Ratio and Proportion With solved examples

Ratio: Ratio is a comparison of two quantities by division. It is a relation that one quantity bears to another concerning magnitude. It represents as a:b or a/b which means a part of b. Here, a is called antecedent, and b is called the consequent.


Types of Ratio:

• Compound Ratio: When two or more ratios are multiplied with each other then it is called compound ratio i.e ab: bc is a compound ratio if a:b and c:d are two given ratios.

• Duplicate Ratio:  square of any ratio is known as the Duplicate ratio. 
Duplicate ratio of a:b = a²:b²

• Triplicate Ratio: Cube of any ratio.
Triplicate ratio of a: b = a³:b³

• Sub Duplicate Ratio: Square root of any ratio.
Sub Duplicate ratio of a:b = a^1/2:b^1/2

• Sub Triplicate Ratio: Cube root of any ratio.
Sub Triplicate ratio of a:b = a^1/3:b^1/3 

• b:a is inverse ratio if a:b is given ratio.

Properties of Ratio:

1. In a ratio, qualities are compared, the qualities must be of the same kind.
2. The ratio of two quantities determines how many times are quantity is contained by the other.

Important Facts:

1. If A:B = x:y, B:C = p:q.

To find A: C

A/B × B/C = A/C or A: C

To find A:B: C,
A:B:C =  xp: py: yq 

2. If A:B = X:y, B:C = p:q  and C:D  = m:n, then
To find,
 A:D = xpm : yqn

To find A:B: C:D,
A:B:C:D = xpm : ypm: yqm: yqn

Ex.1 Find the ratio compounded of the four ratios 4:3,9:13,26:5 and 2:15.

Sol. the required ratio = (4*9*26*2)/(3*13*5*15) = 16/25

Ex.2 If A:B = 3:4, B:C = 8:10, find A:B:C.

Sol. A:B = 3:4

            

           C:D =8:10

A:B:C = 3*8 : 4*8: 4*10 = 24:32:40 = 3:4:5 

Ex 3. If A:B = 3:4,B:C = 8:10 and C:D = 15:17.

Sol. A:B = 3:4

            

           B: C=8:10

               

                C:D=15:17

A:B:C:D = 3*8*15 : 4*8*15 : 4*10*15: 4*10*17

                    

                   9:12:15:17 Ans.

Proportion:

If four quantities in proportion, the product of the extremes are equal to the product of the means.

i. Third proportion:

If a:b:: b:c then c is called the third proportion.

it is calculated as

a/b = b/c 

then c = b² / a

ii. Fourth proportion:

if a:b:: c:d then d is called the fourth proportion.

it is calculated as :

a:b :: c:d

a/b=c/d

d= bc /a 

iii. Mean proportion:

If a:b:: b:c, then b is called mean proportion,

it is calculated as:

a/b = b/c

b²= ab

b = √ab

iv. Continued proportion:

Three quantities a,b,c of the same kind are said to be in continued proportion. when

 a:b :: b:c,
The middle number b is said to be a mean proportion to two extreme numbers.

it is calculated as:

a/b = b/c

b²= ab is called continued proportion

Componendo and Dividendo:

If a:b is equal to c:d

a/b = c/d

i. Componendo Rule:

(a+b)/b = (c+d)/d

ii. Dividendo Rule:

(a-b)/b = (c-d)/d

iii. Componendo and Dividendo:

(a+b)/(a-b) = (c+b)/(c-d)

Direct proportion: The two given quantities are so related to each other that if one of them is multiplied by any number, the other is also multiplied by the same number.

Ex. If 5 balls cost 10 Rs. what do 15 balls cost?

Sol. if the number of balls is increased by 2,3, times, the price will also be increased 2,3 times.

5 balls : 15 balls :: Rs. 10: required cost

the required cost = (15*10)/5 = 30 Rs.

Inverse proportion: The two quantities are so related that if one of them is multiplied by any number, the other is divided by the same number and vice versa.

Ex. If 15 men can reap a field in 28 days, in how many days will 10 men reap it?

Sol. Now the number of man increase n times then the number of days will decrease n times.

1/15: 1/10::28: the required number of days

or 10:15:: 28: the required number of days

number of days = (15*28)/10 = 42

The Rule of Three: The method of finding the 4th term of a proportion when the other three are given Simple proportion or the Rule of Three.

Rule:

1. Denote the quantity to be found by the letter 'x', and set it down as the 4th term.

2. Of the three given quantities set down that for the third term which is of the same kind as the quantity to be found.

3.Now, check whether the quantity to be found will be greater or less than the third term; if greater, make the greater of the two remaining quantities the 2nd term and the other 1st term, but if less, make the less quantity the second tern, and the greater the first term.

4. The required value = multiplication of means/ 1st term

Ex. If 15 men can reap a field in 28 days, in how many days will 5 men reap it?

Sol. Step 1. _:_::_: the required number of days

Step 2. _:_::28:x

Step 3: the required number of days will be more since 5 men will take more time than 15 min. Therefore, 5:15 = 28:x

Step 4. x = (15*28/5) = 84 days

Method 2.

Another method Cane be used is Man-day-Hour formula

M1 D1 = M2 D2

15*28 = 5*x

x = (15*28)/5

X= 84

Compound Proportion or Double Rule of Three:

Ex. If 30 men working 7 hrs a day can do a piece of work in 18 days, in how many days will 21 men working 8 hrs a day do the same piece of work?

Sol. Method 1

        21 men : 30 men 

                                               :: 18 days : the required number of days

        8 hrs:    7 hrs      

Multiplication of means/ Multiplication of 1st terms = (80*36*30)/21*8 = 22.5 days

Method 2

M1D1H1 = M2D2H2

30*18*7 = 21*8*x

x = (30*18*7)/21*8

x= 22.5 days

Proportional Division:

Proportion can be applied to divide a given quantity into parts

Ex. Divide 1350 Rs. into three shares proportional to the numbers 2,3,4

Sol. 1st share = 1350 * 2/(2+3+4)= 1350 * 2/9 = 300 Rs.

2nd share  = Rs. 1350 *3/9 = Rs. 450

3rd share = Rs. 1350 *4/9= Rs. 600 

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