This MCQ module is based on: Similarity and Ratios
TOPIC 20 OF 22
Similarity and Ratios
🎓 Class 8
Mathematics
CBSE
Theory
Ch 7 — Playing with Polynomials
⏱ ~35 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: Similarity and Ratios
Targeting Class 8 level in Algebra, with Basic difficulty.
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7.1 Observing Similarity in Change
We are all familiar with digital images. We often change their size and orientation to suit our needs. Observe the set of five tiger images below — Image A, Image B, Image C, Image D and Image E.
Fig 7.1 — Five images of a tiger at different sizes and shapes
We can see that all the images are of different sizes.
Q. Which images look similar and which ones look different?
Images A, C, and D look similar?, even though they have different sizes.
Images A, C, and D look similar?, even though they have different sizes.
Q. 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.
No, they are slightly distorted. The tiger appears elongated in B and compressed and fatter in E.
Q. Why? You may notice that images A, C, and D are rectangular, but E is square. B is also a rectangle but looks different — its orientation has flipped from the other rectangular images.
Can we observe any pattern to answer this question? Perhaps by measuring the widths and heights of the rectangles holding each image.
| Image | Width (in mm) | Height (in mm) |
|---|---|---|
| Image A | 60 | 40 |
| Image B | 40 | 40 |
| Image C | 30 | 20 |
| Image D | 90 | 60 |
| Image E | 60 | 60 |
Q. 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 of image A have changed by the same factor (namely, \(\frac{1}{2}\)) to obtain image C. Since the widths and heights have changed by the same factor, the images look similar.
When we compare image A with image D, we notice that the width of D is \(\frac{90}{60}=\frac{3}{2}\) times that of A. The height is also \(\frac{60}{40}=\frac{3}{2}\) times that of A. Again, both measures have changed by the same factor — so the images look similar.
Now compare A with B. The height of B is the same as A (40 mm), but the width has been reduced by the factor \(\frac{40}{60}=\frac{2}{3}\). Since width and height did not change by the same factor, image B looks different. Similarly for image E, the width has been kept the same as A but the height has changed — so E looks different.
Key Idea — Proportional Change
Images A, C, and D look similar because their widths and heights have changed by the same factor. We say that the changes in width and height are proportional?.
Activity: Scaling Pictures on a Grid
L3 Apply
Materials: Squared (graph) paper, ruler, pencil, a simple picture (e.g., a house or a kite)
Predict: If you double both the width and the height of a drawing, will the new drawing look like the original or distorted? What if you double only the width?
- Draw a simple picture on a grid — say, a house fitting in a 4 × 3 cell box.
- Make a copy where every measurement (width and height) is doubled (8 × 6 cells). Sketch the same house inside.
- Make another copy where only the width is doubled (8 × 3 cells). Draw the house inside.
- Compare the three figures side by side.
Observation: The first copy (both doubled) looks like a bigger version of the original — the shape is preserved. The second copy (only width doubled) looks stretched. The first change is proportional; the second is not. This is the core idea of similarity.
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.
Definition — Ratio
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 \(\frac12\), we get \(60 \times \frac12 : 40 \times \frac12 = 30:20\), which is the ratio of width to height in image C.
Q. By what factor should we multiply the ratio 60 : 40 (image A) to get 90 : 60 (image D)?
Multiply both terms by \(\frac{3}{2}\): \(60 \times \frac32 : 40 \times \frac32 = 90:60\) ✓
Multiply both terms by \(\frac{3}{2}\): \(60 \times \frac32 : 40 \times \frac32 = 90:60\) ✓
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. The ratios of images A and D are proportional as well.
Q. What is the simplest form of the ratios of images B and E?
The ratio of image B is 40 : 40; in simplest form, it is 1 : 1.
The ratio of image E is 60 : 60; in simplest form, it is also 1 : 1.
The ratio of image B is 40 : 40; in simplest form, it is 1 : 1.
The ratio of image E is 60 : 60; in simplest form, it is also 1 : 1.
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.
Definition — Proportion
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.
Fig 7.2 — Rectangles A, C and D all share the same width : height ratio 3 : 2
7.4 Problem Solving with Proportional Reasoning
Example 1: Testing Proportionality
Are the ratios 3 : 4 and 72 : 96 proportional?
Solution:
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.
HCF of 72 and 96 = 24. Dividing, we get \(72 \div 24 : 96 \div 24 = 3 : 4\).
Since both ratios in simplest form are the same, they are proportional. So \(3:4 :: 72:96\). ✓
To find the simplest form of 72 : 96, we need to divide both terms by their HCF.
HCF of 72 and 96 = 24. Dividing, we get \(72 \div 24 : 96 \div 24 = 3 : 4\).
Since both ratios in simplest form are the same, they are proportional. So \(3:4 :: 72:96\). ✓
Example 2: Kesang's Lemonade
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.
Q. 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, she added 10 spoons of sugar. The ratio of glasses of sugar is 6 : 10. If she needs to make 18 more glasses of lemonade, how many spoons of sugar should she use? We model this as
6 : 10 :: 18 : ?
We know that each term in the ratio must change by the same factor, for the ratios to be proportional.
The first term has increased from 6 to 18. To find the factor of change, we can divide 18 by 6 to get 3.
The second term should also change by the same factor. When 10 increases by a factor of 3, it becomes \(10 \times 3 = 30\). So the ratio becomes 6 : 10 :: 18 : 30. Kesang should use 30 spoons of sugar.
Fig 7.3 — Scaling both terms by the same factor keeps ratios proportional
Example 3: Nitin's Building
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.
Q. Is Nitin correct in his thinking?
Solution:
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 simplest form).
The ratio in Hari's case is 40 : 2, i.e., 20 : 1 (in simplest form).
Since both ratios are proportional, the walls are equally strong. Nitin should not worry!
The ratio in Nitin's case is 60 : 3, i.e., 20 : 1 (in simplest form).
The ratio in Hari's case is 40 : 2, i.e., 20 : 1 (in simplest form).
Since both ratios are proportional, the walls are equally strong. Nitin should not worry!
Competency-Based Questions
Scenario: A photo-printing shop offers four standard print sizes: 4 in × 6 in (small), 5 in × 7 in (medium), 8 in × 12 in (large), and 10 in × 14 in (poster). A customer wants to enlarge a small print without distortion.
Q1. Which of the four sizes is proportional (similar in shape) to the 4 × 6 small print?
L3 ApplyAnswer: (b) 8 × 12. 4 : 6 simplifies to 2 : 3. Check each option: 5 : 7 (already simplest, not 2:3), 8 : 12 = 2 : 3 ✓, 10 : 14 = 5 : 7. Only 8 × 12 is proportional.
Q2. The customer prints the 4 × 6 photo onto the 5 × 7 frame by stretching it to fit. Analyse what visual distortion will occur.
L4 AnalyseAnswer: Width scales by \(\frac{5}{4}=1.25\) but height scales by \(\frac{7}{6}\approx 1.17\). Since the two factors are different, the change is not proportional. The image will appear horizontally stretched relative to the original — faces will look slightly wider and flatter. Similar to the distortion seen between tiger images A and E in Fig 7.1.
Q3. A tailor claims: "My paper pattern at scale 6 : 9 and your fabric at scale 12 : 18 are in the same proportion, so the dress will fit." Evaluate whether the tailor's claim is correct.
L5 EvaluateAnswer: Simplify 6 : 9 → divide by HCF 3 → 2 : 3. Simplify 12 : 18 → divide by HCF 6 → 2 : 3. Both ratios are 2 : 3 in simplest form, so \(6:9::12:18\). The tailor's claim is correct. The fabric is just a proportional enlargement of the pattern (factor of 2).
Q4. Design three banner sizes for a school event so that all three banners look similar. The smallest banner is 30 cm × 20 cm. Give the next two sizes and the scale factor for each. Justify using ratios in simplest form.
L6 CreateSample design: Smallest 30 × 20 → ratio 3 : 2. Medium: scale factor 2 → 60 × 40 (still 3 : 2). Large: scale factor 3 → 90 × 60 (still 3 : 2). All three banners have the same ratio 3 : 2 in simplest form, so they are in proportion and appear similar — a bigger version of the smaller one. Many answers possible as long as the simplified ratio is 3 : 2.
Assertion–Reason Questions
Assertion (A): The ratios 4 : 6 and 10 : 15 are proportional.
Reason (R): Both ratios reduce to 2 : 3 in their simplest form.
Reason (R): Both ratios reduce to 2 : 3 in their simplest form.
Answer: (a) — 4:6 ÷2 = 2:3, 10:15 ÷5 = 2:3. Both true; reducing to the same simplest form is precisely what proves proportionality.
Assertion (A): Tiger images A (60 × 40) and E (60 × 60) look similar.
Reason (R): Both images share the same width of 60 mm.
Reason (R): Both images share the same width of 60 mm.
Answer: (d) — A is false: A has ratio 3 : 2 while E has ratio 1 : 1. They are not proportional and so look different. R is true (both do have width 60) but a common width alone does not make images similar.
Assertion (A): If two rectangles have the same width-to-height ratio in simplest form, they are similar in shape.
Reason (R): Rectangles are similar when both their width and height change by the same factor.
Reason (R): Rectangles are similar when both their width and height change by the same factor.
Answer: (a) — Both true. Equal simplest-form ratios mean one rectangle is obtained from the other by multiplying both sides by the same factor, which is the definition of proportional change / similarity.
Frequently Asked Questions
What makes two figures similar?
Two figures are similar if their corresponding angles are equal and the ratios of their corresponding sides are the same. This means one figure is a scaled version of the other.
What is a scale factor?
A scale factor is the ratio of the length of any side of one figure to the corresponding side of a similar figure. If the scale factor is 2, every side of the larger figure is twice the corresponding side of the smaller.
Are all squares similar?
Yes. Every square has four 90-degree angles and four equal sides. Any two squares differ only by a constant scale factor.
Are all rectangles similar?
No. Although all rectangles have four right angles, their length-to-width ratios can differ, so their shapes need not match. Only rectangles with the same side ratio are similar.
How is similarity used in real life?
Similarity is used in maps, photographs, architectural scale models, screen resolutions and shadow problems to relate small and large versions of the same shape.
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