# Percentage

## Importance of Percentange

Practice and expertise is essential for solving percentage based questions because in every chapter of arithmetic, percentage based questions are asked. Moreover by solving percentage questions you get idea of many other basic concepts.

## Percentage - Scope of Questions

Percentage, based questions are mainly arithmetic and from sale, purchase, Profit & Loss, Discount, Interest, Number system, Alligation, Reduction in cost, Population based chapters.

## Percentage - Way to Success

Deep study of percentage is required with complete accuracy and rechecking habit. Rechecking of answers is must for this chapter.

## Percentage - Important Points

Percentage : Percentage refers to “Per hundred” i.e,

5 % means

• 5 out of 100
• $\frac{5}{100}$

Percentage is denoted by ‘%’.

• a represented as the percent of b as,

$\frac{a}{b}\times 100$

• b represented as the percent of a as,

$\frac{b}{a}\times 100$

To Convert a fraction/Decimal into percentage multiply it by 100.

Example:

$0.25 = \frac{25}{100} = \frac{25}{100} \times 100\% = 25\%$

To convert a percent into fraction divide it by 100

Example:

$12.5\% = \frac{12.5}{100} = \frac{1}{8}$

## Percentage - Important Rules and Shortcut Tricks

Rule Description
1

If x is reduced to x0, then,

Reduce % = $\frac{x-x_o}{x} \times 100$

2

If x is reduced to x1, then,

Reduce % = $\frac{x_1-x}{x} \times 100$

3

If an amount is increased by a% and then it is reduced by a% again, then percentage change will be a decrease of

$\frac{a^2}{100} \times 100$

4

If a number is increased by a% and then it is decreased by b%, then resultant change in percentage will be

$\left(a-b-\frac{ab}{100}\right)\%$

5

If a number is decreased by a% and then it is increased by b%, then net increase or decrease per cent is

$\left(-a+b-\frac{ab}{100}\right)\%$

6

If a number is first decreased by a% and then by b%, then net decrease per cent is

$\left(-a-b+\frac{ab}{100}\right)\%$

7

If a number is first increased by a% and then again increased by b%, then total increase per cent is

$\left(a+b+\frac{ab}{100}\right)\%$

8

If the cost of an article is increased by A%, then how much to decrease the consumption of article, so that expenditure remains same is given by

OR

If the income of a man is A% more than another man, then income of another man is less in comparison to the 1st man by

$\left(\frac{A}{\left(100+A\right)}\times100\right)\%$

9

If the cost of an article is decreased by A%, then the increase in consumption of article to maintain the expenditure will be?

OR

If ‘x’ is A% less than ‘y’, then ‘y’ is more than ‘x’ by

Required % = $\left(\frac{A}{\left(100-A\right)}\times100\right)\%$

10

If the length of a rectangle is increased by a% and breadth is increased by b%, then the area of rectangle will increase by

Required Increase = $\left(a+b+\frac{ab}{100}\right)\%$

11

If the side of a square is increased by a% then, its area will increase by

$\left(2a+\frac{a^2}{100}\right)\%$

12

If the side of a square is decreased by a%, then the area of square will decrease by

$\left(-2a+\frac{a^2}{100}\right)\%$

13

If the length, breadth and height of a cuboidare increased by a%, b% and c% respectively, then, Increase% in volume =

$\left[a+b+c+\frac{ab+bc+ca}{100}+\frac{abc}{\left(100\right)^2}\right]\%$

14

If every side of cube is increased by a%, then increase % in volume =

$\left(3a+\frac{3a^2}{100}+\frac{a^3}{\left(100\right)^2}\right)\%$

15

If a% of a certain sum is taken by 1st man and b% of remaining sum is taken by 2nd man and finally c% of remaining sum is taken by 3rd man, then if ‘x’ rupeeis the remaining amount then,

Initial amount =

$\frac{100\times100\times100\times x}{\left(100-a\right)\left(100-b\right)\left(100-c\right)}$

16

If an amount is increased by a% and then again increased by b% and finally increased by c%, So, that resultant amount is ‘x’ rupees, then,

Initial amount =

$\frac{100\times100\times100\times x}{\left(100+a\right)\left(100+b\right)\left(100+c\right)}$

17

If the population of a certain town, is P and annual increament rate is r%, then

• After ‘t’ years population =

$P\left(1+\frac{r}{100}\right)^t$

• Before ‘t’ years population =

$\frac{P}{\left(1+\frac{r}{100}\right)^t}$

18

If the population of a certain town, is P and annual decreament rate is r%, then

• After ‘t’ years population =

$P\left(1-\frac{r}{100}\right)^t$

• Before ‘t’ years population =

$\frac{P}{\left(1-\frac{r}{100}\right)^t}$

19

On increasing/decreasing the cost of a certain article by x%, a person can buy ‘a’ kg article less/more in‘y’ rupees, then

• Increased/decreased cost of the article =

$\frac{xy}{100\times a}$

• Initial cost =

$\frac{xy}{\left(100\pm x\right)a}$

Negative sign when decreasing and positive sign when increasing.

20

If a person saves ‘R’ rupees after spending x% on food, y% on cloth and z% on entertainment of his income then,

Monthly income = $\frac{100}{100-\left(x+y+z\right)}\times R$

21

The amount of milk is x% in ‘M’ litre mixture. How much water should be mixed in it so that percentage amount of milk would be y%?

Amount of water = $\frac{M\left(x-y\right)}{y}$

22

An examinee scored m% marks in an exam, and failed by p marks. In the same examination another examinee obtained n% marks and passed with q more marks than minimum, then

Maximum marks = $\frac{100}{\left(n-m\right)}\times\left(p+q\right)$

23

In an examination, a% candidates failed in Maths and b% candidates failed in English. If c% candidate failed in both the subjects, then,

• Percentage of candidates who passed in both the subjects =

$100 – (a + b – c)\%$

• Percentage of candidates who failed in either subject =

$(a + b – c)\%$

24

In a certain examination passing marks is a%. If any candidate obtains ‘b’ marks and fails by ‘c’ marks, then,

Total marks =

$\frac{100\left(b+c\right)}{a}$

25

In a certain examination, ‘B’ boys and ‘G’ girls participated. b% of boys and g% of girls passed theexamination, then,

Percentage of passed students of the total students =

$\left(\frac{B.b+G.g}{B+G}\right)\%$

26

If a candidate got A% votes in a poll and he won or defeated by ‘x’ votes, then,

$\frac{50\times x}{\left(50-A\right)}$

27

If a number ‘a’ is increased or decreased by b%,

then the new number will be

$\left(\frac{100\pm b}{100}\right)\times a$

28

If the present population of a town is P and the population increases or decreases at rate of R1%, R2% and R3% in first, second and third year respectively, then

• population of town after 3 years =

$P\left(1\pm\frac{R_1}{100}\right)\left(1\pm\frac{R_2}{100}\right)\left(1\pm\frac{R_3}{100}\right)$

• population of town after n years =

$P\left(1\pm\frac{R_1}{100}\right)\left(1\pm\frac{R_2}{100}\right).............\left(1\pm\frac{R_n}{100}\right)$

29

If two numbers are respectively x% and y% more than the third number, then

first number as a percentage of second is

$\left(\frac{100+x}{100+y}\times100\right)\%$

30

If two numbers are respectively x% and y% less than the third number, then

first number as a percentage of second is

$\left(\frac{100-x}{100-y}\times100\right)\%$

31

If the price of an article is reduced by a% and buyer gets c kg more for some Rs. b, then

the new price per kg of article =

$\frac{ab}{100\times c}$