Percentage

If x is reduced to x₀, then,

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

If x is reduced to x₁, then,

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

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

\[\frac{a^2}{100} \times 100\]

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

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

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

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

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

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

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

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

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

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

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 % = \[\left(\frac{A}{\left(100-A\right)}\times100\right)\%\]

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 = \[\left(a+b+\frac{ab}{100}\right)\%\]

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

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

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

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

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

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

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

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

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 =

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

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 =

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

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 =

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

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 =

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

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.

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

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

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 = \[\frac{M\left(x-y\right)}{y}\]

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 = \[\frac{100}{\left(n-m\right)}\times\left(p+q\right)\]

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 =

  \[(a + b – c)\%\]

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

Total marks =

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

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 =

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

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

Total no. of votes =

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

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

then the new number will be

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

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 =

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

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

first number as a percentage of second is

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

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

first number as a percentage of second is

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

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 =

\[\frac{ab}{100\times c}\]