PROPORTIONAL
7 REASONING-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 —
Image A Image B Image C
Image D Image E
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.
Reprint 2026-27
, Ganita Prakash | Grade 8
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! Math
Why does it look different from the other rectangular images? Talk
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), 12 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.
160
Reprint 2026-27
, Proportional Reasoning-1
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 12, we get
60 × 12 : 40 × 12
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.
161
Reprint 2026-27
7 REASONING-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 —
Image A Image B Image C
Image D Image E
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.
Reprint 2026-27
, Ganita Prakash | Grade 8
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! Math
Why does it look different from the other rectangular images? Talk
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), 12 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.
160
Reprint 2026-27
, Proportional Reasoning-1
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 12, we get
60 × 12 : 40 × 12
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.
161
Reprint 2026-27
