Ratio and Proportion: Formula and Tricks with Problem (Question) and Solutions

Ratio

The comparative relation between two quantities of same type is called ratio.
The ratio between a and b is denoted by a:b. Here, a is called antecedent and b is called consequent.

Type of Ratio 

Ex: If a:b=7:9 and b:c = 15:7, then a:c=

(a) 5:3
(b) 7:21
(c) 3:5
(d) 7:15

Proportion

If a,b,c and d are such that if a :b and c:d are same, then a,b,c and d are in proportion and is denoted by a:b::c:d. Here a and d are known as extremes and b and c are known as means
a x d= b x c

Basic Operations on Proportion

If a,b,c and d are in proportion, then

(i) Invertendo
b : a :: d : c

(ii) Alternendo
a : c :: b : d

(iii) Componendo
(a + b) : b :: (c + d) : d

(iv) Dividendo
(a – b) : b :: (c – d) : d

(v) Componendo and Dividendo
(a + b) : (a – b) :: (c + d) : (c – d)

Important Formulae

Formula 1

If A:B = a:b and B :C =m:n, then A: B:C is

equal to am: mb: nb

Ex: If a:b =5:14 and b:c=7:3, then find a:b::c.
(a) 2:3:5
(b) 8:7:15
(c) 6:18:7
(d) 5:14:6

Sol: a:b:c=(5×7):(7×14):(14 x 3)

[dividing the ratio by 7]

= 5:14: (3 x 2) = 5:14:6

Formula 2

If A: B =a:b, B:C = m:n and C : D = r :s, then
A:B:C:D is equal to amr: pmr: bnr : bns

Ex: The ratio of A: B=1:3, B:C = 2:5 and C:D = 2:3.

Find the value of A: B:C:D.

(a) 7:16:9:40
(c) 8:11:6:7
(b) 4:12:30: 45
(d) 5:7: 3:2

Sol. (b) A:B =1:3,B:C =2:5, C 😀 =2:3

A:B:C:D = (1x2x2): (3 x2 x2): (3 x5 x2): (3 x5 x3)

= 4:12:30:45

Formula 3

Formula 4

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