UPSKILL MATH PLUS
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Learn moreObserving Similar Images
We often resize digital images. We may make an image smaller or larger, but we usually want it to look like the original image.
Two images can have different sizes and still look similar.
Images that look similar have their corresponding dimensions changed by the same factor.
Observe carefully:
- Images \(A\), \(C\) and \(D\) look similar even though their sizes are different.
- Images \(B\) and \(E\) look different because they appear stretched or compressed.
Why?
| Image | Dimensions | Change | Multiplication Factor | Result |
| Image \(A\) | \(120\ W\) & \(80\ H\) | Original image | Original | - |
| Image \(B\) | \(120\ W\) & \(60\ H\) | \(120\ W\) \(\rightarrow\) \(120\ W\) \(80\ H\) \(\rightarrow\) \(60\ W\) |
\(W\) - \(1\) \(H\) - \(\frac{1}{2}\) |
Distorted |
| Image \(C\) | \(60\ W\) & \(40\ H\) | \(120\ W\) \(\rightarrow\) \(60\ W\) \(80\ H\) \(\rightarrow\) \(40\ W\) |
\(W\) - \(\frac{1}{2}\) \(H\) - \(\frac{1}{2}\) |
Similar |
| Image \(D\) | \(180\ W\) & \(120\ H\) | \(120\ W\) \(\rightarrow\) \(180\ W\) \(80\ H\) \(\rightarrow\) \(120\ W\) |
\(W\) - \(\frac{3}{2}\) \(H\) - \(\frac{3}{2}\) |
Similar |
| Image \(E\) | \(60\ W\) & \(60\ H\) | \(120\ W\) \(\rightarrow\) \(60\ W\) \(80\ H\) \(\rightarrow\) \(600\ W\) |
\(W\) - \(\frac{1}{2}\) \(H\) - \(\frac{3}{4}\) |
Distorted |
What is Ratio?
A ratio compares two quantities of the same kind. It is written using the sybol \(" : "\).
The ratio of two quantities \(a\) and \(b\) are termed as \(a:b\), where \(a\) is called the first term or antecedent and \(b\) is called the second term or consequent.
Any ratio \(a:b\) can be pronounced as \(a\) is to \(b\).
It also can be written as \(a\) to \(b\) or \(a/b\).
Important!
The order of quantities matters.
For example:
\(7:5\) \(\neq 5:7\).
If \(7:5\) represents boys \(:\) girls, then \(5:7\) represents girls \(:\) boys.
Equivalent Ratios:
In the above images, images \(A\), \(C\) and \(D\) multiplied by the same factor \(\frac{3}{2}\).
The ratios multiplied by the same factor are called an equivalent ratios.
For example, \(25 : 15\) and \(20 : 12\) are equal ratios since both have same factor \(\frac{5}{3}\).
The ratios which are not multiplied by the same factor are called not equivalent ratios.
Ratios in Their Simplest Form:
Sometimes, different ratios represent the same comparison. To compare them easily, we write each ratio in its simplest form.
A ratio is said to be in its simplest form when its two terms have no common factor other than \(1\).
How to Find the Simplest Form?
- Find the Highest Common Factor (HCF) of both terms.
- Divide both terms of the ratio by the HCF.
Examples
| Ratio | HCF | Simplest form |
| \(60 : 40\) | \(20\) | \(3 : 2\) |
| \(30 : 20\) | \(10\) | \(3 : 2\) |
| \(45 : 15\) | \(15\) | \(3 : 1\) |
Proportional Ratios
When two ratios have the same simplest form, they are said to be in proportion or proportional.
We use the symbol \(::\) to show proportion.
\(a : b :: c : d\)
It is read as, "\(a\) is to \(b\) as \(c\) is to \(d\)".
How to check whether two ratios are proportional?
- Reduce both ratios to their simplest forms.
-
Compare the simplest forms.
- If the simplest forms are the same, the ratios are proportional.
- If the simplest forms are different, the ratios are not proportional.
Examples
| Two Ratios | Simplest form | Proportional or not |
| \(60 : 40\) & \(30 :20\) | \(3:2\) & \(3:2\) | Yes. They are proportional |
| \(30 : 20\) & \(90 : 60\) | \(3:2\) & \(3:2\) | Yes. They are proportional |
| \(40 : 20\) & \(90 : 60\) | \(2:1\) & \(3:2\) | No. They are not proportional |
| \(40:20\) & \(60:60\) | \(2:1\) & \(1:1\) | No. They are not proportional |
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