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Ratio and Proportion 1

The document explains the concepts of ratio and proportion, highlighting the importance of comparison by division. It defines commensurable and incommensurable quantities, various types of ratios such as compound, duplicate, and reciprocal ratios, and introduces the concept of proportion among four quantities. Additionally, it outlines important properties of proportion, including Invertendo, Alternendo, Componendo, and Dividendo.

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Piupa Banerjee
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15 views3 pages

Ratio and Proportion 1

The document explains the concepts of ratio and proportion, highlighting the importance of comparison by division. It defines commensurable and incommensurable quantities, various types of ratios such as compound, duplicate, and reciprocal ratios, and introduces the concept of proportion among four quantities. Additionally, it outlines important properties of proportion, including Invertendo, Alternendo, Componendo, and Dividendo.

Uploaded by

Piupa Banerjee
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Ratio and Proportion

In certain situations, comparison by division makes better sense than comparison by taking the
difference. The comparison by division is called the ratio of the two numbers.
For example, for most mammals the ratio of legs to noses is 4:1, but for humans, the ratio of
legs to noses is 2:1.
Increase (or decrease) in a ratio
 If a quantity increases or decreases in the ratio a : b, then the new (resulting) quantity
𝑎
is times the original quantity.
𝑏
6
 For example if 20 is increased in the ratio 6 : 5, then the new quantity is 20 × = 24
5

Commensurable and incommensurable quantities


If the ratio between two quantities can be expressed exactly by the ratio of two integers, then
the two quantities are said to be commensurable, otherwise they are incommensurable.
 2.5 : 5.0 can be expressed as 1 : 2, hence the two quantities 2.5 and 5.0 are
commensurable.
 (2 – √2): 3 cannot be expressed as a ratio of integers. So 2–√2 and 3 are
incommensurable quantities.
Composition of ratios

When two or more ratios are For ratios a : b and c : d,


Compound
multiplied term wise, the ratio thus the compound ratio is
ratio
obtained is called compound ratio. (a × b) : (c × d)

Duplicate It is the compound ratio of two


𝑎2 : 𝑏2
ratio equal ratios.

Triplicate It is the compound ratio of three


𝑎3 : 𝑏3
ratio equal ratios.

1
Sub-duplicate It is the ratio of the square root of
√𝑎: √𝑏
ratio the terms of the original ratio.

Sub-triplicate It is the ratio of the cube root of the 3 3


√𝑎: √𝑏
ratio terms of the original ratio.

Reciprocal It is the ratio of the reciprocal of the 1 1


∶
ratio terms of the original ratio. 𝑎 𝑏

Proportion: Four non-zero quantities, a, b, c and d are said to be in proportion (or, are
proportional).
If a : b = c : d
 This is often expressed as a : b :: c : d
 It is read as "a is to b is the same as c is to d"
Continued proportion: Three non-zero quantities of the same kind and in the same unit are
said to be in continued proportion, if the ratio of the first to the second is the same as the ratio of
the second to the third.
 a, b, c are in continued proportion if a : b = b : c
 a, b, c, d, e, … are in continued proportion if a : b = b : c = c : d = d : e = …
Important properties of proportion

Invertendo a:b=c:d=b:a=d:c

Alternendo a:b=c:d=a:c=b:d

Componendo a:b=c:d=a+b:b=c+d:d

Dividendo a:b=c:d=a−b:b=c−d:d

2
Componendo and Dividendo a:b=c:d=a+b:a−b=c+d:c−d

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