** Ratio:** Correlation of any two values, from which it is realized that the second amount is the number of times of the first amount, is called the ratio. The ratio is expressed by the (:) sign, such as the ration of 5 meter and 7 meters can be written as 5:7 or ^{ 5}/_{7} .

Here the first term 5 of this ratio is called Antecedent and the second term 7 is called the consequent.

**Inverse or Reciprocal Ratio: **The proportion in which the second type of amount is decreased due to the increase of the first type.

**Compound Ratio: **The multiplication of antecedent and consequent of two or more terms of any rations are known as complex ratio

such as 2: 3, 4: 5 and 3: 5

= (2 × 4 × 3): (3 × 5 × 5) = 8: 25

** The quadratic ratio or the Duplicate Ratio: **The quadratic ratio of a: b: c is a^{2} : b^{2} : c^{2 }

example – What would be the quadratic ratio of 5: 7: 8?

Solution – The quadratic ratio of 5: 7: 8 = 5^{2} : 7^{ 2} : 8^{2} = 25: 49: 64

**Triplet ratio or concentration ratio**: The Triplet ratio of a: b: c is a^{3} : b^{3} : c^{3}.

Example – What is the triplet ratio of 3: 5: 4?

Solution – 3: 5: 4 concentration of

= 3^{3} : 5^{3} : 4^{3} = 27: 125: 64

Type-1

Example -1. If A: B = 4: 5 and B: C = 6: 7, then what will be the value of A: C?

Tricky Solution: A: C = 4 × 6: 5 × 7 = 24: 35 Answer.

Example-2. If Part B is ^{2}/_{3} of part A and part B is ^{3}/_{4} of Part C, then what will be the value of A:C?

Tricky Solution: A:C = 2 × 3:3 × 4 = 1:2 Answer.

[^{A}/_{B} = ^{2}/_{3} ⇒ A:B = 2:3 Similarly, B: C = 3: 4]

Example-3. If A:B = 7: 9 and B: C = 3: 5, then what will be the value of A:B:C?

Tricky Solution: A: B: C = (7: 9) × 3:9 × 5 = 21:27:45 = 7:9:15 Answer.

Type-2

Example-1. If A: B = 3: 4, B: C = 8: 9 and C: D = 12: 13, then what is the value of A: D ?

Tricky Solution: A: D = 3 × 8 × 12: 4 × 9 × 13 = 8: 13 Answer.

Example-2. If the value of A is ^{ 2}/_{5} of part B of The value of B is ^{2}/_{3} of Part C and the value of C is equal to ^{ 1}/4 of D, So what will be the value of A: D?

Tricky Solution: A: D = 2 × 2 × 1: 5 × 3 × 4 = 1: 15

Example-3. If A: B = 4: 5, B: C = 3: 4 and C: D = 6: 7, then what will be the value of A: B: C: D?

Tricky Solution: A: B: C = (4: 5) × 3: 5 × 4 = 12: 15: 20

A: B: C: D = (12:15 to 20) × 6: 20 × 7 = 36: 45: 60: 70 Answer.

Type-3

TRICK –

**Proportion: **When two proportions are equal in number, then that expression is called proportionately.

Example- a: b = c: d ⇒ a: b :: c: d

In this proportion, a, b, c and d are respectively called first, second, third, and fourth proportional. A and D are the external amounts and B and C are called Middle Amounts. Then

Product of external amounts (a × d) = Product of Middle Amount (b × c)

Type-4

**Continuous proportion: **In three Numbers a, b and c, The ratio of a and b is equal to the ratio of b and c, then this expression is called continuous proportion. Such as: a:b = b: c ⇒ a:b::b:c In the above expression, a, b and c are respectively called first, middle, and third proportion.

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