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Ratio :
Ratio is a comparision of two or more quantities of the same kind. It gives idea how many times the one quantity is greater or less than the other , it is obtained by division. If a and b are two quantities then the fraction a/b is called the ratio of a and b . It is written as a:b
The quantities a and b are called the terms of the ratio. a is called antecedent (first term) and b is called consequent (second term).

Properties of Ratios :

(1) Generally a ratio is expressed in reduced form.(which has no common multiplier other than 1.)

(2) The order of the terms in a ratio is important.

(3) Quantities to be compared must be in the same units .

(4) Two or more ratios may be compared by reducing them to the same denominator .

(5) The value of a ratio is unaltered if the antecedent and the consequent are divided or multiplied by the same quantity.

Important Points :

(1) b:a is called inverse ratio of a:b .

(2) A compounded ratio of two ratios a:b and u:v is au:bv (antecedent multiplied by antecedent and consequent multiplied by consequent.)

(3) A ratio a:b is said to be greater inequality or less inequality if a>b or a<b respectively .

(4) Duplicate ratio of a:b is a² : b² (A ratio compounded itself.)

(5) Sub-duplicate ratio of a:b is √a : √b .

(6) Triplicate ratio of a:b is a³:b³ .

(7) Sub= triplicate ratio of a:b is ∛a: ∛b .

(8) Continued ratio is the relation between the magnitudes of three or more quantities of the same kind . The continued ratio of three similar quantities a,b,c is represented by a:b:c.

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Proportion :

An equality of two or more ratios is called proportion . Four quantities a,b,c,d are said to be in proportion if
a:b = c:d or a:b :: c:d
⇒ = ⇔ ad = bc

Note 1 : First and last terms are called the extremes,(a and b) while the two middles terms b and c are called the means.

Note 2 : Four quantities (a,b,c and d) are in proportional if the product of extremes is equal to the product of the means .
= ⇔ ad = bc
ad (product of extermes), bc (product of means)

Note 3 : Three quantities a,b,c (of same kind) will be in continuous proportion if = ⇒ b² = ac ⇒ b = √ac

Properties of proportion (of four Quantities)

(i) = ⇔ = (is known Altenendo)

(ii) = ⇔ = (is known Invertendo).

(iii) = ⇔ = (is known Componendo).

(iv) = ⇔ = (is known Dividendo)

(v) = ⇔ = (is known Componendo Dividendo)

(vi) = ⇔ (is known Addendo)

(vii) = ⇔ (is known Subtranendo)