**Ratio**

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

B:C=2:7=(2x2:7x2)=4:14

A:B:C=3:4:14

**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

10(2x+10)=3(9x+10)

7x=70

x=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 **

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.

x²=ab

x=√ab

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)

=(7x9):(5x9):(5x11)

= 63: 45: 55

**Variation **

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.

**Example**

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

h=8

**Practice Questions**