RATIO AND PROPORTION

BASIC AND IMPORTANT CONCEPTS:

1. The ratio of two quantities of the same kind is the fractions that one

Quantity is of the other. The ratio of ‘a’ to ‘b’ is usually written as a : b.

The quantities ‘a’ and ‘b’ are called the terms of the ratio.

The first term is called the Antecedent and the second term is called

Consequent.

2. Ratio has no unit and can be written as a : b. the second term ‘b’ can not

be zero.

3. If both the terms of a ratio are multiplied or divided by the same non-

zero

Number, the ratio remains unchanged.

4. A ratio must be expressed in its lowest terms. the ratio is in its lowest

terms

if the H.C.F. of its both the terms is unity (1).

e.g.- ratio 5 :8 is in its lowest terms as the H.C.F. of its terms 5 and 8 is

1.

The ratio 5 :20 is not in its lowest terms as the H.C.F. of its terms 5 and

20

is 5 and not 1.

5. Ratios a : b and b : a cannot be equal unless a = b

i.e.- a : b b : a unless a = b.

1

Composition of Ratios:

Compound Ratio :

When two or more ratios are multiplied term wise, the ratio obtained

is

called compound ratio.

e.g.- compound ratio of a : b and c : d is ac : bd

compound ratio of a : b , c : d and e : f is ace : bdf

Duplicate Ratio:

it is the compound ratio of two equal ratios.

e.g.- duplicate ratio of a : b = a2 : b2

so, duplicate ratio of 3 : 4 = 32 : 42

= 9 : 16

Triplicate Ratio:

it is the compound ratio of three equal ratios.

e.g.- triplicate ratio of a : b = a3 : b3

so, triplicate ratio of 3 : 4 = 33 : 43

= 27 : 64

Sub – duplicate Ratio:

Sub – duplicate ratio of a : b is

e.g.- Sub – duplicate ratio of 25 : 49 =

= 5 : 7

Sub – triplicate Ratio:

Sub – triplicate ratio of a : b is

2

e.g.- Sub – triplicate ratio of 64 : 125 =

= 4 : 5

Reciprocal Ratio:

Reciprocal ratio of a : b is or b : a

e.g.- reciprocal ratio of 5 : 7 = = 7 : 5

PROBLEMS:

QUESTION1:

a) Find Ratio between 35 min and 1 hrs.

b) If 3A = 5 B = 6 C . Find A : B : C

SOLUTION:

a) 1 hrs = 60 = 105 min [ 1 hour = 60 min ]

Ratio between 35 min and 1 hrs.

= Ratio between 35 min 105 min

= 35 : 105

= 1 : 3

b) If 3A = 5B = 6C . Find A : B : C

3

Let 3A = 5B = 6C = K

then , 3 A = K

A =

5 B = K

B =

6 C = K

C =

A : B : C = : :

A : B : C = : : [L . C . M. of 3, 5 and 6 is 30]

= 30 : 30 : 30

Hence, A : B : C = 10 : 6 : 5

Example2:

If A : B = 6 : 5 and B : C = 11 : 9 find i) A : C ii) A : B : C

If a : b = 4 : 5 find the value of

Solution:

i) and

4

ii) A : B = 6 : 5 , B : C = 11: 9

A : B = 6 : 5 , B : C = 1 :

A : B = 6 : 5 , B : C = 1 :

A : B = 6 : 5 , B : C = 5 :

A : B : C = 6 : 5 :

A : B : C = 66 : 55 : 45

Let a : b = 4 : 5

If a = 4x then , b = 5x

=

=

=

=

=

Example3:

If = Find

5

SOLUTION:

= (cross multiply)

3(a + b) = 2(a + 3b)

3a + 3b = 2a + 6b

a = 3b

=

Let b = x , then a = 3x

=

=

=

=

QUESTION4:

a.) If x = = Find x : y

SOLUTION:

a.) x = =

= ( Using componendo and Dividendo)

13

=

6

1

x : y = 13 : 1

The work done by ( x – 3 ) men in ( 2x + 1 ) days and the work done by (2x + 1 ) men in ( x – 4 ) days are in the ratio 3 : 10 find the value of ‘ x ’.

SOLUTION:

Amount of work done by ( x – 3 ) men in ( 2x + 1 ) days = ( x – 3 ) ( 2x + 1 ) units of

work

Similarly , Amount of work done by ( 2x + 1 ) men in ( x – 4 ) days = ( 2x + 1 )(x – 4)

units

of work

As per given conditions:

=

=

= ( Cross Multiply.)

10 ( 2x2 – 5x – 3 ) = 3 ( 2x2 – 7x – 4 )

20x2 – 50x – 30 = 6x2 – 21x – 12

20x2 – 6x2 – 50 + 21x – 30 + 12 = 0

14x2 – 29x – 18 = 0

14x2 + 7x – 36x – 18 = 0 (splitting middle term.)

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7x ( 2x + 1 ) – 18 ( 2x + 1 ) = 0

( 2x + 1 ) ( 7x – 18 ) = 0

2x + 1 = 0 Or 7x – 18 = 0

2x = –1 Or 7x = 18

x = Or x =

Negative value cannot be possible in this case.

x = = 2.57

b.) If $ 680 be divided among A, B and C in such a way that A gets of what B

gets and B gets of what C gets . Find their respective shares of money.

SOLUTION:

Let C Get $ x

B Gets of what C Gets

then,

B Gets $

A Gets of what B Gets

A Gets X = = $

= 680

C gets $ x = $ 480

8

B gets $

A gets $

Example7:

Two numbers are in the ratio 2 : 3 . if 5 is added to each

number then

new ratio becomes 5 : 7. find the numbers.

What are the terms which differ by 50 and the measure of

which is ?

Solution:

Let the common multiple be x.

So, the required numbers be 2x and 3x.

As per given condition:

=

35 – 25 =

required numbers are 2x =

9

3x = 3

What are the terms which differ by 50 and the measure of

which is ?

Solution:

The measure of the terms is

So, the ratio of the terms is 2 : 7

Let the common multiple be x .

So, the required numbers be 2x and 7x.

The terms differ by 50

As per given condition:

7

Example9:

Find the

Duplicate ratio of 3 : 4

Triplicate ratio of 4 : 5

Sub duplicate ratio of 16 : 25

Sub triplicate ratio of 27 : 125

Solution:

10

Duplicate ratio of 3 : 4 = 32 : 42 = 9 : 16

Triplicate ratio of 4 : 5 = 43 : 53 = 64 : 125

Sub duplicate ratio of 16 : 25 = = 4 : 5

Sub triplicate ratio of 27 : 125 =

= 3 : 5

Example10:

If (2x+5) : (8x+9) is the duplicate ratio of 3 : 5 find the value of x.

If 4x+9 = 9x -10 is the triplicate ratio of 4 : 5 find x.

Solution:

duplicate ratio of 3 : 5 = 9 : 25

duplicate ratio of 3 : 5 = (2x+5) : (8x+9) (given)

(2x+5) : (8x+9) = 9 : 25

= (cross multiply)

If 4x + 4 = 9x -10 is the triplicate ratio of 4 : 5 find x.

(cross multiply)

11

1140 =

Example11:

Compare the ratios 14 : 23 , 5 : 12 and 61 : 92 . Arrange the ratios

in ascending order.

Solution:

L.C.M. of 23 , 12 and 92 is = 23 X 12 = 276

115 < 168 < 183

Proportion:

When two ratios are equal , the four terms involved taken in order are

called

Proportions . a , b , c and d are said to be in proportion if a : b = c : d

The terms ‘a’ and ‘d’ are called extremes and ‘b’ and ‘c’ are called the

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means.

This is often expressed as a : b :: c : d and say as “ a is to b as c is to d”.

a : b = c : d i.e. i.e. a

product of extremes = product of means

In a : b = c : d , the fourth term ‘d’ is called the fourth proportional to a

, b , c.

Continued proportion:

Four quantities are said to be in continued proportion if the ratio of the

first to the second quanity is the same as the ratio of the second to the

third and third to the fourth.

Thus, are said to be in continued proportion.

If a , b and c are in continued proportion then or b2 = ac

here ‘b’ is called mean proportion between a and c and ‘c’ is called

third proportion to a and b.

Example12:

Find x (i) 1 : 4 : : x : 9 (ii) x : 2.5 : : 5 : 7

Solution:

(i) 1 : 4 : : x : 9

(cross multiply)

13

(ii) x : 3.5 : : 5 : 7

(cross multiply)

Example13:

Find the fourth proportional to

2.5 , 7.5 and 6.5 (ii) (), ()

and ()

Solution:

Let the fourth proportional to 2.5 , 7.5 and 6.5 be x.

(cross multiply)

2.5

Let the fourth proportional to (), ()

and () be x.

14

(cross multiply)

(), ()

and () is (

Example14:

Find the third proportional to

20 and 30 (ii)

Solution:

Let the third proportional be x.

, 30 and x are in continued proportion

(cross multiply)

15

Let the third proportional be x.

then , and x are in continued proportion.

(cross multiply)

Example16:

if a , b and c are in continued proportion , prove that :

a2 – b2 : a2 + b2 = a – c : a + c .

Solution:

a , b and c are in continued proportion .

then , b2 = ac .

proved.

16

Example17:

Two numbers are in the ratio 2 : 3 . if 5 is added to each number

then

new ratio becomes 5 : 7.find the numbers.

What are the terms which differ by 40 and the measure of

which is ?

Solution:

Let the number be x.

The required number be 2x and 3x.

As per given condition:

= (cross multiply)

35 – 25 =

required numbers are 2x =

3x = 3

What are the terms which differ by 50 and the measure of

which is ?

Solution:

The measure of the terms is

17

So, the ratio of the terms is 2 : 7

Let the common multiple be x .

So, the required numbers be 2x and 7x.

The terms differ by 50

As per given condition:

7

Example18:

a.) If a : b = c : d , show that:

4a + 3b : 4a – 3b = 4c + 3d : 4c – 3d.

b.) Find two numbers such that the mean proportional

between

them is 14 and the third proportional to them is 112.

Solution:

a : b = c : d then

and dividendo

18

4a + 3b : 4a – 3b = 4c + 3d : 4c – 3d

Let the required numbers be x and y.Then, 14 is mean proportional between x and y.

x : 14 = 14 : y and xy = 14 = 196 ……… (1)

112 is third proportional to x and y

x : y = y : 112 and y2 = 112x …….. (2)

from eq. (1) , xy = 196 and x =

from eq. (2) , y2 = 112 substituting value of x

then, y3 = 112 196

y

= 2

x =

hence, the required numbers are 7 and 28 .

1. If a, b, c and d are in proportion, show that

=

SOLUTION:

a, b, c and d are in proportion

19

=

Let = = k

a = bk , c = dk

L. H. S =

=

=

=

= k

R. H. S =

=

=

=

= k

L. H. S = R. H. S

=

2. If a, b, c and d are in proportion, show that

=

20

SOLUTION:

a, b, c and d are in proportion

Let = = k

Then, a = bk , c = dk

L. H. S =

=

=

=

=

R. H. S =

=

=

21

=

=

L. H. S = R. H. S

=

5. If a, b and c are in continued proportion,

Prove that: a2 b2 c2 ( a-3 + b-3 + c-3 ) = a3 + b3 + c3 . (Use k – method)

SOLUTION:

=

Let = = k

b = ck and a = bk = (ck )k = ck2

L. H. S = a2 b2 c2 ( a-3 + b-3 + c-3 )

= (ck2)2 (ck)2 c2 [ ( ck2)-3 + ( ck) -3 + c-3 ]

= c6 k6

= ( k6 + k3 + 1 )

= c3 ( 1 + k3 + k6 )

22

R. H. S = a3 + b3 + c3

= (ck2)3 + (ck)3 + c3

= c3 k6 + c3 k3 + c3

= c3 (k6 + k3 + 1)

L. H. S = R. H. S

a2 b2 c2( a-3 + b-3 + c-3 ) = a3 + b3 + c3

1.

If a : b : : c : d then b : a : : d : c

Proof: a : b : : c : d

=

= {∵ Reciprocals of equals are equal}

b : a : : d : c

2.

23

If a : b : : c : d then a : c : : b : d

Proof: a : b : : c : d

=

= {∵ Reciprocals of equals are equal}

a : c : : b : d

3.

If a : b : : c : d then ( a + b ) : b : : ( c + d ) : d

Proof: a : b : : c : d

=

+ 1 = + 1 {Adding 1 to both sides}

=

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

24

4.

If a : b : : c : d then ( a - b ) : b : : ( c - d ) : d

Proof: a : b : : c : d

=

- 1 = - 1

=

(a - b ) : b : : (c - d ) : d

5.

If a : b : : c : d then ( a + b ) : ( a - b ) : : ( c + d ) : ( c - d )

Proof: a : b : : c : d

=

= (I) (By Componendo)

And = (II) (By Dividendo)

= divide (I) by (II)

( a + b ) : ( a - b ) : : ( c + d ) : ( c - d )

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6.

If = = then, each ratio is equal to

Proof: Let = = = k

Then a = bk , c = dk , e = fk

= = = k

Hence, each ratio =

1. If a : b = 3 : 5 find the value of

i) b : a ii) 7b : 2a

SOLUTION:

i) =

= {By Invertendo}

b : a = 5 : 3

ii) =

X = X by multiplying both sides

=

26

7b : 2a = 35 : 6

2. If = find the value of

SOLUTION:

=

= (Cubing both sides)

=

=

3. If = find the value of

i) ii) iii)

SOLUTION:

i) =

27

= (Cubing both side)

ii)

=

= (By componendo)

=

iii)

=

= (By Componendo)

= (By Dividendo)

4. If = find the value of

SOLUTION:

28

=

X = X {Multiplying on both sides}

= =

=

=

5. If 7x + y = 2x + 8x + 8y find the value of

SOLUTION:

7x + y = 2x + 8x + 8y

7x - 2x = 8y - y

5x = 7y

=

= (Squaring both sides)

= (By Componendo & Dividendo)

29

=

6. If = find the value of

SOLUTION:

=

= ( by Dividendo)

=

SUMMMARY AND KEY POINTS

1. The ratio of two quantities of the same kind is the fractions that one

Quantity is of the other. The ratio of ‘a’ to ‘b’ is usually written as a : b.

30

The quantities ‘a’ and ‘b’ are called the terms of the ratio.

The first term is called the Antecedent and the second term is called

Consequent.

2. Ratio has no unit and can be written as a : b. the second term ‘b’ can not

be zero .

3. If both the terms of a ratio are multiplied or divided by the same non-

zero

Number, the ratio remains unchanged.

4. A ratio must be expressed in its lowest terms. the ratio is in its lowest

terms

if the H.C.F. of its both the terms is unity (1).

e.g.- ratio 5 : 8 is in its lowest terms as the H.C.F. of its terms 5 and 8 is

1.

The ratio 5 : 20 is not in its lowest terms as the H.C.F. of its terms 5

and 20

is 5 and not 1.

5. Ratios a : b and b : a cannot be equal unless a = b

i.e.- a : b b : a unless a = b.

Composition of Ratios:

Compound Ratio :

When two or more ratios are multiplied term wise, the ratio obtained

is

31

called compound ratio.

e.g.- compound ratio of a : b and c : d is ac : bd

compound ratio of a : b , c : d and e : f is ace : bdf

Duplicate Ratio:

it is the compound ratio of two equal ratios.

e.g.- duplicate ratio of a : b = a2 : b2

so, duplicate ratio of 3 : 4 = 32 : 42

= 9 : 16

Triplicate Ratio:

it is the compound ratio of three equal ratios.

e.g.- triplicate ratio of a : b = a3 : b3

so, triplicate ratio of 3 : 4 = 33 : 43

= 27 : 64

Sub – duplicate Ratio:

Sub – duplicate ratio of a : b is

e.g.- Sub – duplicate ratio of 25 : 49 =

= 5 : 7

Sub – triplicate Ratio:

Sub – triplicate ratio of a : b is

e.g.- Sub – triplicate ratio of 64 : 125 =

= 4 : 5

Reciprocal Ratio:

32

Reciprocal ratio of a : b is or b : a

e.g.- reciprocal ratio of 5 : 7 = = 7 : 5

Proportion:

When two ratios are equal , the four terms involved taken in order are

called

Proportions . a , b , c and d are said to be in proportion if a : b = c : d

The terms ‘a’ and ‘d’ are called extremes and ‘b’ and ‘c’ are called the

means.

This is often expressed as a : b :: c : d and say as “ a is to b as c is to d”.

a : b = c : d i.e. i.e. a

Product of Extremes = Product of Means

In a : b = c : d , the fourth term ‘d’ is called the fourth proportional to a

, b , c.

Continued proportion:

Four quantities are said to be in continued proportion if the ratio of the

first to the second quanity is the same as the ratio of the second to the

third and third to the fourth.

Thus, are said to be in continued proportion.

If a , b and c are in continued proportion then or b2 = ac

here ‘b’ is called mean proportion between a and c and ‘c’ is called

33

third proportion to a and b.

Properties of Ratio:

Invertendo:

If a : b : : c : d then b : a : : d : c

2. Alternendo:

If a : b : : c : d then a : c : : b : d

3.

If a : b : : c : d then ( a + b ) : b : : ( c + d ) : d

4.

If a : b : : c : d then ( a - b ) : b : : ( c - d ) : d

5.

If a : b : : c : d then ( a + b ) : ( a - b ) : : ( c + d ) : ( c - d )

6.

If = = then each ratio is equal to

34