 
know the difference 
in lengths. He was also 
curious to know how 
little the difference 
was because the 
screws looked nearly 
the same.
In the following figure, screws are placed above a scale. Measure 
them and write their length in the space provided.
A PEEK BEYOND 
THE POINT
3
Chapter-3.indd   46 Chapter-3.indd   46 4/12/2025   11:39:13 AM 4/12/2025   11:39:13 AM
A Peek Beyond the Point
Which scale helped you measure the length of the screws accurately? 
Why? 
What is the meaning of 2 
7
10
  cm (the length of the first screw)?
As seen on the ruler, the unit length between two consecutive 
numbers is divided into 10 equal parts. To get the length 2 
7
10
 cm, we 
go from 0 to 2 and then take seven parts of 
1
10
. The length of the screw 
is 2 cm and 
7
10
 cm. Similarly, we can make sense of the length 3 
2
10
 cm.
We read 2 
7
10
 cm as two and seven-tenth centimeters, and 3 
2
10
 cm 
as three and two-tenth centimeters.
Can you explain why the unit was divided into smaller parts to measure 
the screws?
Measure the following objects using a scale and write their 
measurements in centimeters (as shown earlier for the lengths of the 
screws): pen, sharpener, and any other object of your choice.
Write the measurements of the objects shown in the picture:
Between 2 cm
 and 3 cm
_______________
_______________
_______________
More than 2 
1
2
 
cm but less than
3 cm
2 
7
10
 cm
47
Chapter-3.indd   47 Chapter-3.indd   47 4/12/2025   11:39:14 AM 4/12/2025   11:39:14 AM
Ganita Prakash | Grade 7 
As seen here, when exact measures are required we can make use of 
smaller units of measurement.
3.2 A Tenth Part
The length of the pencil shown in the figure below is 3 
4
10
 units, which can 
also be read as 3 units and four one-tenths, i.e., (3 × 1) + 
(
4 × 
1
10
)
 units.
= 10 times 
1
10
 = 10 × 
1
10
 = 1 unit 
This length is the same as 34 one-tenths units because 10 one-tenths 
units make one unit.
1
10
 + 
1
10
 + 
1
10
 + 
1
10
 + 
1
10
 + 
1
10
 + 
1
10
 + 
1
10
 + 
1
10
 + 
1
10
48
Chapter-3.indd   48 Chapter-3.indd   48 4/12/2025   11:39:14 AM 4/12/2025   11:39:14 AM
A Peek Beyond the Point
34 × 
1
10
 = 
34
10
 = 
10
10
 + 
10
10
 + 
10
10
 + 
4
10
 (34 one-tenths)
     = 1 + 1 + 1 + 
4
10
 (3 and 4 one-tenths)
A few numbers with fractional units are shown below along with 
how to read them.
 4 
1
10
  ‘four and one-tenth’
    
4
10
  ‘four one-tenths’ or ‘four-tenths’
    
41
10
  ‘forty-one one-tenths’ or ‘forty-one tenths’
41 
1
10
  ‘forty-one and one-tenth’
For the objects shown below, write their lengths in two ways and 
read them aloud. An example is given for the USB cable. (Note that the 
unit length used in each diagram is not the same).
The length of the USB cable is 4 and 
8
10
 units or 
48
10
 units.
Arrange these lengths in  increasing order:
(a)  
9
10
(b) 1 
7
10
 (c) 
130
10
(d) 13 
1
10
 
(e) 10 
5
10
(f) 7 
6
10
(g) 6 
7
10
(h) 
4
10
49
Chapter-3.indd   49 Chapter-3.indd   49 4/16/2025   4:28:12 PM 4/16/2025   4:28:12 PM
Page 5


3.1 The Need for Smaller Units
Sonu’s mother was fixing a 
toy. She was trying to join 
two pieces with the help of 
a screw. Sonu was 
watching his mother with 
great curiosity. His mother 
was unable to join the 
pieces. Sonu asked why. 
His mother said that the 
screw was not of the 
right size.
She brought another screw from the box and was able to fix the toy. 
The two screws looked the same to Sonu. But when he observed them 
closely, he saw they were of slightly different lengths.
Sonu was fascinated 
by how such a small 
difference in lengths 
could matter so much. 
He was curious to 
know the difference 
in lengths. He was also 
curious to know how 
little the difference 
was because the 
screws looked nearly 
the same.
In the following figure, screws are placed above a scale. Measure 
them and write their length in the space provided.
A PEEK BEYOND 
THE POINT
3
Chapter-3.indd   46 Chapter-3.indd   46 4/12/2025   11:39:13 AM 4/12/2025   11:39:13 AM
A Peek Beyond the Point
Which scale helped you measure the length of the screws accurately? 
Why? 
What is the meaning of 2 
7
10
  cm (the length of the first screw)?
As seen on the ruler, the unit length between two consecutive 
numbers is divided into 10 equal parts. To get the length 2 
7
10
 cm, we 
go from 0 to 2 and then take seven parts of 
1
10
. The length of the screw 
is 2 cm and 
7
10
 cm. Similarly, we can make sense of the length 3 
2
10
 cm.
We read 2 
7
10
 cm as two and seven-tenth centimeters, and 3 
2
10
 cm 
as three and two-tenth centimeters.
Can you explain why the unit was divided into smaller parts to measure 
the screws?
Measure the following objects using a scale and write their 
measurements in centimeters (as shown earlier for the lengths of the 
screws): pen, sharpener, and any other object of your choice.
Write the measurements of the objects shown in the picture:
Between 2 cm
 and 3 cm
_______________
_______________
_______________
More than 2 
1
2
 
cm but less than
3 cm
2 
7
10
 cm
47
Chapter-3.indd   47 Chapter-3.indd   47 4/12/2025   11:39:14 AM 4/12/2025   11:39:14 AM
Ganita Prakash | Grade 7 
As seen here, when exact measures are required we can make use of 
smaller units of measurement.
3.2 A Tenth Part
The length of the pencil shown in the figure below is 3 
4
10
 units, which can 
also be read as 3 units and four one-tenths, i.e., (3 × 1) + 
(
4 × 
1
10
)
 units.
= 10 times 
1
10
 = 10 × 
1
10
 = 1 unit 
This length is the same as 34 one-tenths units because 10 one-tenths 
units make one unit.
1
10
 + 
1
10
 + 
1
10
 + 
1
10
 + 
1
10
 + 
1
10
 + 
1
10
 + 
1
10
 + 
1
10
 + 
1
10
48
Chapter-3.indd   48 Chapter-3.indd   48 4/12/2025   11:39:14 AM 4/12/2025   11:39:14 AM
A Peek Beyond the Point
34 × 
1
10
 = 
34
10
 = 
10
10
 + 
10
10
 + 
10
10
 + 
4
10
 (34 one-tenths)
     = 1 + 1 + 1 + 
4
10
 (3 and 4 one-tenths)
A few numbers with fractional units are shown below along with 
how to read them.
 4 
1
10
  ‘four and one-tenth’
    
4
10
  ‘four one-tenths’ or ‘four-tenths’
    
41
10
  ‘forty-one one-tenths’ or ‘forty-one tenths’
41 
1
10
  ‘forty-one and one-tenth’
For the objects shown below, write their lengths in two ways and 
read them aloud. An example is given for the USB cable. (Note that the 
unit length used in each diagram is not the same).
The length of the USB cable is 4 and 
8
10
 units or 
48
10
 units.
Arrange these lengths in  increasing order:
(a)  
9
10
(b) 1 
7
10
 (c) 
130
10
(d) 13 
1
10
 
(e) 10 
5
10
(f) 7 
6
10
(g) 6 
7
10
(h) 
4
10
49
Chapter-3.indd   49 Chapter-3.indd   49 4/16/2025   4:28:12 PM 4/16/2025   4:28:12 PM
Ganita Prakash | Grade 7 
Arrange the following lengths in increasing order: 4 
1
10
, 
4
10
, 
41
10
, 41 
1
10
.
Sonu is measuring some of his body parts. The length of Sonu’s lower 
arm is 2 
7
10
 units, and that of his upper arm is 3 
6
10
 units. What is the 
total length of his arm?
To get the total length, let us see the lower and upper arm length as 
2 units and 7 one-tenths, and 3 units and 6 one-tenths, respectively.
So, there are (2 + 3) units and (7 + 6) one-tenths. Together, they make 
5 units and 13 one-tenths. But 13 one-tenths is 1 unit and 3 one-tenths. 
So, the total length is 6 units and 3 one-tenths.
(a)     (2 + 3) + 
(
7
10
 + 
6
10
)
    = (2 + 3) + 
(
13
10
)
    = 5 + 
13
10
    = 5 + 
10
10
 + 
3
10
  = 5 + 1 + 
3
10
    = 6 + 
3
10
    = 6 
3
10
?
Or, both the lengths can be converted to tenths and then added:
(c)  27 one-tenths and 35 one-tenths is 62 one-tenths
27
10
 + 
35
10
 = 
62
10
62
10
 is the same as 60 one-tenths 
(
60
10)
 and 2 one-tenths 
(
2
10)
, which is 
(b)     2   
7
10
 
          +  3   
6
10
          =  5   
13
10
          =  6   
3
10
?
50
Chapter-3.indd   50 Chapter-3.indd   50 4/12/2025   11:39:14 AM 4/12/2025   11:39:14 AM