Ratio Proportion and Variation


The ratio is the relationship between the quantities of the same kind. In ratio, the quantities are compared as the multiple or parts of other quantities. If the ratio is a : b then a is called antecedent and b is called the consequent. The ratio of the quantities is expressed after removing the common factor between the quantities. Generally, the ratio is useful in comparison. If the 

quantities A & B are compared & their ratio comes out  as p:q 

We can say A/B=p/q

or A & B can be expressed as pK, qk respectively. 

This provides a scope of comparison in terms of multiples of p & q.

If a ratio is given as a : b & a quantity x is added in both antecedents. & consequent then


 $$ \frac{a+x}{b+x}>\frac{a}{b}\rightarrow if a<b  .... (i) $$

$$ \frac{a+x}{b+x}<\frac{a}{b}\rightarrow if a>b   .... (Ii)$$

$$ \frac{a+x}{b+x}=\frac{a}{b}\rightarrow if a=b $$


(i) and (ii) was considering x = +ve 

If x = -ve (i) and (ii) inequalities will reverse. 

➤ If two quantities A & B are in the ratio p : q then 


$$\frac{a}{b}=\frac{p}{q}\Rightarrow A=pK, B=qK$$

So  A+B = (p+q)K 

So  \( \frac{A}{A+B} =\frac{p}{p+q} \)

Or \( A = \frac{p}{p+q} \left ( A+B \right ) \)

So A will be \(\frac{p}{p+q}\) part of the total sum of the quantities similarly B will be \( \frac{q}{p+q} \) part of the total sum of the quantities.

Types of Ratios

Duplicate ratio a² : b² is called the duplicate ratio of a : b.

Triplicate ratio a²: b² is called the triplicate ratio of a : b.

Sub-duplicate ratio √a : √b is called the sub-duplicate ratio of a : b.

Sub-triplicate ratio  ³√a : ³√b is called the sub-triplicate ratio of a : b.

Compound ratio ab : cd is the compound ratio of a:c and b : d. It is the ratio of the products of the antecedents to that of the consequences of two or more given ratios.

Inverse ratio ¹⁄ₐ :¹⁄ₑ is the inverse ratio of a : e.

Example 1

If A:B=3:4 and B:C=2:7, Find A:B:C, Find A:B:C.

Solution: A:B=3:4



Example 2

Two numbers are in the ratio of 2:9. If 10 is added to each, they are in the ratio of 3:10. Find the number?

Solution: (2x+10)/(9x+10)=3/10




Two numbers are 20 and 90

Example 3

The ratio of two quantities A & B is 4: 9 what is the triplicate ratio of sub-duplicate ratio of A & B. 

Solution: The sub-duplicate ratio of A& B is √4:√9 = 2:3

The triplicate ratio of 2 : 3 is 2³ : 3³ = 8 : 27 


Proportion is the comparison of two equal ratios when the two ratios are equal then all the quantities comprising the ratios are called in proportion. Example: a/b = c/d then a, b, c & d are in proportion. This is represented as a:b::c:d

Continued Proportion

If three quantities a,b and c are such that a:b::b:c, then b²=ac and a, b and c are in continued proportion. Also, the quantity c is called the third proportion of a and b.

Fourth Proportion

If four quantities a, b, c, and x are such that a:b::c:x, then ax=bc and x is called the fourth proportion of a band c.

Mean or Second Proportion

If three quantities a, b and x are such that a:x::x:b, then  x²=ab and x is called the mean of a and b.



So Mean proportion is the geometric mean of the two quantities. Here the quantities b and o will be in ‘continued proportion’ similarly if a, b, ¢ d are in continued proportion then 

a/b = b/c = c/d

or in another way, we can say that if the terms are in G.P. then they will be in continued proportion. 

If a:b = c:d, then following properties are:

b:a=d:c    (Invertendo)

a:c = b:d   (Alter nendo)

(a+b):b = (c+d):d  (Componendo)

(a-b):b = (c-d):d  (Dividendo)

(a+b):(a-c) = (c+d):(c-d)  (Componendo-Dividendo {alertInfo}

Example 1

Find a fourth proportional to the numbers 60, 48, and 30. 

Solution: 60:48:38:x =24

Example 2

If A:B=7:5 and B:C =9: 11, thenA:B:C is equal to 

Solution: n1 = 7 n2 = 9, d1= S and d2=11.

A:B:C =(n1xn2) : (d1xn2) : (d1 x d2)


= 63: 45: 55


If two quantities x and y are related in such a way that as the quantity x change it also brings a change in the second quality y, then two quantities are in variation.

Let us assume there are two quantities A & B if any change is done ‘A’ & that change results in the change in B then the quantities are called in variation. One thing has to be remembered the change we are counting is multiple changes,s not the differential change i.e. If a quantity is 70 and then it becomes 80 then the change will not be taken as (+10). It will be taken as 80/70 = 4/7

Direct Variation

The quantity x is in direct variation to y if an increase in x makes y increase proportionally. Also, a decrease in x makes y decrease proportionally. It can be expressed as x = ky, where k is called the constant of proportionality.

e.g ., the cost is directly proportional to the number of articles bought.

Inverse Variation

The quantity x is an inverse variation to y if an increase in x makes y decrease proportionally. Also, a decrease in x makes y increase proportionally. It can be expressed as k

x =k/y, where k is called the constant of proportionality.

e.g. The time taken by a vehicle in covering a certain distance is inversely proportional to the speed of the vehicle.

Joint Variation

If there are more than two quantities x, y, and z, and x varies with both y and z, then x is in joint variation to y and z. It can be expressed as x=kyz, where k is the constant of proportionality.

E.g men do work some number of days working certain hours a day.


lf 4 examiners can examine a certain number of answer books in 8 days by working 5 hours a day, for how many hours a day would 2 examiners have to work in order to examine twice the number of answer books in 20 days. 

Solution: Man. Day. Hour = constant = 4.8.5= 160

Next –> Man. day. hour = 160 x 2 (twice as earlier)

2.20.h=160x 2


Practice Questions