**Percentages - Concept**

The
word ‘Percent ‘is abbreviated form of the Latin word per centum which means per
hundred or hundredths.

Thus
the term percent means per hundred or for every hundred. The symbol ‘% ‘ is
used for the term percent. Thus 52 percent is written as 52% and it means that ‘52
out of 100 ‘.

**Definition :**A percent is a fraction whose denominator is 100 and the numerator of the fraction is called the rate percent.

**Conversion of a fraction into percentage:**

To convert a fraction
into percentages , multiply the fraction by 100 and put % sign.

To
convert

a
b

into percentage ,
a
b

x 100%**Example :**

7
10

=(
7
10

)x 100% = 70%
1
7

= (
1
7

) x 100% = 14
2
7

%**Conversion of a percentage into fraction :**

To convert a percent into fraction, divide it by 100
and remove % sign

To convert a% into fraction .

a
100

**Example :**32% =

32
100

=
8
25

16

2
3

% =
50
3

% =
50
300

=
1
6

**Conversion of a ratio into a percentage:**

To convert a ratio into a percentage, first convert the
given ratio into a fraction then to a
percentage

To
convert a: b into percentages , a:b =

a
b

=
a
b

x 100 % **Example :**Convert 1 : 4 into percentage

1
:4 => ¼ x 100 = 25%

**Conversion of a percentage into ratio :**

To convert a percentage into a ratio, first convert the
given percentage into fraction in simplest form and then to a ratio.

Example : 40%
=

40
100

=
2
5

=> 2 :5**Conversion of a decimal into a percentage:**

To convert a decimal into percentage ,first change int
to fraction for the removal of decimal and then multiply it by 100 and put %
sign.

**Example :**3.2 =

32
10

= (
32
10

x 100)% =32%**Percentage – Fraction conversion Table**

100% =1 50% =

1
2

33
1
3

%
=
1
3

25%=
1
4

20%=

1
5

16
2
3

% =
1
6

14
2
7

% =
2
7

12 ½% =
1
8

11

1
9

%=
1
9

10%=
1
10

9
1
11

%=
1
11

8
1
3

% =
1
12

6

2
3

% =
1
15

6
¼% =
1
16

7
9
13

%=
1
13

66
2
3

% =
2
3

**To find a% of b:**

a
100

x b**Example :**30% of 300 =>

30
100

x 300 = 90**Example 2 :**Raju scored 68% marks. Total marks were 400.How much did he score?

Explanation
: Marks scored =
68 % of 400

=

68
100

x 400
= 272

**To find one value is how much percentage of another value:**

*To a is how percentage of b =>*

a
b

x 100%**Example :**28 is what percentage of 56 ?

Explanation
:

To know 28 is what percentage of 56

=>
To know 28 is what percentage of 56

28
56

x 100
= 50%

**Example 2 :**Ramesh obtained 480 marks out of 600 and Suresh obtained 560 marks out of 800.Whose performance is better?

Explanation:

Percentage marks of Ramesh =

Percentage marks of Ramesh =

480
600

x
100 % = 80%
Percentage marks of Suresh =

560
600

x 100 = 70%
So, obviously Ramesh’s performance is better than that
of Suresh even though getting less absolute marks

**Percentage increase or decrease in a value :**

**Percentage change in value ={**
Change in value
Original Value

}
x 100%**Example :**The total expenses of Sharma were Rs 8000 per month . Now his expenses come down by Rs 1000. Find the percentage decrease in expense of Sharma?

Explanation
:

Percentage decrease =

Percentage decrease =

Decrease in Salary
Initial Salary

x 100 %
=

1000
8000

X 100% = 12.5%**Example 2 :**The marks of Sudheer are increased from 450 to 500. What is the percentage increase in his marks?

Explanation: Increase in marks = 500 -450 = 50

Percentage increase =

increase
Inital Marks

x 100 %
=

50
450

x 100%
=11.11 %**Important Formula 1:**

**I****f a value n is increased by x% , then the new value ( value after increase ) is**

**N x**
100+x
100

**Example :**The price of sugar is Rs 2000. If its price is increase by 15% , find its new price ?

Explanation:
Present price = Rs 2000

The percentage increase = 15%

New price = Rs 2000
x

(100+15)
100

= Rs 2300**Important Formula 2:**

**If a value n is decreased by x% , then the new value ( value after decrease ) is**

**N x**
(100 - x )
100

**Example :**The price of sugar is Rs 2000. If its price is decrease by 15% , find its new price ?

Explanation:
Present price = Rs 2000

The percentage decrease = 15%

New price =
Rs 2000 x

(100-15)
100

= Rs 1700**Important Formula 3:**

**If a is x% more than b, the b is less than by a by**
x
(100 + x)

x 100%**Example :**The salary of A is 20% more than the salary of B. Then by what percentage the salary of B is less than A ?

Explanation : Salary of A is 20% more than salary of B.

Then B’s
salary is

20
(100 +20)

x 100% =
20
120

x 100%
= 16.66% less than A

**Important Formula 4:**

**If a is x% less than b, the b is more than by a by**
x
100-x

x 100%**Example :**The salary of A is 20% less than the salary of B. Then by what percentage the salary of B is more than A ?

Explanation : Salary of A is 20% more than salary of B.

Then B’s
salary is (

20
100-20

) x 100% =
20
80

x 100%
= 25% more than A

**Important Formula 5:**

**If the value of a number is first increase by x% and later decrease by x% , then the net effect is always a decrease which is equal to x% of x and is written as**
x

% ^{2}100**Example:**The price of an article is increase by 20% , later decreased by 20%.What is the overall percentage change in its price ?

Explanation
: The price is
first increased by 20% and again decreased by 20% , then overall percentage
change in its price is => -(

20

) = - 4%^{2}100
4% decrease

**Important Formula 6:**

**If the value of a number is first changed ( increased or decreased) by x% and then changed (increased or decreased ) by y% ,**

**then net change % = { ± x ± y +**
±x±y
100

} %**Example :**The price of an article is first increased by 20% , later decreased by 10%. What is the total % change in its price ?

Explanation : In the above
formula, we take + sign for increase and – sign for decrease

If a value
is first increased by x% and later decreased by y% , then overall percentage
change = (x – y –

xy
100

) %
= { 20 – 10 –

(20 x 10)
100

} %
= 8%

**Important Formula 7:**

**If the population of a town is P and in increases ( or decreases ) at the rate of R% per annum , then**

*i)**Population , after n years = $P \times \left[1+\frac{R}{100}\right] ^{N}$*

*ii)**Population , n years ago = $ \frac{P}{\left[1+\frac{R}{100}\right]^{N}} $***Example :**The population of a village has been increasing at the rate of 10% every year . If the present population is 6000, what will its population after 2 years ?

Explanation
:

The population after 2 years = 6000 x ( 1 +

The population after 2 years = 6000 x ( 1 +

10
100

)^{2}
= 6000 x

100
100

x
110
100

= 7260

**Example 2 :**The price of an article has been decreasing at the rate of 10% every year. If its present price is Rs 8100 , What was its price 2 years ago ?

Explanation :
The price 2 years ago was = Rs $ \frac{8000}{(1-\frac{10}{100})^{2}}$

= Rs 8100 x

100
90

x
100
90

= Rs 10000

**Important Formula 7:**

**If the price of a commodity increases or decreases by x% , then the decrease or increase in consumption so as not to increase or decrease the expenditure is**

**(**
x
100±x

) x 100% **Example 1**: If food prices go up by 20% , by how much should a person reduce his consumption so as not to increase his expenditure ?

Explanation : Price is increased by 20%. To keep the
same expenditure as before, the percentage decrease in his consumption
=>

20
120

x 100 %
=>16.66%

**Example 2 :**If price of kerosene be reduced by 20% , find by how much percent a house holder must raise his consumption of kerosene so that not to increase his expenditure?

Explanation :
Price of kerosene is reduced by 20%

The percentage increase in his consumption to keep the
expenditure constant ,

=.

= 25%
20
80

x 100%**To practice the problems on percentages with different kinds of difficulties , and online tests**

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