## Ratio, Proportion

### Ratio :-

Ratio is the comparison between two quantities in terms of their magnitudes. The ratio of two quantities is equivalent to a fraction that one quantity is of the other.
The ratio between p and q can be represented as p : q, where p is called antecedent and q is called consequent.

 p or p : q q

For example there can be a ratio between ₹ 300 and ₹ 500, but there cannot be the ratio between ₹ 200 and 900 apples. Hence, the unit of quantity for the comparison of ratio should be same.

### Types of Ratio :-

#### Duplicate Ratio :-

If two numbers are in ratio, then the ratio of their squares is called duplicate ratio. If p and q are two numbers, then the duplicate ratio of p and q would be p2 : q2
For example- Duplicate ratio of 7 : 8 = 72 : 82 = 49 : 64

#### Sub-duplicate Ratio :-

If two numbers are in ratio, then the ratio of their square roots is called sub-duplicate ratio. If p and q are two numbers, then the sub-duplicate ratio of p and q would be p : √q
For example- Sub-duplicate ratio of 9 : 16 = √9 : √16 = 3 : 4

#### Triplicate Ratio > :-

If two numbers are in ratio, then the ratio of their cubes is called triplicate ratio. If p and q are two numbers, then the triplicate ratio of p and q would be p3 : q3
For example Triplicate ratio of 4 : 5 = 43 : 53 = 64 : 125

#### Sub-triplicate Ratio :-

If two numbers are in the ratio, then the ratio of their cube roots is called sub-triplicate ratio. If p and q are two numbers, then the sub-triplicate ratio of p and q would be 3p : 3q
For example sub-triplicate ratio of 8 : 27 = 38 : 327 = 2 : 3

#### Inverse Ratio :-

If two numbers are in ratio, then their antecedent and consequent are interchanged and the ratio obtained is called inverse ratio. If p and q are two numbers and their ratio p : q, then its inverse ratio will be q : p.
For example Inverse ratio of 6 : 7 is 7 : 6

#### Compound Ratio :-

If two or more ratios are given, then the antecedent of one is multiplied with antecedent of other and respective consequents are also multiplied. If p : q , r : s and t : u are three ratios, then their compound ratio will be prt : qsu.
For example compound ratio of 2 : 3, 4 : 5 and 6 : 7 = 2 × 4 × 6 : 3 × 5 × 7 = 48 : 105

### Comparison of Ratios :-

Rule - If two ratios are given for comparison, convert each ratio in such a way that both ratios have same denominator, then compare their numerators, the fraction with greater numerator will be greater.

Ex- Find the greater ratio between 7 : 8 and 9 : 10.

 Solution:- 7 and 9 8 10

LCM of 8 and 10 = 40

 ⇒ ( 7 × 5 ) and ( 9 × 4 ) ( 8 × 5 ) ( 10 × 4 )

 ⇒ 35 and 36 40 40

 ⇒ 35 < 36 40 40

∴ 7 : 8 < 9 : 10

Rule - If two ratios are given for comparison, convert each ratio in such a way that both ratios have same numerator, then compare their denominators, the fraction with lesser denominator will be greater.

 Ex- Find the lest fraction between 4 and 5 5 9

 Solution:- 4 and 5 5 9

LCM of 4 and 5 = 20

 = ( 4 × 5 ) and ( 5 × 4 ) ( 5 × 5 ) ( 9 × 4 )

 = 20 and 20 25 36

 = 20 > 20 25 36

 ∴ 4 > 5 5 9

### Properties of Ratios :-

 1. P1 = P2 = P3 = ............. = ( P1 + P2 + P3 + …..) q1 q2 q3 ( q1 + q2 + q3 + …….)

If two or more ratios are equal, then the ratio whose numerator is the sum of the numerators of all the ratios and denominator is the sum of the denominators of all the ratios is equal to the original ratios.

 Since 25 = 5 40 8

 ∴ 25 = 5 40 8

 = ( 25 + 5 ) = 30 ( 40 + 8 ) 48

 2. P1 , P2 , P3 , ........... Pn are unequal ratios ( or fractions ), then ( P1 + P2 + P3 + …..+Pn ) q1 q2 q3 qn ( q1 + q2 + q3 + …….+ qn )
lies between the lowest and the highest of these ratios.

 3. If the ratio p > 1 and k is a positive number, then ( p + k ) < p and ( p - k ) > p q ( q + k ) q ( q - k ) q

 Similarly, if p < 1 , then ( p + k ) > p and ( p - k ) < p q ( q + k ) q ( q - k ) q

 4. if r > p ,then ( P + r ) > p s q ( q + s ) q

 and if r < p ,then ( P + r ) < p s q ( q + s ) q

Ex- Salaries of A and B are in the ratio of 3 : 4. If the salary of each one is increased by ₹ 5000 the new ratio becomes 7 : 8. What is A's present salary?
Solution:- Let the present salaries of A and B 3x and 4x respectively. according to the question,

 ∴ ( 3x + 5000 ) = 7 ( 4x + 5000 ) 8

⇒ 8 ( 3x + 5000 ) = 7 ( 4x + 5000 )
⇒ 24x + 40000 = 28x + 35000
⇒ 40000 - 35000 = 28x - 24x
⇒ 5000 = 4x

 ⇒ x = 5000 = 1250 4

∴ x = 1250
∴ A's present salary = 3x = 3 × 1250 = ₹3750

Ex- The ratio between the present ages of A and B is 4 : 5. After 10 years, the ratio between their ages will be 6 : 7. Find A's present age.
Solution:- Let the present ages of A and B 4x and 5x respectively.
according to the question,

 ∴ 4x + 10 = 6 5x + 10 7

⇒ 7 ( 4x + 10 ) = 6 ( 5x + 10 )
⇒ 28x + 70 = 30x + 60
⇒ 70 - 60 = 30x - 28x
⇒ 10 = 2x

 ⇒ x = 10 = 5 2

∴ A's present age = 4x = 4 × 5 = 20 years

### Proportion :-

When two ratios are equal, the four quantities composing them are said to be proportionals.

 Hence, if p = r or, p : q = r : s, then p, q, r, s are in proportional and is written as p : q :: r : s, q s

where symbol '::' represents proportion and it is read as 'p is to q' as 'r is to s'.
Here, p and s are called 'Extremes' and q and r are called 'Means'.

### Properties of Proportion :-

1.Invertendo :-

 If p = r ,   then q = s q s p r

2. Alternendo :-

 If p = r ,   then p = q q s r s

3. Componendo :-

 If p = r ,   then ( p + q ) = ( r + s ) q s q s

4. Dividendo :-

 If p = r ,   then ( p - q ) = ( r - s ) q s q s

5. Componendo and Dividendo :-

 If p = r ,  then ( p + q ) = ( r + s ) q s ( p - q ) ( r - s )

 Ex– Find the value of x + a +  x + b , If x = 2ab x – a x - b a + b

 Solution:- x = 2ab a + b

Dividing on both sides by a

By componendo and dividendo,

 ⇒ x = 2b a a + b
 ⇒ ( x + a ) = ( 2b + a + b ) ( x – a ) ( 2b – a – b )

 ⇒ ( x + a ) = ( 3b + a ) ……(1) ( x – a ) ( b – a )

 x = 2ab a + b

Dividing on both sides by b

 ⇒ x = 2a b ( a + b )
 ⇒ ( x + b ) = ( 2a + a + b ) ( x - b ) ( 2a – a – b )

 ⇒ ( x + b ) = ( 3a + b ) ..........( 2 ) ( x - b ) ( a – b )

On adding Eq. (1) and Eq. (2), we get

 ⇒ ( x + a ) + ( x + b ) = ( 3b + a ) + ( 3a + b ) ( x + a ) ( x - b ) ( b – a ) ( a – b )

 = -( 3b + a ) + ( 3a + b ) ( a – b ) ( a – b )

 = ( -3b – a + 3a + b ) ( a – b )

 = ( 2a – 2b ) ( a – b )

 = 2( a – b ) = 2 ( a – b )

### Basic Rules of Proportion :-

Rule- If p : q :: q : r, then r is called third proportional to p and q, which are in continued proportion. r will be calculated as
p : q :: q : r
or, p : q = q : r
or, p × r = q × q
or, p × r = q2

 ⇒ r = q2 p

Ex- Calculate the 3rd proportional to 15 and 30.
Solution:- Let 3rd proportional be x.
15 : 30 :: 30 : x

 ⇒ 15 = 30 30 x

 ⇒ x = ( 30 × 30 ) = 60 15

∴ x = 60

Rule- If p : q :: r : s, then s is called the 4th proportional to p, q and r, s will be calculated as
p : q :: r : s
or, p : q = r : s
or, p × s = q × r

 ∴ s = ( q × r ) p

Ex- Find the 4th proportional to 4, 16 and 7.
Solution:- Let the 4th proportional be x.
4 : 16 :: 7 : x
or, 4 : 16 = 7 : x
or, 4 × x = 16 × 7

 ⇒ x = ( 16 × 7 ) = 28 4

∴ x = 28

Rule- If p : x :: x : q, then x is called the mean proportional between p and q, x will be calculated as
p : x :: x : q
or, p : x = x : q
or, x × x = p × q
or, x2 = pq

∴ x = √ pq

Ex- What will be mean proportional between 36 and 49?
Solution:- Let mean proportional be x.
36 : x :: x : 49
or, 36 : x = x : 49
or, x × x = 36 × 49
or, x2 = 36 × 49
∴ x = √ 36 × 49 = 6 × 7 = 42

Trick- If P : Q = p : q and Q : R = m : n, then P : Q : R = pm : qm : qn and P : R = pm : qn

Ex- If P : Q = 8 : 15 and Q : R = 3 : 2 then find P : Q : R.
Solution:- P : Q : R = ( 8 × 3 ) : ( 15 × 3 ) : ( 15 × 2 )
= 24 : 45 : 30
dividing the ratio by 3
= 8 : 15 : 10

Ex- If P : Q = 8 : 15, Q : R = 5 : 8 and R : S = 4 : 5, then find P : S

 Solution:- P = P × Q × R S Q R S

 = 8 = 5 × 4 = 4 15 8 5 15

Trick- If, A : B = a : b, B : C = c : d and C : D = e : f, then A : B : C : D = ace : bce : bde : bdf

Ex- The ratio of A : B = 2 : 3, B : C = 5 : 7 and C : D = 3 : 10. Find the value of A : B : C : D.
Solution:- A : B = 2 : 3, B : C = 5 : 7, C : D = 3 : 10
A : B : C : D = ( 2 × 5 × 3 ) : ( 3 × 5 × 3 ) : ( 3 × 7 × 3 ) : ( 3 × 7 × 10) = 30 : 45 : 63 : 210
= 10 : 15 : 21 : 70

 Trick- If x is divided in p : q, then 1st part = px and 2nd part = qx ( p + q ) ( p + q )

Ex- Divide 132 in the ratio 5 : 6.
Solution:- Let the 1st part be 5x and 2nd part be 6x.
according to the question,
5x + 6x = 132
or, 11x = 132
or, x = 132/11
∴ x = 12
1st part = 5x = 5 × 12 = 60
By trick

 1st part = 5 × 132 = 5 × 132 = 60 ( 5 + 6 ) 11

 2nd part = 6 × 132 = 6 × 132 = 72 ( 5 + 6 ) 11

 Trick- If x is divided in p : q : r, then 1st part = px , 2nd part = qx and 3rd part = qx ( p + q + r ) ( p + q + r ) ( p + q + r )

Ex- Divide 13257 in the ratio 2 : 3 : 4.
Solution:- Let the 1st part 2x, 2nd part = 3x and 3rd part = 4x
according to the question,
2x + 3x + 4x = 13257
or, 9x = 13257

 ⇒ x = 13257 = 1473 9

∴ x = 1473
1st part = 2x = 2 × 1473 = 2946
2nd part = 3x = 3 × 1473 = 4419
3rd part = 4x = 4 × 1473 = 5892
By trick

 1st part = 2 × 13257 = 2 × 13257 = 2 × 1473 = 2946 ( 2 + 3 + 4 ) 9

 2nd part = 3 × 13257 = 3 × 13257 = 3 × 1473 = 4419 ( 2 + 3 + 4 ) 9

 3rd part = 4 × 13257 = 4 × 13257 = 4 × 1473 = 5892 ( 2 + 3 + 4 ) 9

Trick- The incomes of two persons are in the ratio p : q and their expenditures are in the ratio r : s. If each of them saves ₹ x, then their incomes are given by

 Income of first person = x ( s - r ) × p ( ps - qr )

 Income of second person = x ( s - r ) × q ( ps - qr )

and their expenditures are given by

 Expenditure of first person = x ( q - r ) × r ( ps - qr )

 Expenditure of second person = x ( q - r ) × s ( ps - qr )

Ex- The ratio of incomes of A and B is 4 : 5 and ratio of their expenditures is 3 : 4. If each person saves ₹ 2000, then find their incomes and expenditure.
Solution:- Let the income of A be ₹ 4x and that of B be ₹ 5x. Expenditures of A = ₹ ( 4x - 2000 )
Expenditures of B = ₹ ( 5x - 2000 )
according to the question,

 ⇒ ( 4x - 2000 ) = 3 ( 5x - 2000 ) 4

⇒ 4 ( 4x - 2000 ) = 3 ( 5x - 2000 )
⇒ 16x - 8000 = 15x - 6000
⇒ 16x - 15x = 8000 - 6000
∴ x = 2000
Income of A = 4x = 4 × 2000 = ₹ 8000
Income of B = 5x = 5 × 2000 = ₹ 10000
Expenditure of A = 4x - 2000 = ₹ 6000
Expenditure of B = 5x - 2000 = ₹ 8000
By Trick
Here, p = 4 , q = 5 , r = 3 , s = 4 and x = ₹ 2000

 Income of A = x ( s - r ) × p = 2000 ( 4 - 3 ) × 4 = 2000 × 4 = ₹ 8000 ( ps - qr ) ( 16 - 15 )

 Income of B = x ( s - r ) × q = 2000 ( 4 - 3 ) × 5 = 2000 × 5 = ₹ 10000 ( ps - qr ) ( 16 - 15 )

 Expenditure of A = x ( q - p ) × r = 2000 ( 5 - 4 ) × 3 = ₹ 6000 ( ps - qr ) ( 16 - 15 )

 Expenditure of B = x ( q - p ) × r = 2000 ( 5 - 4 ) × 4 = ₹ 8000 ( ps - qr ) ( 16 - 15 )

Trick- If two numbers are in ratio p : q and x is added to the numbers, then the ratio becomes r : s. The two numbers will be

 First number = xp ( r - s ) ( ps - qr )

And

 Second number = xq ( r - s ) ( ps - qr )

Ex- Two numbers are in the ratio of 3 : 4. If 20 is added to both the numbers, then the ratio becomes 7 : 8. Find the smallest number.
Solution:- Let the numbers be 3x and 4x.
according to the question,

 ( 3x + 20 ) = 7 ( 4x + 20 ) 8

or, 8 ( 3x + 20 ) = 7 ( 4x + 20 )
or, 24x + 160 = 28x + 140
or, 160 - 140 = 28x - 24x
or, 20 = 4x

 ⇒ x = 20 = 5 4

∴ smallest number = 3x = 3 × 5 = 15
By trick
Here, p = 3 , q = 4 , r = 7, s = 8 and x = 20

 First number = xp ( r - s ) = ( 20 × 3) ( 7 - 8 ) ( ps - qr ) ( 24 - 28 )

 = ( 60 × -1) = 15 -4

 Second number = xq ( r - s ) = ( 20 × 4) ( 7 - 8 ) ( ps - qr ) ( 24 - 28 )

 = ( 80 × -1) = 20 -4

∴ smallest number = 15

Trick- If two numbers are in ratio p : q and x is subtracted from the numbers, then the ratio becomes r : s. The two numbers will be

 First number = xp ( s - r ) ( ps - qr )

And

 Second number = xq ( s - r ) ( ps - qr )

Ex- Two numbers are in the ratio of 6 : 7. If 10 is subtracted from each, the ratio becomes 4 : 5. Find the greater number.
Solution:- Let the numbers 6x and 7x.
according to the question,

 ( 6x - 10 ) = 4 ( 7x - 10 ) 5

⇒ 5 ( 6x - 10 ) = 4 ( 7x - 10 )
or, 30x - 50 = 28x - 40
or, 30x - 28x = 50 - 40
or, 2x = 10

 ∴ x = 10 = 5 2

∴ greater number = 7x = 7 × 5 = 35
By trick
Here, p = 6 , q = 7 , r = 4 , s = 5 and x = 10

 First number = xp ( s - r ) = 10 × 6 ( 5 - 4 ) ( ps - qr ) ( 30 - 28 )

 = ( 60 × 1 ) = 30 2

 Second number = xq ( s - r ) = 10 × 7 ( 5 - 4 ) ( ps - qr ) ( 30 - 28 )

 = ( 70 × 1 ) = 35 2

∴ greater number = 35