**PERCENTAGE**

**Importance : **For percentage, it may be mentioned that in every chapter of arithmetic, percentage based questions are asked, hence practice and expertise is essential. Moreover by solving percentage questions we get idea of many other basic concepts.

**Scope of questions : **Percentage, based questions are mainly arithmetic and from sale, purchase, profit & Loss, Discount, Interest, Number system, Allegation, Reduction in cost, Population based chapters.

**Way to success : **Deep study of percentage is required with complete accuracy and rechecking habit. Rechecking of answers is must for this chapter.

**Percentage : **Percentage refers to “Per hundred” i.e. 8% means 8 out of hundred $$or\quad \frac { 8 }{ 100 } .$$

Percentage is denoted by ‘%’.

a represented as the per cent of b as, $$\frac { a }{ b } \times 100$$

$$b\%\quad of\quad a\quad =\quad \frac { a }{ b } \times 100$$

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

As $$0.35=\frac { 35 }{ 100 } =\frac { 35 }{ 100 } \times \quad 100\quad =\quad 35%$$

To covert a per cent into fraction, divide it by 100

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

**Rule 1 : **If x is reduced to x_{0}, then,

$$Reduce \%\quad =\frac { x-{ x }_{ 0 } }{ x } \times 100$$

**Rule 2 : **If x is increased to x_{1}, then,

$$Increment\%\quad =\frac { { x }_{ 1 }-{ x } }{ x } \times 100$$

**Rule 3 : **If an amount is increased by a%and then it is reduced by a% again, then percentage change will be a dcrease of $$\frac { { a }^{ 2 } }{ 100 } \%$$

**Rule 4 : **If a number is increased by a% and then it is reduced by b% , then resultant change in percentage will be $$\left( a-b-\frac { ab }{ 100 } \right) \%$$

**Rule 5 : **If a number is decreased by a% and then it is increased by b%, then net increase or decrease per cent is

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

$$(\begin{matrix} Negative\quad sign\quad for\quad decrease \\ Positive\quad sign\quad for\quad increase \end{matrix})$$

**Rule 6 : **If a number is first decrease by a% and then by b%, then net decrease per cent is $$\left( -a-b-\frac { ab }{ 100 } \right) \%\quad $$ (-ve sign for decrease)

**Rule 7 : **If a number is first increased by a% and then again increased by b%, then total increase per cent is $$\left( a+b+\frac { ab }{ 100 } \right) \%$$

**Rule 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 $$\left( \frac { A }{ \left( 100+A \right) } \times 100 \right) \%$$

**Rule 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 $$Required\%=\left( \frac { A }{ \left( 100-A \right) } \times 100 \right) \%$$(i.e. increase)

**Rule 10 : **If the length of a rectangle is increased by a% and breadth is increased by b%, then the area of rectangle will be increased by $$Required\quad Increase=\left( a+b+\frac { ab }{ 100 } \right) \%$$

**Note : **If a side is increased, take positive sign and if it is decreased, take negative sign. It is applied for two dimensional figures.

**Rule 11 : **If the side of a square is increased by a% then, its area will increase by $$\left( 2a+\frac { { a }^{ 2 } }{ 100 } \right) \%=\left( a+a+\frac { a.a }{ 100 } \right) \%$$

The above formula is also implemented for circle where radius is used as side. This formula is used for two dimensional geometrical figures having both length and breadth equal.

**Rule 12 : **If the side of a square is decreased by a% , then the area of square will decrease by $$∴\quad Decrease\quad =\quad \left( -2a+\frac { { a }^{ 2 } }{ 100 } \right) \%$$

**Rule 13 : **If the length, breadth and height of a cuboid are increased by a%,b% and c% respectively, then Increase% in volume $$=\left[ a+b+c+\frac { ab+bc+ca }{ 100 } +\frac { abc }{ { \left( 100 \right) }^{ 2 } } \right] \%$$

**Rule 14 : **If every side of cube is increased by a%, then increased % in volume $$=\left( 3a+\frac { 3{ a }^{ 2 } }{ 100 } +\frac { { a }^{ 3 } }{ { \left( 100 \right) }^{ 2 } } \right) \%$$

This formula will also be used in calculating increase in volume of sphere, where increase in radius is given.

**Rule 15 : **If a% of a certain sum is taken by 1st man and b% of remaining is taken by 2nd man and finally c% of remaining sum is taken by 3rd man, then if ‘x’ rupee is the remaining amount, then Initial amount $$=\frac { 100\times 100\times 100x }{ \left( 100-a \right) \left( 100-b \right) \left( 100-c \right) } $$

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

**Rule 17 : **If the population/cost of a certain town/article, is P and annual increment rate is r%, then

(i) After ‘t’ years population/cost $$=P{ \left( 1+\frac { r }{ 100 } \right) }^{ t }$$

(ii) Before ‘t’ years population/cost $$=\frac { P }{ { \left( 1+\frac { r }{ 100 } \right) }^{ t } } $$

**Rule 18 : **If the population/cost of a town/article is P and it decreases/reduces at the rate of r% annually, then

(i) After ‘t’ years population/cost $$=P{ \left( 1−\frac { r }{ 100 } \right) }^{ t }$$

(ii) Before ‘t’ years population/cost $$=\frac { P }{ { \left( 1−\frac { r }{ 100 } \right) }^{ t } } $$

**Rule 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 $$=\left( \frac { xy }{ 100\times a } \right) $$

And Initial cost $$=\left( \frac { xy }{ \left( 100\pm x \right) a } \right) $$

[Negative sign when decreasing and positive sign when increasing]**Rule 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 $$=\frac { 100 }{ 100-\left( x+y+z \right) } \times R$$

**Rule 21 : **The amount of acid/milk is x% in ‘M’ litre mixture. How much water should be mixed in it so that percentage amount of acid/milk would be y%?

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

**Rule 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

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

**Rule 23 : **In an examination, a% candidates failed in Maths and b% candidates failed in English. If c% candidates failed in both the subjects , then

(i) Passed candidates in both the subjects =100 − (a + b − c)%

(ii) Percentage of candidates who failed in either subject = (a + b − c)%

**Rule 24 : **In a certain examination passing marks is a%. If any candidate obtains ‘b’ marks and fails by ‘c’ marks, then,

$$Total\quad marks=\frac { 100\left( b+c \right) }{ a } $$

**Rule 25 : **In a certain examination, ‘B’ boys and ‘C’ girls participated. b% of boys and g% of girls passed the examination, then, Percentage of passed students of the total students $$=\left( \frac { B.b+G.g\quad }{ B+G } \right) %$$

**Rule 26 : **If a candidate got A% votes in a poll and he won or defeated by ‘x’ votes, then what was the total no of votes which was casted in poll?

$$∴Total\quad no.\quad of\quad votes=\frac { 50\times x }{ \left( 50-A \right) } $$

**Rule 27 : **If a number ‘a’ is increased or decreased by b%, then the new number will be $$\left( \frac { 100\pm b }{ 100 } \right) \times a$$

**Rule 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 the population of town after 3 years =$$P\left( 1\pm \frac { { R }_{ 1 } }{ 100 } \right) \left( 1\pm \frac { { R }_{ 2 } }{ 100 } \right) \left( 1\pm \frac { { R }_{ 3 } }{ 100 } \right) $$

‘+’ is used when population increases.

‘−’ is used when population decreases.

The above formula may be extended for n number of years.

⇒ Population after ‘n’ years=

$$P\left( 1\pm \frac { { R }_{ 1 } }{ 100 } \right) \left( 1\pm \frac { { R }_{ 2 } }{ 100 } \right) …\left( 1\pm \frac { { R }_{ n } }{ 100 } \right) $$

**Rule 29 : I**f two numbers are respectively x% and y% less than the third number, first number as a percentage of second is $$\frac { 100-x }{ 100-y } \times 100\%$$

**Rule 30 : **If two numbers are respectively x% and y% more than a third number, the first as percentage of second is $$\frac { 100+x }{ 100+y } \times 100\%$$

**Rule 31 : **If the price of an article is reduced by a% and buyer gets c kg more for some Rs. b, the new price per kg of article $$\frac { ab }{ 100\times c } $$

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