##
** ****Ratio and Proportion Concept**

**Ratio :**The comparison between two quantities in terms of magnitude is called ratio i.e, it tells that the one quantity is how many times the other quantity.

For example, Sunil has
750 rupees and Ramesh has 1400 rupees. It means that ratio of the money Sunil
and Ramesh 750 is to 1400. It can be expressed as 750: 1400

**Important note :**In a ratio , the order of terms is very important . For example, in the above example, the required ratio is 750: 1400 while 1400:750 is wrong .

So ratio of any two quantities is expressed as
a :b or

a
b

. The two quantities that are being compared are called terms.
In a:b , the first
term a is called antecedent and second term b is called consequent .

**Properties of Ratio :**

1.
A ratio is a number , so to find the ratio
of two quantities , they must be expressed in the same units (weight , length ,
volume or currency etc)

Example: We cannot compare 8 kg of rice and 3 toys.

2.
Ratio has no units

3.
A ratio does not change if all of its terms
are multiplied or divided by the same number, thus 2 : 3
= 4 : 6 = 6 :9

4.
The ratio of two fractions can be expressed
in ratio of integers.

Example: (

2
3

): (
5
3

) =
2
3

x
3
5

= 2 : 5
5.
The terms of the ratio are always expressed
is smaller values.

**Types of Ratios :**

1.

**Duplicate Ratio:**The ratio of the squares of two numbers is called the duplicate ratio of the two numbers or When the ratio is compounded with itself, it is called as duplicate ratio.
Duplicated Ratio of a : b => a

^{2}: b^{2}
The duplicate ratio of 3 : 4 is 3

^{2}: 4^{2 }= 9 :16
2.

**Sub Duplicate Ratio**: The ratio of the square roots of two numbers is called the sub-duplicate ratio of two numbers .
Duplicate Ratio of a : b => $ \sqrt{a}$
: $\sqrt{b}$

Duplicate Ratio of 16: 49 => 4 :7

3.

**Triplicate Ratios :**The ratio of the cubes of two numbers is called the triplicate ratio of the two numbers .
Triplicate Ratio of a : b => a

^{3}: b^{3}
The Triplicate ratio of 3 : 4 is 3

^{3}: 4^{3 }= 27 :64
4.

**Sub Triplicate Ratio :**The ratio of the cube roots of two numbers is called the sub-duplicate ratio of two numbers .
Sub - Triplicate Ratio of a : b =>Cuberoot

Sub Triplicate Ratio of 64: 125 => 4 :5

5.

**Inverse Ratio or Reciprocal Ratio:**If first term (antecedent) and second term (consequent) of a ratio interchange their places , the new ratio is called the inverse ratio of the given ratio .
If a :b be the given ratio , then

1
a

:
1
b

or b:a is
its inverse ratio
Example: If 3 : 5 is the ratio, then its inverse ratio
=> 5 : 3

6.

**Compound Ratio:**The ratio of the product of the first terms ( antecedent) to the product of the second terms (consequent) of two or more ratios is called the compound ratio.
Thus a:b , c:d
and e:f are three given ratios , then a x c x e : b x d x f is the compound
ratio of the given ratios.

Example:
The compound ratio of 2 : 3, 4 : 5 and
6:1 is

ð 2 x 4 x 6 : 3 x 5 x 1

ð 48
: 15

ð 16
: 5

**Proportion:**

The equality of two
ratios is called a proportion and we say that the four numbers are in
proportion.

If

a
b

=
c
d

, then
a,b,c and d are said to be in proportion
and we write a:b:: c:d . This is
read as a is to b as c is to d
Here all terms a,b,
c and d are called proportional’s . a,b,c and d respectively called first , second (mean), third and fourth proportional.

Here a,d are known
as extremes and b and c called means.

For example

3
5

=
6
10

, we write 3 :5 :: 6 : 10 and say
3,5 , 6 and 10 are in proportion .**Properties of Proportion :**

1) If
four numbers are in proportion then product of the extremes is equal to the
product of the means

If these are not in proportion, then product of
extremes is not equal to the product of means.

If a:b::c:d
=> a x d = b x c

2)

**Continued Proportion :**If a,b and c are three numbers such that a:b = b:c , then these numbers a,b and c are said to be in continued proportion
3)

**Fourth Proportion :**The fourth proportion of a, b and c is $ \frac{bc}{a}$
Example: What is the fourth proportion of 6,12 and 8

Here a =6 , b = 12 and c=8

Fourth proportion = $\frac{bc}{a}$=

12 x 8
6

=16
4)

**Third Proportion :**The third proportion of a and b is
b

^{2}a
Example: What is the third proportion of 12 and 6

Here a= 12 and b =6

Third proportion =

b

= ^{2}a
6

= 3
^{2}12
5)

**Mean Proportion :**The mean proportion of a and b is √ a x b
Example: Find the mean proportion of 81,144

A= 81 and b=144

Mean Proportion =
√ 81 x 144

**Variation:**

Two quantities a
and b are said to be varying with each other if there exists some relationship
between a and b.

**Direct Proportion:**

Two quantities are
said to be directly proportional, if the increase (or decrease) in one quantity
causes increase (or decrease) in the other quantity by same the proportion.

In other words, the
ratio of a and b is a constant. The statement b varies as a is written as b ∞ a
.

**Examples:**

I.
The price of articles varies directly to the
number of articles. More articles more cost and less articles less cost

II.
The word done varies directly to the number
of men at work. Fewer men at work, less work is done in same time. More men at
work, more done in same time.

**Inverse Proportion:**

Two quantities a
and b are said to be vary inversely if the increase (or decrease) in one
quantity causes decrease (or increase) in the other quantity by the same
proportion.

The statement b
varies inversely as a is symbolically written as b ∞

1
a

**Example**

I.
The price of an article is varies inversely
to the demand of the article. When the price of the articles goes up , the
demand for the article comes down

II.
The time taken to finish a work varies
inversely to the number of men at work

More men at work, less time taken to finish the work

Fewer
men at work, more time taken to finish the work.