# Percentage Topic-Wise Short Notes, Solutions, Methods, Tips, Tricks & Techniques To Solve Problems

## Percentage Formulas, Shortcuts, Rules, Tricks & Tips - Quantitative Aptitude

#### Useful For All Competitive Exams Like UPSC, SSC, BANK & RAILWAY

Posted By Careericons Team

### Introduction to Percentage:

The term per cent means ‘for every hundred’ or ‘out of hundred’. A fraction whose denominator is 100 is called a percentage, and the numerator of the fraction is called the rate per cent. The term ‘per cent’ is denoted by the symbol ‘%’.

1 per cent means 1 part out of 100 or $1/100.$

25 per cent means 25 parts out of 100 or ${25}/{100}$ = $1/4$.

So per cent is really a fraction whose denominator is 100, and the numerator of this fraction is the 'rate per cent'.

The percentage is important in interpreting the data given in graphs and tables. It is necessary to understand clearly how to work it out.

$\text"Percentage (value)" = \text"Rate % × Base number"$

$\text"Base number" = \text"Percentage (value)"/\text" Rate %"$

$\text"Rate %" = \text"Percentage value"/\text" Base number"$

### "31" - Important Aptitude Rules, Formulas & Quick Tricks to Solve Percentage Problems

In this list of rules, you will get an idea that How to solve all different types & kinds of Percentage based aptitude problems asked in various competitive exams like UPSC, SSC, Bank, and Railway examinations at all levels.

By using this method, you can able to solve all problems from basic level to advanced level of questions asked based on Percentage in a faster approch.

Let's discuss the rules one by one with all Percentage formulas with examples,

Rule 1:

If x is reduced to $x_0$, then,

Reduce % = ${x - x_0}/x × 100$

Rule 2:

If x is increased to $x_1$,then,

Increment % = ${x_1 - x}/x × 100$

Rule 3:

If an amount is increased by a% and then it is reduced by a% again,

Then percentage change will be a decrease of $a^2/100%$

Rule 4:

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

$(a - b -{ab}/100) %$

(Negative for decrease, Positive for increase)

Rule 5:

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

$(-a + b -{ab}/100)%$

Negative sign for decrease; Positive sign for increase

Rule 6:

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

$(- a - b +{ab}/100)%$

(–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

$(a + b +{ab}/100)%$

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

$(A/{(100 + A)} × 100)%$

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 % = $(A/{(100 - A)} × 100)%$ (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 increase by

Required Increase = $(a + b +{ab}/100)%$

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

$(2a + {a^2}/100)% = (a + a + {a.a}/100)$%

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

Decrease % = $(-2a + {a^2}/100)%$

This formula is also applicable for circles. where decrease % of radius is given.

Rule 13:

If the length, breadth and height of a cuboid are increased by a%, b% and c% respectively, then,

Increase% in volume = $[a + b + c + {ab + bc + ca}/100 + {abc}/{(100)^2}]%$

Rule 14:

If every side of cube is increased by a%, then

Increase % in volume = $(3a + {3a^2}/100 + a^3/{(100)^2})%$

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 sum 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 = ${100 × 100 × 100x}/{(100 - a)(100 - b)(100 - c)}$

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 = ${100 × 100 × 100x}/{(100 + a)(100 + b)(100 + c)}$

Rule 17:

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

1. After 't' years population/cost = P$(1 + {r}/100)^t$

2. Before 't' years population/cost = $P/{(1 + {r}/100)^t}$

Rule 18:

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

1. After 't' years population/cost = P$(1 - {r}/100)^t$

2. Before 't' years population/cost = $P/{(1 - {r}/100)^t}$

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 = $({xy}/{100 × a})$ &

Initial cost = ${xy}/{(100 ± x)a}$

[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 = ${100}/{100 - (x + y + z) × R}$

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 of water = ${M(x - y)}/y$

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 marks = $100/{(n - m)} × (p + q)$

Rule 23:

In an examination, a% candidates failed in Maths and b% candidates failed in English. If c% candidate 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 marks = ${\text"100 (b + c)"}/a$

Rule 25:

In a certain examination, 'B' boys and 'G' girls participated. b% of boys and g% of girls passed the examination, then,

Percentage of passed students of the total students =$({B.b + G.g}/{B + G})%$

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 no. of votes = ${50 × x}/{(50 - A)}$

Rule 27:

If a number 'a' is increased or decreased by b%,

Then, the new number will be $({100 ± b}/100)$ × 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$(1 ± {R_1}/100)(1 ± {R_2}/100)(1 ± {R_3}/100)$

'+' 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$(1 ± {R_1}/100)(1 ± {R_2}/100)...(1 ± {R_n}/100)$

Rule 29:

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

Percentage of second is ${100 - x}/{100 - y} × 100%$

Rule 30:

If two numbers are respectively x% and y% more than a third number the first as,

Percentage of second is ${100 + x}/{100 + y} × 100%$

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 = ${ab}/{100 × c}$

### 11 - Types of Percentage Based Aptitude Questions and Answers Practise Test

Click the below links & Learn the specific model from Percentage problems that you have to practice for upcoming examination

Refer: Get all Topic-wsie Quantitatiive aptitude problems for upcoming competitive exams

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