NCERT Textbook: Proportional Reasoning-1 | Mathematics Class 8- New NCERT (Ganita Prakash) PDF Download

 Page 1


7.1 Observing Similarity in Change
We are all familiar with digital images. We often change the size and 
orientation of these images to suit our needs. Observe the set of images 
below —
We can see that all the images are of different sizes.
Which images look similar and which ones look different?
Images (A, C, and D) look similar, even though they have different sizes.
PROPORTIONAL 
REASONING-1
7
Image A
Image D Image E
Image B Image C
Chapter 7.indd   159 Chapter 7.indd   159 10-07-2025   15:14:27 10-07-2025   15:14:27
Page 2


7.1 Observing Similarity in Change
We are all familiar with digital images. We often change the size and 
orientation of these images to suit our needs. Observe the set of images 
below —
We can see that all the images are of different sizes.
Which images look similar and which ones look different?
Images (A, C, and D) look similar, even though they have different sizes.
PROPORTIONAL 
REASONING-1
7
Image A
Image D Image E
Image B Image C
Chapter 7.indd   159 Chapter 7.indd   159 10-07-2025   15:14:27 10-07-2025   15:14:27
Ganita Prakash | Grade 8 
160
Do images B and E look like the other three images?
No, they are slightly distorted. The tiger appears elongated in B, and
compressed and fatter in E!
Why?
You may notice that images A, C, and D are rectangular, but E is 
square. Maybe that is why E looks different. But B is also a rectangle! 
Why does it look different from the other rectangular images? 
Can we observe any pattern to answer this question? Perhaps by 
measuring the rectangles?
Image Width (in mm) Height (in mm)
Image A 60 40
Image B 40 20
Image C 30 20
Image D 90 60
Image E 60 60
What makes images A, C, and D appear similar, and B and E different?
When we compare image A with C, we notice that the width of C is 
half that of A. The height is also half of A. Both the width and height 
have changed by the same factor (through multiplication), 
1
2
 in this 
case. Since the widths and heights have changed by the same factor, the 
images look similar.
When we compare image A with image B, we notice that the width of 
B is 20 millimetre (mm) less than that of A. The height too is 20 mm less 
than the height of A. Even though the difference (through subtraction) is 
the same, the images look different. Have the width and height changed 
by the same factor? The height of B is half the height of A. But the width of 
B is not half the width of A. Since the width and height have not changed 
by the same factor, the images look different.
Can you check by what factors the width and height of image D change 
as compared to image A? Are the factors the same?
Images A, C, and D look similar because their widths and heights have 
changed by the same factor. We say that the changes to their widths and 
heights are proportional.
Math 
Talk
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Page 3


7.1 Observing Similarity in Change
We are all familiar with digital images. We often change the size and 
orientation of these images to suit our needs. Observe the set of images 
below —
We can see that all the images are of different sizes.
Which images look similar and which ones look different?
Images (A, C, and D) look similar, even though they have different sizes.
PROPORTIONAL 
REASONING-1
7
Image A
Image D Image E
Image B Image C
Chapter 7.indd   159 Chapter 7.indd   159 10-07-2025   15:14:27 10-07-2025   15:14:27
Ganita Prakash | Grade 8 
160
Do images B and E look like the other three images?
No, they are slightly distorted. The tiger appears elongated in B, and
compressed and fatter in E!
Why?
You may notice that images A, C, and D are rectangular, but E is 
square. Maybe that is why E looks different. But B is also a rectangle! 
Why does it look different from the other rectangular images? 
Can we observe any pattern to answer this question? Perhaps by 
measuring the rectangles?
Image Width (in mm) Height (in mm)
Image A 60 40
Image B 40 20
Image C 30 20
Image D 90 60
Image E 60 60
What makes images A, C, and D appear similar, and B and E different?
When we compare image A with C, we notice that the width of C is 
half that of A. The height is also half of A. Both the width and height 
have changed by the same factor (through multiplication), 
1
2
 in this 
case. Since the widths and heights have changed by the same factor, the 
images look similar.
When we compare image A with image B, we notice that the width of 
B is 20 millimetre (mm) less than that of A. The height too is 20 mm less 
than the height of A. Even though the difference (through subtraction) is 
the same, the images look different. Have the width and height changed 
by the same factor? The height of B is half the height of A. But the width of 
B is not half the width of A. Since the width and height have not changed 
by the same factor, the images look different.
Can you check by what factors the width and height of image D change 
as compared to image A? Are the factors the same?
Images A, C, and D look similar because their widths and heights have 
changed by the same factor. We say that the changes to their widths and 
heights are proportional.
Math 
Talk
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Proportional Reasoning-1
161
7.2 Ratios
We use the notion of a ratio to represent such proportional relationships 
in mathematics.
We can say that the ratio of width to height of image A is
60 : 40.
The numbers 60 and 40 are called the terms of the ratio.  
The ratio of width to height of image C is 30 : 20, and that of image D 
is 90 : 60.
In a ratio of the form a : b, we can say that for every ‘a’ units of the 
first quantity, there are ‘ b’ units of the second quantity.
So, in image A, we can say that for every 60 mm of width, there are 40 
mm of height.
We can say that the ratios of width to height of images A, C, and D 
are proportional because the terms of these ratios change by the same  
factor. Let us see how.
Image A — 60 : 40
Multiplying both the terms by 
1
2
, we get
60 × 
1
2
 : 40 × 
1
2
 
which is 30 : 20, the ratio of width to height in image C.
By what factor should we multiply the ratio 60 : 40 (image A) to get 90 : 60 
(image D)?
A more systematic way to compare whether the ratios are proportional 
is to reduce them to their simplest form and see if these simplest forms 
are the same.
7.3 Ratios in their Simplest Form
We can reduce ratios to their simplest form by dividing the terms by 
their HCF.
In image A, the terms are 60 and 40. What is the HCF of 60 and 40? It 
is 20. Dividing the terms by 20, we get the ratio of image A to be 3 : 2 in 
its simplest form.
The ratio of image D is 90 : 60. Dividing both terms by 30 (HCF 
of 90 and 60), we get the simplest form to be 3 : 2. So the ratios of  
images A and D are proportional as well.
What is the simplest form of the ratios of images B and E?
The ratio of image B is 40 : 20; in its simplest form, it is 2 : 1.
The ratio of image E is 60 : 60; in its simplest form, it is 1 : 1.
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Page 4


7.1 Observing Similarity in Change
We are all familiar with digital images. We often change the size and 
orientation of these images to suit our needs. Observe the set of images 
below —
We can see that all the images are of different sizes.
Which images look similar and which ones look different?
Images (A, C, and D) look similar, even though they have different sizes.
PROPORTIONAL 
REASONING-1
7
Image A
Image D Image E
Image B Image C
Chapter 7.indd   159 Chapter 7.indd   159 10-07-2025   15:14:27 10-07-2025   15:14:27
Ganita Prakash | Grade 8 
160
Do images B and E look like the other three images?
No, they are slightly distorted. The tiger appears elongated in B, and
compressed and fatter in E!
Why?
You may notice that images A, C, and D are rectangular, but E is 
square. Maybe that is why E looks different. But B is also a rectangle! 
Why does it look different from the other rectangular images? 
Can we observe any pattern to answer this question? Perhaps by 
measuring the rectangles?
Image Width (in mm) Height (in mm)
Image A 60 40
Image B 40 20
Image C 30 20
Image D 90 60
Image E 60 60
What makes images A, C, and D appear similar, and B and E different?
When we compare image A with C, we notice that the width of C is 
half that of A. The height is also half of A. Both the width and height 
have changed by the same factor (through multiplication), 
1
2
 in this 
case. Since the widths and heights have changed by the same factor, the 
images look similar.
When we compare image A with image B, we notice that the width of 
B is 20 millimetre (mm) less than that of A. The height too is 20 mm less 
than the height of A. Even though the difference (through subtraction) is 
the same, the images look different. Have the width and height changed 
by the same factor? The height of B is half the height of A. But the width of 
B is not half the width of A. Since the width and height have not changed 
by the same factor, the images look different.
Can you check by what factors the width and height of image D change 
as compared to image A? Are the factors the same?
Images A, C, and D look similar because their widths and heights have 
changed by the same factor. We say that the changes to their widths and 
heights are proportional.
Math 
Talk
Chapter 7.indd   160 Chapter 7.indd   160 10-07-2025   15:14:27 10-07-2025   15:14:27
Proportional Reasoning-1
161
7.2 Ratios
We use the notion of a ratio to represent such proportional relationships 
in mathematics.
We can say that the ratio of width to height of image A is
60 : 40.
The numbers 60 and 40 are called the terms of the ratio.  
The ratio of width to height of image C is 30 : 20, and that of image D 
is 90 : 60.
In a ratio of the form a : b, we can say that for every ‘a’ units of the 
first quantity, there are ‘ b’ units of the second quantity.
So, in image A, we can say that for every 60 mm of width, there are 40 
mm of height.
We can say that the ratios of width to height of images A, C, and D 
are proportional because the terms of these ratios change by the same  
factor. Let us see how.
Image A — 60 : 40
Multiplying both the terms by 
1
2
, we get
60 × 
1
2
 : 40 × 
1
2
 
which is 30 : 20, the ratio of width to height in image C.
By what factor should we multiply the ratio 60 : 40 (image A) to get 90 : 60 
(image D)?
A more systematic way to compare whether the ratios are proportional 
is to reduce them to their simplest form and see if these simplest forms 
are the same.
7.3 Ratios in their Simplest Form
We can reduce ratios to their simplest form by dividing the terms by 
their HCF.
In image A, the terms are 60 and 40. What is the HCF of 60 and 40? It 
is 20. Dividing the terms by 20, we get the ratio of image A to be 3 : 2 in 
its simplest form.
The ratio of image D is 90 : 60. Dividing both terms by 30 (HCF 
of 90 and 60), we get the simplest form to be 3 : 2. So the ratios of  
images A and D are proportional as well.
What is the simplest form of the ratios of images B and E?
The ratio of image B is 40 : 20; in its simplest form, it is 2 : 1.
The ratio of image E is 60 : 60; in its simplest form, it is 1 : 1.
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Ganita Prakash | Grade 8 
162
These ratios are not the same as 3 : 2. So, we can say that the ratios of 
width to height of images B and E are not proportional to the ratios of 
images A, C, and D.
When two ratios are the same in their simplest forms, we say that the 
ratios are in proportion, or that the ratios are proportional. We use 
the ‘::’ symbol to indicate that they are proportional.
So a : b :: c : d indicates that the ratios a : b and c : d are proportional.
Thus, 
60 : 40 :: 30 : 20 and 60 : 40 :: 90 : 60.
7.4 Problem Solving with Proportional Reasoning
Example 1: Are the ratios 3 : 4 and 72 : 96 proportional?
3 : 4 is already in its simplest form.
To find the simplest form of 72 : 96, we need to divide both terms by 
their HCF.
What is the HCF of 72 and 96?
The HCF of 72 and 96 is 24. Dividing both terms by 24, we get 3 : 4.
Since both ratios in their simplest form are the same, they are proportional.
Example 2: Kesang wanted to make lemonade for a celebration. She 
made 6 glasses of lemonade in a vessel and added 10 spoons of sugar to 
the drink. Her father expected more people to join the celebration. So he 
asked her to make 18 more glasses of lemonade. 
To make the lemonade with the same sweetness, how many spoons of 
sugar should she add? 
To maintain the same sweetness, the ratio of 
the number of glasses of lemonade to the number 
of spoons of sugar should be proportional. For 6 
glasses of lemonade, she added 10 spoons of sugar. 
The ratio of glasses of lemonade to spoons of 
sugar is 6 : 10. If she needs to make 18 more glasses 
of lemonade, how many spoons of sugar should 
she use? We can model this problem as —
6 : 10 :: 18 : ?
We know that each term in the ratio must change 
by the same factor, for the ratios to be proportional.
How can we find the factor of change in the ratio?
The first term has increased from 6 to 18. To find the factor of change, 
we can divide 18 by 6 to get 3.
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Page 5


7.1 Observing Similarity in Change
We are all familiar with digital images. We often change the size and 
orientation of these images to suit our needs. Observe the set of images 
below —
We can see that all the images are of different sizes.
Which images look similar and which ones look different?
Images (A, C, and D) look similar, even though they have different sizes.
PROPORTIONAL 
REASONING-1
7
Image A
Image D Image E
Image B Image C
Chapter 7.indd   159 Chapter 7.indd   159 10-07-2025   15:14:27 10-07-2025   15:14:27
Ganita Prakash | Grade 8 
160
Do images B and E look like the other three images?
No, they are slightly distorted. The tiger appears elongated in B, and
compressed and fatter in E!
Why?
You may notice that images A, C, and D are rectangular, but E is 
square. Maybe that is why E looks different. But B is also a rectangle! 
Why does it look different from the other rectangular images? 
Can we observe any pattern to answer this question? Perhaps by 
measuring the rectangles?
Image Width (in mm) Height (in mm)
Image A 60 40
Image B 40 20
Image C 30 20
Image D 90 60
Image E 60 60
What makes images A, C, and D appear similar, and B and E different?
When we compare image A with C, we notice that the width of C is 
half that of A. The height is also half of A. Both the width and height 
have changed by the same factor (through multiplication), 
1
2
 in this 
case. Since the widths and heights have changed by the same factor, the 
images look similar.
When we compare image A with image B, we notice that the width of 
B is 20 millimetre (mm) less than that of A. The height too is 20 mm less 
than the height of A. Even though the difference (through subtraction) is 
the same, the images look different. Have the width and height changed 
by the same factor? The height of B is half the height of A. But the width of 
B is not half the width of A. Since the width and height have not changed 
by the same factor, the images look different.
Can you check by what factors the width and height of image D change 
as compared to image A? Are the factors the same?
Images A, C, and D look similar because their widths and heights have 
changed by the same factor. We say that the changes to their widths and 
heights are proportional.
Math 
Talk
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Proportional Reasoning-1
161
7.2 Ratios
We use the notion of a ratio to represent such proportional relationships 
in mathematics.
We can say that the ratio of width to height of image A is
60 : 40.
The numbers 60 and 40 are called the terms of the ratio.  
The ratio of width to height of image C is 30 : 20, and that of image D 
is 90 : 60.
In a ratio of the form a : b, we can say that for every ‘a’ units of the 
first quantity, there are ‘ b’ units of the second quantity.
So, in image A, we can say that for every 60 mm of width, there are 40 
mm of height.
We can say that the ratios of width to height of images A, C, and D 
are proportional because the terms of these ratios change by the same  
factor. Let us see how.
Image A — 60 : 40
Multiplying both the terms by 
1
2
, we get
60 × 
1
2
 : 40 × 
1
2
 
which is 30 : 20, the ratio of width to height in image C.
By what factor should we multiply the ratio 60 : 40 (image A) to get 90 : 60 
(image D)?
A more systematic way to compare whether the ratios are proportional 
is to reduce them to their simplest form and see if these simplest forms 
are the same.
7.3 Ratios in their Simplest Form
We can reduce ratios to their simplest form by dividing the terms by 
their HCF.
In image A, the terms are 60 and 40. What is the HCF of 60 and 40? It 
is 20. Dividing the terms by 20, we get the ratio of image A to be 3 : 2 in 
its simplest form.
The ratio of image D is 90 : 60. Dividing both terms by 30 (HCF 
of 90 and 60), we get the simplest form to be 3 : 2. So the ratios of  
images A and D are proportional as well.
What is the simplest form of the ratios of images B and E?
The ratio of image B is 40 : 20; in its simplest form, it is 2 : 1.
The ratio of image E is 60 : 60; in its simplest form, it is 1 : 1.
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Ganita Prakash | Grade 8 
162
These ratios are not the same as 3 : 2. So, we can say that the ratios of 
width to height of images B and E are not proportional to the ratios of 
images A, C, and D.
When two ratios are the same in their simplest forms, we say that the 
ratios are in proportion, or that the ratios are proportional. We use 
the ‘::’ symbol to indicate that they are proportional.
So a : b :: c : d indicates that the ratios a : b and c : d are proportional.
Thus, 
60 : 40 :: 30 : 20 and 60 : 40 :: 90 : 60.
7.4 Problem Solving with Proportional Reasoning
Example 1: Are the ratios 3 : 4 and 72 : 96 proportional?
3 : 4 is already in its simplest form.
To find the simplest form of 72 : 96, we need to divide both terms by 
their HCF.
What is the HCF of 72 and 96?
The HCF of 72 and 96 is 24. Dividing both terms by 24, we get 3 : 4.
Since both ratios in their simplest form are the same, they are proportional.
Example 2: Kesang wanted to make lemonade for a celebration. She 
made 6 glasses of lemonade in a vessel and added 10 spoons of sugar to 
the drink. Her father expected more people to join the celebration. So he 
asked her to make 18 more glasses of lemonade. 
To make the lemonade with the same sweetness, how many spoons of 
sugar should she add? 
To maintain the same sweetness, the ratio of 
the number of glasses of lemonade to the number 
of spoons of sugar should be proportional. For 6 
glasses of lemonade, she added 10 spoons of sugar. 
The ratio of glasses of lemonade to spoons of 
sugar is 6 : 10. If she needs to make 18 more glasses 
of lemonade, how many spoons of sugar should 
she use? We can model this problem as —
6 : 10 :: 18 : ?
We know that each term in the ratio must change 
by the same factor, for the ratios to be proportional.
How can we find the factor of change in the ratio?
The first term has increased from 6 to 18. To find the factor of change, 
we can divide 18 by 6 to get 3.
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Proportional Reasoning-1
163
The second term should also change by the same factor. When 10 
increases by a factor of 3, it becomes 30. Thus,
6 : 10 :: 18 : 30.
So, she should use 30 spoons of sugar to make 18 glasses of lemonade 
with the same sweetness as earlier.
Example 3: Nitin and Hari were constructing a compound wall around 
their house. Nitin was building the longer side, 60 ft in length, and Hari was 
building the shorter side, 40 ft in length. Nitin used 3 bags of cement but 
Hari used only 2 bags of cement. Nitin was worried that the wall Hari built 
would not be as strong as the wall he built because she used less cement. 
Is Nitin correct in his thinking?
In Nitin and Hari’s case, we should compare the ratio of the length 
of the wall to the bags of cement used by each of them and see whether 
they are proportional.
The ratio in Nitin’s case is 60 : 3, i.e., 20 : 1 (in its simplest form).
The ratio in Hari’s case is 40 : 2, i.e., 20 : 1 (in its simplest form).
Since both ratios are proportional, the walls are equally strong. Nitin 
should not worry!
Example 4: In my school, there are 5 teachers and 170 students. The 
ratio of teachers to students in my school is 5 : 170. Count the number 
of teachers and students in your school. What is the ratio of teachers to 
students in your school? Write it below.
______ : ______
Is the teacher-to-student ratio in your school proportional to the one in 
my school?
Example 5: Measure the width and height (to the nearest cm) of the 
blackboard in your classroom. What is the ratio of width to height of the 
blackboard?
______ : ______
Can you draw a rectangle in your notebook whose width and height are 
proportional to the ratio of the blackboard?
Compare the rectangle you have drawn to those drawn by your 
classmates. Do they all look the same? 
Math 
Talk
Note to the Teacher: Give more such examples that students can relate to and ask 
them to give reasons why they think they are correct. Engaging with these problems 
and finding solutions through a process of proportional reasoning should go along 
with learning procedures and methods to solve the problems.
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