NCERT Class 8 Mathematics Ganita Prakash Part 1 Chapter 07 Proportional Reasoning 1 PDF Download

Chapter 7: Proportional 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 -

[Figure: Five images of a tiger of different sizes and proportions, labeled A through E, See in your textbook]

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.

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?

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), \( \frac{1}{2} \) 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.

Teacher's Note

When two quantities are proportional, they change by the same factor (through multiplication), not by the same difference (through subtraction). Image A and Image B both have their dimensions reduced by 20 mm, but they do not look similar because the multiplication factor is different for width and height. Always check the factor, not the difference.

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 \( \frac{1}{2} \), we get
\( 60 \times \frac{1}{2} : 40 \times \frac{1}{2} \)
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.

Teacher's Note

To check if two ratios are proportional, always reduce both to their simplest form using the HCF. If the simplest forms are identical, the ratios are proportional. For example, 60 : 40 and 90 : 60 both reduce to 3 : 2, so they are proportional.

Key Points

• Two quantities are proportional when they change by the same factor through multiplication, not when they change by the same difference.
• A ratio \( a : b \) means for every \( a \) units of the first quantity, there are \( b \) units of the second quantity.
• To compare whether ratios are proportional, reduce them to their simplest form by dividing both terms by their HCF. Identical simplest forms mean the ratios are proportional.
• Images or shapes that have proportional widths and heights look similar even if they have different sizes.

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