p a g e study notes for all competative...

1 | P a g e

STUDY NOTES FOR ALL COMPETATIVE EXAM

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CONCEPT OF PERCENTAGE

Percentages are special types of fractions in

which the denominator is always hundred.

e.g.

7

100

It is written as 10% or its simplified form is

1

10

PERCENTAGE INCREASE /

DECREASE

If the price of a commodity increases by a%,

then the reduction in consumption so as not

to increase the expenditure is:

100 %100

a

a

If the price of a commodity decreases by a%,

then the increase in consumption so as not to

decrease the expenditure is:

100 %100

a

a

Results on Population:

Populations after n year = 1

100

nr

p

Populations before n years = 1

100

n

p

r

If the value of a number is first increased by

a% and later decreased by a%. Then the net

effect is always decreased which is equal to

a% of a and is written is

2

%100

a

SOME IMPORTANT FRACTION

VALUES

1 100%

150%

2

1 133 % 33.33%

3 3

125%

4

120%

5

or

1 214 %

7 7

112.5%

8

1 111 %

9 9

110%

10

1 19 %

11 11

1 18 %

12 3

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1 97 %

13 13

1 17 %

14 7

1 26 %

15 3

1 16 %

16 4

1 155 %

17 17

1 55 %

18 9

1 55 %

19 19

15%

20

1 14 %

24 6

14%

25

1 12 %

40 2

3 137 %

8 2

5 162 %

8 2

4 157 %

7 7

SOME QUESTIONS BASED ON

PERCENTAGE

EX: - If ram earns 25% more than Shyam

then what % less does Shyam earn than

ram?

125%

4

It means Shyam income is 4 and Ram

income is 5

So ram is % less than Shyam

=

1100 20%

5

EX: A's income is 10% more than B's

income. How much per cent is B's income

less than A's income?

Sol: Let B's Income = Rs. 100

A's income =

100 110110

100

Required percentage =

10 1100 9 %

110 11

EX: If ram earns 𝟏𝟔𝟐

𝟑% more than

Shyam what % less does Shyam earn than

Ram?

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2 116 %

3 6

Shyam = 6 and Ram = 7

Shyam is % less than Ram

=

1 2100 14 %

7 7

EX: Sixty – five percent of a number is 21

less than three – fourth of that number.

What is the number?

SOL:

Three fourth = ¾ = 75%

75% - 65% = 21

10% = 21

100% = 210

EX: If 𝟔𝟔𝟐

𝟑% of a no. is added with itself

then result becomes 3900. Find the

original number?

SOL:

2 266 %

3 3

It means 3 is the original number

And 2 is added to the number then it

becomes 5 and

5 unit = 3900

1 unit = 780

3 unit = 2340

EX: If 96 is added in a number then

number becomes 157𝟏

𝟕% of itself. Find the

numbers?

SOL:

1 1 114157 % 100 57 1

7 7 7 7

It means 7 is the original number and after

adding 96 it becomes 11 that means

4 unit = 96

1 unit = 16

7 unit = 112

EX: If the sides of a square is increased by

40% find the percentage change in its

area ?

100

xyx y

40 4040 40

100

80 16

96

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Ex: If the radius of a circle is increased by

25% then find the percentage change in

its area ?

125%

4

It mean we can say radius of circle is 4

And new radius is 5

So area of original circle = 16

So new area of original circle = 25

Percentage change in area =

25 16 9100 100 56.25%

16 16

Ex: If the base radius of a right circular

cylinder is increased by 40% and its

height is reduce by 37.5% then find the

percentage change its volume?

2 340% ,37.5%

5 8

It means original radius is 5 and original

height is 8

Volume of cylinder = 25 8 200

New radius is 7 and new height is 5

New volume of cylinder = 49 5 245

Percentage change in area

245 200100 22.5%

200

Ex : If the length of the rectangle is

increased by 37𝟏

𝟐% and its breadth is

decreased by 16𝟐

𝟑%. Find the

approximate % change in its area ?

Sol:

1 3 2 137 % ,16 %

2 8 3 6

It means original length of rectangle is 8 and

original breadth of rectangle is 6

Area of rectangle = 48

New length of rectangle is 11

New breadth of rectangle is 5

Area of rectangle is 55

So percentage change in area

=

55 48100 14.58%

48

EX: A man saves a certain parts of his

income every month, so that he can

purchase a car in one year . By what

percent he must increase in his saving, so

that he can purchase the same car in nine

month?

SOL:

Let the amount of the car is 36

So he saves 3 rupees per month if he has to

purchase to it in 12 month

If he has to purchase it in 9 month

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So he saves 4 rupees per month

His saving increased by

4 3 1100 33 %

3 3

EX: The price of sugar is increased by

16𝟐

𝟑% and the consumption of a family is

decreased by 20% . Find the % change in

expenditure?

SOL:

2 116 %

3 6

120%

5

It means the original price of sugar is 6 and

consumption of family is 5

Total expenditure is 30

And new price of sugar is 7 and new

consumptions is 4

Total expenditure is 28

Percentage change in expenditure is

30 28 200 2100 6 %

30 30 3

Ex: The sale of a cinema ticket is

increased by 57𝟏

𝟕% and the price of ticket

is increased by 16𝟐

𝟑%. Find the change in

his revenue?

Sol:

1 457 %

7 7

2 116 %

3 6

It means no. of sale of cinema ticket is 7 and

original price of ticket is 6

Therefore total revenue is 42

New sale of ticket is 11

New price of ticket is 7

New revenue is 77

Percentage change in its revenue is

77 42 35100 100 83.33%

42 42

Ex: A man can type 20 line in 10 minute

but he leaves 8% margin on each line. In

how many time he will type 23 page with

40 lines on each page on which he leaves

25% more margin of before

Sol:

We know that

20 line in 10 minute

It means 2 lines per minute

If he leaves 8% margin that means

2 × 92 line/minute

But he has to type 23 page with 40 lines

leaves 25% more margin of before

Which means

23 × 40 × 90 = 2 × 92 × x

x = 450 min

Ex : If the income tax is increased by 19%

then net income is reduced by 6%. Find

the rate of income tax?

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Sol:

19 6

100 100

6

19

tax NI

tax

NI

Total income is 6 + 19 = 25

Rate of income tax is

6100 24%

25

Ex : Deepak spends 40% of his salary on

house rent, on remaining 10% spends on

travels , on remaining 16𝟐

𝟑 % spends on

food and remaining is saved. If he saved

Rs. 6750 what amount he spent on food ?

Sol:

Let the income be 100

House rent = 40

Remaining = 60

Travel = 60 × 10% = 6

Remaining = 54

Food = 54 × 16.66% = 9

Remaining = 45

45 = 6750

1 = 150

Amount spent on food

150 × 9 = 1350

Ex: A man spends 75% of his income. His

income is increased by 20% and he

increases his expenditure by 10%. His

saving are increased by :

Sol:

75% = ¾

Let the income be = 4

Expenditure = 3

Saving = 1

New income = 4 × 120% = 4.8

New expenditure = 3 ×110% = 3.3

New saving = 1.5

Percentage change in saving

1.5 1100 50%

1

A reduction of 25% in the price of sugar

enables a person to purchase 4 kg more

for Rs. 800. Find the original and current

price per kg ?

Sol :

25% = ¼

Original price of sugar = 4

New price of sugar = 3

We know that quantity is inversely

proportional to price

Original quantity of sugar = 3

New quantity of sugar = 4

1 unit = 4

3 unit = 12 (original quantity)

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4 unit = 16 (new quantity)

Original price per kg =

Amount / original quantity

= 800/12 = 66.67

Current price per kg

Amount / current quantity

800/16 = 50

Ex: In an examination 35% is passing

marks one candidate obtain 135 marks

and failed by 40 marks then find the

maximum marks ?

Sol: 35% = 135 + 40

100% =

175100 500

35

Ex: Rohit obtained 480 marks out of 600

and Mohit obtained 560 marks out of 800.

Whose performance is better and how

much %?

Sol:

Rohit got =

480100 80%

600

Mohit got =

560100 70%

800

Rohit performance is better.

% better =

10 2100 14 %

70 7

Ex: In an election between two

candidates, 65 votes were declared invalid

one candidate get 52% votes and win by

98 votes .Find the total number of votes?

Winner 52% Looser 48%

4% = 98

100% = 2450

Valid vote = 2450

Total vote = 2450 +65 = 2515

Ex: The length and breadth of rectangle

is increased by 30% and 20%

respectively, so find the net % change in

its area?

Sol:

Net % change = x + y + X × Y

100

= 30 + 20 + 30×20

100

= 56%

Ex: A batsman scored 120 runs which

included 3 boundaries and 8 sixes. What

percent of his total score did he make by

running between the wickets?

Number of runs made by running

= 110 - (3 × 4 + 8 × 6)

= 120 - (60)

= 60

Now, we need to calculate 60 is what percent

of 120.

=> 60/120 × 100 = 50 %

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Ex: Rahul's Mathematics test had 75

problems, 10 arithmetic, 30 algebra, 35

geometry problems. Although he

answered 70% of arithmetic, 40% of

arithmetic and 60% of geometry

problems correctly, still he got less than

60% problems right. How many more

questions he would have to answer more

to get passed

Sol:

Number of questions attempted correctly =

(70% of 10 + 40% of 30 + 60% of 35)

= 7 + 12 + 21 = 40.

Questions to be answered correctly for 60%

= 60% of total questions

= 60 % of 75 = 45.

He would have to answer 45 - 40 = 5

Ex: 10% of inhabitants of a village having

died of cholera, a panic set in, during

which 25% of the remaining inhabitants

let the village. The population is then

reduced to 4050. Find the original

inhabitants

Sol:

Let the total number is x,

then,

(100 - 25)% of (100 - 10)% x = 4050

75% of 90% of x = 4050

75/100 × 90/100 × x = 4050

x = (4050 × 50)/27 = 6000

Two numbers are less than third number

by 30% and 37% respectively. How much

percent is the second number less than by

the first?

Let the third number is x.

Then first number = (100-30)% of x

= 70% of x = 7x/10

Second number is (63x/100)

Difference = 7x/10 - 63x/100 = 7x/10

So required percentage is, difference is what

percent of first number

=> (7x/100 × 10/7x × 100 )% = 10%

Ex: In an election between two

candidates, one got 55% of the total valid

votes, 20% of the votes were invalid. If the

total number of votes was 7500, the

number of valid votes that the other

candidate got, was:

Total number of votes = 7500

Given that 20% of Percentage votes were

invalid

=> Valid votes = 80%

Total valid votes = 7500 × (80/100)

1st candidate got 55% of the total valid

votes.

Hence the 2nd candidate should have got

45% of the total valid votes

=> Valid votes that 2nd candidate got = total

valid votes x (45/100)

7500 × (80/100) × (45/100) = 2700

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Ex: 405 sweets were distributed equally

among children in such a way that the

number of sweets received by each child

is 20% of the total number of children.

How many sweets did each child receive?

Let the total number of children be x.

Then, x × (20% of x) = 405

=> x × 20x/100 = 405

1/5 x2 = 405 => x = 45

Number of sweets received by each child =

20% of 45 = 9.

Ex: Entry fee in an exhibition was Rs. 1.

Later, this was reduced by 25% which

increased the sale by 20%. The

percentage increase in the number of

visitors is :

Let the total original sale be Rs. 100. Then,

original number of visitors = 100.

New number of visitors = 120/0.75 = 160.

Increase % = 60 %.

Ex: In a City, 35% of the population is

composed of migrants, 20% of whom are

from rural areas. Of the local population,

48% is female while this figure for rural

and urban migrants is 30% and 40%

respectively. If the total population of the

city is 728400, what is its female

population?

Total Population = 728400

Migrants = 35 % of 728400 = 254940

Local population

= (728400 - 254940) = 473460.

Rural migrants = 20% of 254940 = 50988

Urban migrants

= (254940 - 50988) = 203952

Female population

= 48% of 473460 + 30% of 50988 + 40% of

203952

= 324138

Ex: In an examination, 34% of the

students failed in Mathematics and 42%

failed in English. If 20% of the students

failed in both the subjects, then the

percentage of students who passed in both

the subjects was :

n (A) = 34, n (B) = 42, n (A ∩ B) = 20.

So, n (A U B) = n (A) + n(B) - n(A ∩B) =

34 + 42 - 20 = 56.

Percentage failed in either or both the

subjects = 56.

Hence, percentage passed

= (100 - 66)% = 44%.

Ex: The ratio of earnings of A and B is

4:5. If the earnings of A increase by 20%

and the earnings of B decrease by 20%,

the new ratio of their earnings becomes

6:5. What are A's earnings?

Sol:

Cannot be determined

Ex: 30% of the men are more than 25

years old and 80% of the men are less

than or equal to 50 years old. 20% of all

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men play football. If 20% of the men

above the age of 50 play football, what

percentage of the football players are less

than or equal to 50 years?

Let total number of men = 100

Then,

20 men play football.

80 men are less than or equal to 50 years old.

Remaining 20 men are above 50 years old.

Number of football players above 50 years

old

2020 4

100

Number of football players less than or equal

to 50 years old

= 20 − 4 = 16

Ex: A number is mistakenly divided by

10 instead of being multiplied by 10.What

is the percentage error in the result?

Sol:

Let the number is 10.

Actual result = 10 × 10 = 100

Wrong result = 10

10 = 1

% change = 100−1

100 × 100 = 99%

Ex: The numerator of a fraction is

increased by 400% and denominator is

increased by 500%, so the resultant

fraction is 15/22 . Find the original

fraction?

Sol:

500% 15

600% 22

ofx

ofy

5 15

6 22

x

y

Original Fraction =

15 6

22 5

x

y

Original Fraction = 9/11

Ex: A, B and C are three persons. A

earned 40% more than B and B earned

20% less than C, so what % more earned

A than C.

Sol:

A B C

112 80 100

A more earned than C = 112 – 100 = 12

% more earned =

12 10012%

100

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Ex: Due to reduction of 20% in the price

of apples enables a person to purchase 16

apples more for Rs.320, so find the

reduced rate of 10 apples?

Sol:

Reduced rate = 320×20

100 = 64

Reduced rate of 1 apple = 64

16

= Rs. 4

Reduced rate of 10 apples = 4 × 10

= Rs.40

Ex: A mixture of 40 litres of milk and

water contains 10% water. How much

water should be added, so water may be

20% in the new mixture?

Sol:

40×10

100= 4 lit.

80% = 36

1% = 36

80

20% = 36

80× 20

= 9 Lit.

req. water = 9 – 4

= 5 lit.

Ex: Fresh fruit contains 68% water and

dry fruit contains 20% water. How much

dry fruit can be obtained from 100 kg. of

fresh fruit ?

Sol:

fruit content in 100 kg. of fresh fruits

= 100×32

100 = 32 kg

Since , dry fruits contain 80% fruit

content,

80% = 32

1% = 32

80

100% = 32

80× 100

= 40 kg.

Ex: In an examination, 35% of the

candidates failed in Mathematics and

38% failed in English. 28% passed in both

subjects, then how much percent failed in

both the subjects?

Sol: Failed in Maths = 35%

Passed in Maths = 100% − 35% = 65%

Failed in English = 38%

Passed in Maths = 100% − 38% = 62%

Failed in both subjects

= 100% – (37%+28%+34%)

= 1%

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Ex: Deepak salary is first increased by

25% and then decreased by 20%. The

resultant salary of Deepak is same as the

resultant salary of Naimish which was

first decreased by 25% and increased by

20%. Find the ratio of the salary of

Naimish and Deepak?

Sol:

125 80 75 120

100 100 100 100

20 18

D N

D N

N : D = 10 : 9

EX: A get 30% of the maximum mark and

get failed by 25 mark whereas B get 40%

of total marks in the same exam and get

25% of the passing marks more than the

passing marks. Find the passing marks of

the exam?

B’s marks =

10040 32%

125

32% - 30 % = 25

2% = 25

32% = 25 × 16

= 400

Ex: In a class 25% students were absent

for an exam. 30% failed in exam and

there average marks are 20 less then the

passing marks, 10% of the student passed

with 5 marks of grace and the remaining

student pass with the average marks of 60.

If 33 is the passing marks. Find the

average score of the class?

Sol:

25% are absent it means they got 0 marks

Average marks of the class =

25 0 30 13 10 33 35 602820

100

= 28.20

The sum of salary of A,B and C is

72000.They spend 80% ,85% and 75% of

their salary. If the ratio of their savings is

8 : 9 : 20. What is the salary of A ?

Sol:

20% = 8 100% = 40

15% = 9 100% = 60

25% = 20 100% = 80

Ratio of their income is

40 : 60 : 80

2 : 3 : 4

Salary of A =

272000 16000

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In an examination in which maximum

marks were 500, A got 10% less than B .

B got 25% more than C. C got 20% less

than D. If A got 360 marks, What

percentage of maximum marks was

obtained by D ?

Sol:

According to questions

D = 100

C = 80

B = 100

A = 90

So we can say that

90 = 360

100 = 400

Maximum marks obtained by D = 400

Percentage of marks obtained by D

400100 80%

500

A train has a capacity of 500 seats of

which 10% are in AC and rest in sleeper

class. On the particular day when it is

booked up to the capacity of 85% if AC

class was booked 96% of its capacity, then

how many sleeper class seats were vacant

on that particular day.

Sol:

Capacity of train = 500

Seats in AC = 50

Seats in Sleeper = 450

On particular day

Total passengers in train

500 ×85% = 425

AC seats = 50 × 96% = 48

It means sleeper seats are

425 – 48 = 377

So sleeper seats were vacant on that

particular day

= 450 – 377 = 73

A solution of salt and water contains 20%

salt by weight. If 20 litre of water is

evaporated from the solution, the solution

contains 25% salt. What is the original

quantity ( in litre ) of the solution ?

Sol:

Salt : water = 20 : 80


After evaporation


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1 unit = 20 litre

Original quantity 5 = 5 × 20 = 100

Fresh grapes contains 80% of water ,

while dry grapes contains 10% of water.

If the weight of dry grapes is 500 kg. What

is the total weight when it is Fresh ?

When grapes Fresh

Water = 80%

Pulp = 20%

When fruit is dry

Water = 10%

Pulp = 90%

500 90% 20%

2250

x

x

x = 30

Fresh fruit contains 68% water and dry

fruit contains 20% water. How many kg

of dry fruits can be made from 75 kg of

fresh fruits ?

When fruit is fresh

Water = 68%

Pulp = 32

When fruit is dry

Water = 20

Pulp = 80

75 32% 80%x

x = 30

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