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Let the population of a city 2006 be x

The population of city after two years 12% increases every year = x × 112/100 × 112/100

⇒ 1.2544x

The population decreases by 10% = 1.2544x × 90/100

⇒ 1.128x

The population increases 12% in the year 2011 = 1.128x × 112/100× 112/100

⇒ 1.414x

The population decreses 2012 10% decreases = 1.414x × 0.9

⇒ 1.272

∴ The percentage change approximately  = 27.2%

Given:

One student score 9 marks more than the other student.

The same student gets marks 56% of the sum of their marks.

Calculation:

Let the marks of one student be x.

Let the marks of other student be (x + 9).

The sum of their marks = x + x + 9 = 2x + 9

According to the question;

⇒ (x + 9) = 56/100 × (2x + 9)

⇒ (x + 9) = 14/25 × (2x + 9)

⇒ 25x + 225 = 28x + 126

⇒ 3x = 99

⇒ x = 33

The marks of the one student = 33

The marks of the other student = 33 + 9 = 42

∴ The marks of the both students is 33, 42.

Given:

Rishi secured 66 marks in the test out of 80.

Concept Used:

Concept of percentage. It is calculated on the basis of 100

For example, x% means x out of 100

Calculated:

Rishi secured 66 marks in the test out of 80.

⇒ Out of 1, Rishi secured 66/80 marks

⇒ Out of 100, Rishi secured (66/80) × 100 marks

⇒ 82.5 marks

∴ The percentage of marks obtained by Rishi was 82.5%.

Given:

A and B divided a profit of Rs. 1210

2/5 part of A =1/3 part of B

Concept used:

Total parts of A and B is equal to Rs. 1210

Calculation:

A + B =1210

Also, as per the question

2/5 × A =1/3 × B

A = 5/6 B

⇒ 5/6 × B + B = 1210

⇒ 11/6 × B =1210

⇒ B = (1210 × 6)/11

⇒ B = 660

∴ B got an amount of Rs. 660

Given:

Speed of Tappu's ball = 20km/h 

The required speed of the ball to break the glass = 35 km/h 

Calculations:

Increase in speed = 35 – 20 = 15 

Increase % = (15/20) × 100 = 75% 

∴ Required increase % is 75%

Given:

Three-tenth of total voters promised to vote for A and the rest promised to vote for B.

On the last day of the election, 20% of voters went back of their promise to vote for A and 25% of voters went back of their promise to vote for B.

A lost the election by 850 votes. 

Calculation:

Let the total number of voters be x.

Three-tenth of total voters promised to vote for A and the rest promised to vote for B. 

Voters who promised to vote for A = 3x/10

Total voters = Voters who promised to vote for A + Voters who promised to vote for B

⇒ x = 3x/10 + Voters who promised to vote for B

⇒ Voters who promised to vote for B = x – 3x/10 = 7x/10

On the last day of the election, 20% of voters went back of their promise to vote for A and 25% of voters went back of their promise to vote for B.

Number of voters who backed out for A = 20% of 3x/10 = 3x/50

Number of voters who backed out for B = 25% of 7x/10 = 7x/40

Number of votes obtained by A = Voters who promised to vote for A + Number of voters who backed out for B - Number of voters who backed out for A

⇒ 3x/10 + 7x/40 – 3x/50 = 83x/200

Number of votes obtained by B = Voters who promised to vote for B + Number of voters who backed out for A - Number of voters who backed out for B

⇒ 7x/10 + 3x/50 – 7x/40 = 117x/200

A lost the election by 850 votes. 

⇒ (117x/200) – (83x/200) = 850

⇒ 34x/200 = 850

⇒ x = 850 × 200/34

⇒ x = 5,000 

∴ Total number of voters were equal to 5,000.

Given:

Monthly savings of Sagar = ₹ 40800

Formula used:

Percentage change = [(Initial value – Final value)/Initial value] × 100

Calculation:

Let the monthly income of Sagar = x

Amount left after spending on rent = x × (100 – 36)/100

⇒ x × 64/100 = x(16/25)

Amount left after spending on food = x(16/25) × (100 – 25)/100

⇒ x(16/25) × (75/100) = x(16/25) × (3/4)

Amount left after spending on clothes = x(16/25) × (3/4) × (100 – 15)/100

⇒ x(16/25) × (3/4) × (85/100) = x(16/25) × (3/4) × (17/20)

Sagar savings = x(16/25) × (3/4) × (17/20) = 40800

⇒ x[(3 × 17)/(25 × 5)] = 40800

⇒ x(51/125) = 40800

⇒ x = (40800 × 125)/51

⇒ x = 100000

∴ The monthly income of Sagar is ₹ 100000.

Given:

Salary of A = 14.28% more than salary of B

Salary of C = 6.25% more than salary of B

Calculations:

Let salary of A, B and C be x, y, z respectively.

x = y + 14.28% of y

⇒ y + (1/7) of y

⇒ 8y/7

z = y + 6.25% of y

⇒ y + (1/16) of y

⇒ 17y/16

x : z = (8y/7) : (17y/16)

⇒ 128 : 119

∴ Ratio of salary of C and A is 119 : 128

Given 

X is less than Z = 20%

Y is less than Z = 28% 

Formula Used 

Percentage = (actual/total) × 100

Calculation 

⇒ Let the Z be 100 

⇒ so, x = 100 - 20% of 100 = 80 

⇒ y = 100 - 28% of 100 = 72 

⇒ Percentage the number Y less than the X = [(80 - 72)/80] × 100 = 10%

∴ percentage is the number Y less than the number X is 10%

Given:

30% of the population in a village is literate.

The total population of the village = 6600.

Concepts used:

Number of literate people = Percentage of literal people × Total population

Calculation:

Percentage of literate people in village = 30

⇒ Number of literate people in a village = 30% × 6,600

⇒ 0.30 × 6,600 = 1,980

∴ The number of literate people in the village is 1,980. 

Given:

Speed of Rahul = 30 m/s

Speed of Sagar = 35 m/s

Increase  in speed of Sagar = 14.28%

 Calculations:

New speed of Sagar = 35 + {35 ×(1/7)}

⇒ 35 + 5

⇒ 40 m/s

Percentage difference between new speed of Sagar and speed of Rahul = {(40 - 30)/30} × 100

⇒ (10/30) × 100

⇒ 33.33%

∴ Rahul needs to increase his speed by 33.33% to match Sagar's new speed.

Given:

The average salary of male employees in a firm = Rs. 5000

The average salary of female employees in a firm = Rs. 7400

The mean salary of all the employees = 6600

Formula used:

Average = Sum of values/number of values

Percentage = (Part value/Original value) × 100

Calculation:

Let the number of employees = P   

Let the number of female employees = Q    

The total salary of employees = 6600 × P

⇒ 6600 × P = 5000 (P - Q) + 7400 × Q

⇒ 6600 × P = 5000 × P - 5000 × Q + 7400 × Q

⇒ P(6600 - 5000) = Q(- 5000 + 7400)

⇒ P/Q = 2400/1600

The toal employees in a firm = 2400

The female employees in a firm = 1600

Percentage of female employees = 1600/2400 × 100

⇒ 66.66%

∴ The required percentage is 66.66%.

Given:

Numerator is increased by 0%

Denominator is increased by 40%

Calculation:

Let numerator be x and denominator be y

According to the question,

100x/140y = 16/63

⇒ 5x/7y = 16/63

⇒ 45x = 16y

∴ \(\frac{{\rm{x}}}{{\rm{y}}} = \frac{{16}}{{45}}\)

Given:

Numerator increased by 200%

Denominator increased by 150%

Resultant fraction = 9/35

Calculation:

Let us take the numerator be x

Let us take the denominator be y

According to the question

(x + 200% of x)/(y +150% of y) = 9/35

⇒ (x+2x)/(y+1.5y) = 9/35

⇒ 35(3x) = (2.5y)9 

⇒ 105x = 22.5y

⇒ x/y = 3/14

∴ The original fraction = 3/14

Let the original fraction be x/y. 

Then (115%of x)/(92% of y)=15/16 

=> (115x/92y)=15/16 

((15/16)*(92/115))=3/4

Population after 2 years = 176400*[1+(5/100)]² 

=[176400*(21/20)*(21/40)] 

= 194481. 

Population 2 years ago = 176400/[1+(5/100)]² 

=[716400*(20/21)*(20/21)]

= 160000.

Given:

Price of tea increase = 25%

Formula Used:

Reduction in consumption = (Increase% × 100)/(100 + Increase%)

Calculation:

Reduction in consumption = (Increase% × 100)/(100 + Increase%)

= (25 × 100)/(100 + 25)

= 20% 

∴ The consumption should be reduced by 20%.

Reduction in consumption = [((R/(100+R))*100]% 

[(25/125)*100]%

=20%.

Given:

The price of petrol increased by 25%

Formula Used:

% reduction = Change in consumption/Initial consumption × 100

Calculation:

Let the initial price of petrol be 4x

⇒ The price is increased by 25%

⇒ 4x × 125/100

⇒ 4x × 5/4

⇒ 5x

Let say a car owner consumes 25 liters 

⇒ His initial expenditure = 25 × 4x 

⇒ His initial expenditure = 100x

To maintain this expenditure,

⇒ His consumption = 100x/5x

⇒ His consumption = 20 liters

⇒ Reduction in the consumption = 25 – 20

⇒ Reduction in the consumption = 5 liters

⇒ % Reduction = 5/25 × 100

⇒ % Reduction = 20% 

∴ The % reduction in consumption to maintain his expenditure is 20%.

Formula Used:

Savings = Salary – Expenditure

Calculation:

Let the man's monthly salary be x

So, the initial savings = (20/100) × x = 0.2x

Hence, the initial expenses= x – 0.2x = 0.8x

If the expenses increased by 20%, the new expenses = [(100 + 20)/100] × 0.8x = 0.96x

So, the relation for savings becomes:

800 = x – 0.96x

⇒ x = Rs.20000

∴ The man's monthly salary is Rs.20,000

Given:

Initial Expenditure = 75%

Initial Savings = 25%

Increase in income = 25%

Increase in savings = 20%

Concept:

To solve this type of question, consider the initial income to be 100. Then, calculate the rest of the data using simple % rules.

Formula used:

% increase/Decrease = [(Final value – Initial value)/initial value] × 100

Calculation:

Let the income be 100x.

Expenditure = 75% of 100x = 75x

Savings = 25% 0f 100x = 25x

% increase in come = 25%

New income = 125x

% increase in savings = 20% 0f 25x = 5x

∴ New savings = 25x + 5x = 30x

⇒ New expenditure = New income – new savings = 125x – 30x = 95x

∴ % increase in expenditure = [(95x – 75x)/75x] × 100

= 80/3

Calculation:

Let be assume Income of B = I then the income of A = 5I/4

Expenditure of A = E, then the expenditure of B = (2/3) × E

Savings of A = (5I/4) – E = (5I – 4E)/4 = 12 … (1)

Savings of B = I – (2/3)E = (3I – 2E)/3 = 12 … (2)

When we will solve the equation (1) and (2) then

⇒ I = 24, A’s income = (5/4) × 24 = 30

∴ The required result will be 30.

Given:

Part of a number = 300

Percentage = 25%

Formula used:

Percent of a number = number × (percentage/100)

Calculation:

Percent of a number = number × (percentage/100)

⇒ 300 = number × (25/100)

⇒ number = 300 × (100/25)

⇒ number = 300 × 4

⇒ number = 1200

∴ The number is 1200 whose 25% is 300.

Given:

Percentage of expenditure = 80%

Increment in income = 30%

Increment in expenditure = 30%

Solution:

Let the income of A be 100

⇒ Expenditure = 80% of 100 = 80

⇒ Savings = 100 – 80 = 20

When income increased by 30%

⇒ New income = 100 + 30% of 100

⇒ New income = 100 + 30 = 130

When expenditure increased by 30%

⇒ New expenditure = 80 + 30% of 80

⇒ New expenditure = 80 + 24 = 104

⇒ New savings = 130 – 104 = 26

Increment in savings = 26 – 20 = 6

⇒ Percent increment = 6/20 × 100

⇒ Percent increment = 30%

∴ The savings are increased by 30% 

Let the original salary be Rs.100 . New salary = Rs.90. 

Increase on 90=10 , Increase on 100=((10/90)*100)% 

= (100/9)%

Given:

Ram expenditure and saving ratio = 7 ∶ 5

15% increase in income and expenditure increases = 10%

Formula used

Income = expenditure + saving

Calculation

The ratio of expenditure and saving = 7 ∶ 5

Let the expenditure and Saving be 70x and 50x respectively

So Total income = 70x + 50x = 120x

New total income = 120x × (115/100) = 138x

New Expenditure = 70x × (110/100) = 77x

New Saving = 138x – 77x = 61x

Change in saving = 61x – 50x = 11x

Increase % = (11x/50x) × 100 = 22%

∴ Increase in saving is 22%

Given:

A student has to get 40% marks to pass an examination

Calculation:

Let be assume the total marks of the examination is X

⇒ X × (40/100) = 30 + 30

⇒ X = 150

∴ The required result will be 150.

Let the original income=Rs.100 . 

Then , expenditure=Rs.75 and savings =Rs.25 

New income =Rs.120 , New expenditure = Rs.((110/100)*75)=Rs.165/2 

New savings = Rs.(120-(165/2)) = Rs.75/2 

Increase in savings = Rs.((75/2)-25)=Rs.25/2 

Increase %= ((25/2)*(1/25)*100)% = 50%

Given:

Rishu saves x% of her income 

Increase in her income = 26% 

Increase in her expenditure = 20% 

Increase in saving = 50% 

Concept used:

Income = Saving + Expenditure 

Calculations:

Let income of Rishu be 100 

Saving of Rishu = 100 × (x/100) = x

Expenditure of Rishu = 100 – x 

Rishu's salary after increment = 100 × (126/100) = 126

Rishu's expenditure after increment = (100 – x) × 120/100 = (100 – x) × 6/5 

New Saving = 126 – (100 – x) × 6/5      ----(1) 

New saving after increment = x × 150/100

New saving after increment = 3x/2      ----(2)

From eq. (1) and eq. (2)

⇒ 126 – (100 – x) × 6/5 = 3x/2 

⇒ 126 – 120 + 6x/5 = 3x/2 

⇒ 6 = (3x/2) – (6x/5)

⇒ 6 = (15x – 12x)/10

⇒ x = 20

∴ The value of x is 20

Given:

Rajesh saving = 5%

Income increased = 25%

Formula used

Income = expenditure + saving

Calculation

Let income be 100x

So, Saving = 100x × 5/100 = 5x

Expenditure = 100x – 5x = 95x

According to the question,

New income = 100x × (125/100) = 125x

∵ Saving remains same

So, Expenditure = 125x – 5x = 120x

Change in expenditure = 120x – 95x = 25x

Increase % = (25x/95x) × 100 = 26.31% ≈ 26%

∴ Increase in Expenditure is 26%

Given:

 A student got 20% marks and failed by 72 marks. If he scores 40% marks then he gets 8 marks more than the passing marks. 

Concept:

Percentage.

Solution:

Let total marks be 100x

A student scores 20% marks and failed by 72 marks means if he gets 72 marks he would pass

⇒ 20x + 72

A student scores 40% marks and gets 8 marks more than passing marks.

⇒= 40x – 8

Passing marks 

⇒ 20x + 72 = 40x – 8 

⇒ 20x = 80

⇒ x = 4

Passing marks = 20x + 72

⇒ 80 + 72

⇒ 152

∴ The passing marks is 152.

Given:

Students get 60% of maximum marks

Formula used:

Percentage = (Part value/Original value) × 100

Calculation:

Let the total marks = 500

Students scored 60%

⇒ 500 × (60/100)

⇒ 300

Students scored marks = 50% more than the passing marks

⇒ passing marks × (150/100) = 300

⇒ passing marks = 200

Passing marks percentage of maximum marks

⇒ (200/500) × 100

⇒ 40%

∴ The passing mark is 40%.

Given:

Decreasing 110 by x% gives the same result as increasing 50 by x%

Concept Used:

Concept of percentage, it is calculated on the basis of 100

For example, x% means x out of 100

Calculation:

(100 - x) × (110/100) = (100 + x) × (50/100)

⇒ 1100 - 11x = 500 + 5x

⇒ 16x = 600

⇒ x = 75/2

Now, (75/2)% of 650 = 243.75

(75/2 + 20)% of 180 = 103.5

% more than {(243.75 - 103.5)/103.5} × 100%

⇒ 135.5%

⇒ 136% (correct to the nearest integer)

∴ The required percentage is 136%

Given:

Decreasing 110 by x% gives the same result as increasing 50 by x%.

Calculations:

110 × (100 - x)/100 = 50 × (100 + x)/100

⇒ 11 × (100 - x) = 5 × (100 + x)

⇒ 1100 - 11x = 500 + 5x

⇒ 16x = 600

⇒ x = 600/16

⇒ x = 37.5

x% of 650 = 37.5% × 650

⇒ 243.75

(x - 10)% of 780 = (37.5 - 10)% × 780

⇒ 27.5% × 780

⇒ 214.50

Percentage difference = {(243.75 - 214.50)/214.50} × 100

⇒ (29.25/214.50) × 100

⇒ 13.63% ≈ 14%

∴ x% of 650 is 14% more than (x - 10)% of 780.

Given:

Total number of transgender voters = 1500

Percentage of voters who did not cast their votes = 15%

Percentage of casted votes which are invalid = 20%

The ratio of votes get by candidate A and B = 15 : 19

Number of extra votes get by candidate B = 1600

Concept used:

Total number of voters = {Total number of casted votes/(100 – Percentage of voters did not cast their votes)} × 100

Calculation:

Let the votes get by candidate A and B is 15x and 19x respectively.

Then, 19x – 15x = 1600

⇒ 4x = 1600

⇒ x = 400

Total number of valid votes = (15 + 19) × 400

⇒ 34 × 400

⇒ 13600

Total number of casted votes = {13600/(100 – 20)} × 100

⇒ {13600/80} × 100

⇒ 170 × 100

⇒ 17000

Total number of voters in the village = {17000/(100 – 15)} × 100

⇒ {17000/85} × 100

⇒ 200 × 100

⇒ 20000

Percentage of transgender in total valid voters = (1500/20000) × 100

⇒ 150000/20000

⇒ 7.5%

∴ The transgender is 7.5% of total number of voters.

Given:

A candidate who lost got vote = 46%

Lose votes = 15208

Concept used:

Total cast votes = losses votes +  win votes

Calculations:

Let the total number of votes cast be x

The number of votes got by winning candidate = x - 46% of x

⇒ x - (46x/100) = (100x - 46x)/100

⇒ 54x/100

Difference of votes woned to lost = 54% of x - 46% of x

⇒ 8% of x = 15208

⇒ x = 15208 × 100/8

⇒ x = 190100

∴ The total number of votes casted in the election were 190100

Given 

Allowances for 1st person = 65% of his basic 

Allowances for 2nd person = 80% of his basic 

Formula Used 

Total emolument = basic pay + allowances 

Calculation 

Let the basic pay of 1st person and 2nd person be x and y 

⇒ Total emoluments of both person are same 

⇒ x + 65% of x = y + 80% of y 

⇒1.65x = 1.80y 

⇒ x : y = 12 : 11 

∴ The ratio of the basic pay of the former to the basic pay of the letter is 12 : 11

Given :

A is 35% more than B,

B is 25% less than C

C is 20% more than D

Solution:

Let the value of D be 100 

C = D + 20% of D = 120 

B = C - 25% of C = 90 

A = B + 35% of B = 121.5 

Now checking all the options 

Option 1

⇒ {(121.5 - 120)/121.5} × 100 = 1.23%

C is 1.23% less than A, Option 1 is false

Option 2

⇒ {(100 - 90/90} × 100 = 11.11%

D is 11.11% more than B, Option 2 is false

Option 3

⇒ {(121.5 - 120)/120} × 100 = 1.25%

A is 1.25% more than C, Option 3 is true

∴ Option 3 is the correct choice as this is satisfying the given condition

Given: In an election 3 candidates A, B and C got certain no. of votes. If the number of votes with A increases by 5000, A would have got double the number of votes given to B alone. B got a total of 7000 votes and all the 3 candidates together got a total of 17000 votes.

Formula used: Percentage increase/decrease = [(new - actual)/actual] × 100%

Calculation:

Let the number of votes A, B and C received be a, b and c respectively.

According to question

b = 7000 (given)

Then, a + c = 17000 – b

⇒ a + c = 17000 – 7000 = 10000      ----(1)

From the question

a + 5000 = 2b

⇒ a + 5000 = 2 × 7000 = 14000

⇒ a = 9000

And from equation (1) we can also get c = 10000 – 9000 = 1000

Calculation:

Let the third number be 100

First number = 100 + 20% of 100

⇒ 100 + 20

⇒ 120

Similarly,

Second number = 100 + 35% of 100

⇒ 100 + 35

⇒ 135

Now,

Required ratio = 120 ∶ 135

⇒ 8 ∶ 9

∴ The ratio between first and second number will be 8 ∶ 9

Given:

A and B were two candidates in an election. A got 55% of the total votes casted. 10% of the total population did not cast any votes. B got a total of 8100 votes.

Formula used:

Percentage increase/decrease = [(new value – actual value)/actual value] × 100

Calculation:

Let the total population be 100x.

Total no. of votes casted = (90/100) × 100x = 90x

Total no. of votes of A = (55/100) × 90x

Total no. of votes of B = (45/100) × 90x

According to question

(45/100) × 90x = 8100

⇒ x = (8100 × 100)/(45 × 90)

⇒ x = 200

Total population = 100 × 200 = 20000

∴ The total population is 20000.

Given: There are two candidates X and Y in an election. If X receives 5000 votes of Y, X would have 40% of the total votes casted to them. Y initially received 80% of the total votes.

Formula used: Percentage increase/ decrease = [(new - actual)/actual] × 100%

Calculation: Let the total votes be 100x, Initially Y received 80% of the total votes that means X received 20% votes.

Y initial votes = 80x

When X receives 5000 votes of Y, X would have 40% of total votes.

Now, new votes of X = 40x and votes of Y = 60x

Difference of votes 60x – 40x = 20x = 5000 (given)

⇒ x = 250

Total votes = 250 × 100 = 25000

∴ Votes received by X = (20/100) × 25000 = 5000

Given: In an election A secured 48% of the total votes and C secured 18% of the total votes. B got the remaining votes 2720.

Formula used: Percentage increase/ decrease = [(new - actual)/actual] × 100%

Calculation: A and C together secured 48% + 18% = 66% votes

Votes of B = 100% - 66% = 34%

From the question

34% = 2720

⇒ 1% = (2720/34) = 80

⇒ 100% = 8000

Total votes = 8000

Difference in the number of votes of A and C

48% – 18% = 30% of total votes

⇒ (30/100) × 8000 = 2400

Given:

35% of 9/5 of a number = 252

Calculation:

Let the number be x

(35/100) × (9/5) × x = 252

⇒ (7/20) × (9/5) × x = 252

⇒ x = (252 × 20 × 5)/(7 × 9)

⇒ x = 4 × 20 × 5 = 400

∴ The number is 400.

Given:

In an election only 80% of the total number of people casted votes to Tarun and

Vikash. If the remaining 20% of the population would have all casted their votes to

tarun, there would have been a tie between tarun and Vikash. If the remaining 20%

of the population would have all casted their votes to Vikash, Tarun just received

30% of the total votes.

Formula used:

Percentage increase/ decrease = [(new - actual)/actual] × 100%

Calculation:

Let the total population be t.

Let tarun gets x votes and vikash y number of votes.

According to question

80% people casted the votes

x + y = 0.8t      ----(1)

Case 1 : If remaining 20% casted votes to tarun then, there is a tie between tarun and vikash.

0.2t + x = y

⇒ y – x = 0.2t      ----(2)

Case 2: if remaining 20% casted votes to vikash then,

0.2t + y = 0.7t

⇒ y = 0.5t      ----(3)

On adding 1 and 2, we get

2y = t

⇒ y = 0.5t which is same as (3).

Hence, data is insufficient.

Given:

Let male employees of the company be a.

Concept Used:

Concept of percentage. It is calculated on the basis of 100.

For example, x% means x out of 100

Calculation:

The female employees of the company = a × (100 + 40)/100 = 7a/5

Total employees of the company = a + 7a/5 = 12a/5

Accordingly,

7a/5 - 8 = 12a/5 × 1/2

⇒ 7a/5 - 8 = 6a/5

⇒ 7a/5 - 6a/5 = 8

⇒ a/5 = 8

⇒ a = 40

∴ Total male employees of a company = 40