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# Important Maths Formulae for Quant Preparation for Law Entrance Exams

Author : Palak Khanna

April 20, 2022

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Law entrance exams in India for LLB courses like CLAT and SLAT test a candidate's skill in Quantitative Aptitude.

The questions in the Quantitative Aptitude Section for the Law entrance examination are from various topics like

• Number Systems
• Profit, Loss and Discount
• LCM and HCF
• Time and Work
• Averages
• Surds and Indices
• Probability
• Set Theory & Function
• Permutation & Combination
• Coordinate Geometry
• Mensuration, etc.

Thus, it becomes crucial that candidates should be well prepared with the basic formulas required for solving questions with higher accuracy and in the least time possible.

Here is a list of essential formulas candidates need to successfully crack the Common Law Admission Test (CLAT).

This article will learn about the important maths formulas for the Quantitative Aptitude section to prepare for the Law entrance examination.

## List of Important Maths Formulas for Quant Preparation

Here below you will find all the important maths formulas and important topics which a candidate should know and practice to excel in the Law entrance examination.

(1) Number Systems

(2) Profit, Loss, and Discount

(3) LCM (Least common multiple) and HCF (Highest common factor)

(4) Speed, Time, and Distance.

(5) Percentages.

(6) Time and Work

(7) Averages.

(8) Simple and Compound Interest.

(9) Logarithm.

(10) Probability

(11) Surds and Indices

(12) Set Theory and Functions

(13) Permutation and Combination.

(14) Mixtures and Allegations

(15) Trigonometry

(16) Coordinate Geometry.

(17) Mensuration.  ### (1) Number Systems

In the Indian system, numbers are expressed by means of symbols, namely 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. We called them digits. Here, 0 is called an insignificant digit whereas 1, 2, 3, 4, 5, 6, 7, 8, and 9 are called significant digits.

We can express a number in two ways i.e. Notation and Numeration.

Notation - Representing a number in figures is known as Notation. For example, 250, 300, 1000, 12000.

Numeration - Representing a number in words is known as Numeration.

Enhance your CLAT Maths preparation by learning the important formulas that are given in the post below.

#### Different Types of Numbers

(1). Natural Numbers - (N) If N is the set of natural numbers, then we write N ₌ {1, 2, 3, 4, 5, 6,....}. The smallest natural number is 1.

(2). Whole Numbers - (W) If W is the set of whole numbers, then we write W ₌ {0, 1, 2, 3, 4, 5,.....}. The smallest whole number is 0.

(3). Integers - (I) If I is the set of integers, then we write I ₌ {...., -3, -2, -1, 0, 1, 2, 3,..,}. Where, {1, 2, 3, …} is the set of positive integers. {-1, -2, -3,..} is the set of negative integers and 0 is neither positive nor negative.

(4). Rational Numbers - (Q) Any number which can be expressed in the form of p/q, where both p and q are integers and q ≠ 0 is called a rational number.

(5). Irrational Numbers - Non - recurring and non - terminating decimals are called irrational numbers. These numbers cannot be expressed in the form of p/q and q ≠ 0.

(6). Real Number - It includes both rational and irrational numbers.

#### Formulas of Number Systems

⇒ Sum of the first ‘n’ natural numbers i.e. 1 ₊ 2 ₊ 3 ₊ 4 ₊ 5 ₊ .. ₊ n ₌ n (n ₊ 1) / 2.

⇒ Sum of the squares of the first ‘n’ natural numbers i.e. 1² ₊ 2² ₊ 3² ₊ 4² ₊ …. ₊ n² ₌ n (n ₊ 1) (2n ₊ 1) / 6.

⇒ Sum of the cubes of first ‘n’ natural numbers i.e. 1³ ₊ 2³ ₊ 3³ ₊ … ₊ n³ ₌ {n (n ₊ 1) / 2}².

⇒ Sum of first ‘n’ odd numbers ₌ n².

⇒ Sum of first ‘n’ even numbers ₌ n (n ₊ 1).

#### Mathematical Formulas:

⇒ (a – b)² ₌ (a² ₊ b² – 2ab)

⇒ (a ₊ b)² ₌ (a² ₊ b² ₊ 2ab)

⇒ (a ₊ b) (a – b) ₌ (a² – b²)

⇒ (a ₊ b)² ₌ (a² ₊ b² ₊ 2ab)

⇒ (a ₊ b ₊ c)² ₌ a² ₊ b² ₊ c² ₊ 2 (ab ₊ bc ₊ ca)

⇒ (a³ – b³) ₌ (a – b) (a² ₊ ab ₊ b²)

⇒ (a³ ₊ b³) ₌ (a ₊ b) (a² – ab ₊ b²)

⇒ (a³ ₊ b³ ₊ c³ – 3abc) ₌ (a ₊ b ₊ c) (a² ₊ b² ₊ c² – ab – bc – ac)

⇒ When a ₊ b ₊ c ₌ 0, then a³ ₊ b³ ₊ c³ ₌ 3abc

⇒ (a ₊ b) n ₌ an ₊ (nC1) an -1b ₊ (nC2) an - 2b² ₊ … ₊ (nCn-1) abn -1 ₊ bn

### (2) Profit, Loss, and Discount

It is a basic concept of arithmetic in which we study the gain or loss in a business transaction. Profit and loss are the terms related to transactions in trade and business. Whenever a purchase article is sold, then either profit is earned or loss is incurred.

#### Formulas of Profit, Loss, and Discount

⇒ Profit / Gain ₌ Selling Price (SP) - Cost Price (CP).

⇒ Profit Percentage (%) ₌ (Profit / Cost Price (CP) x 100)

⇒ Selling Price (SP) ₌ (100 ₊ Profit Percentage / 100) x Cost Price.

⇒ Cost Price ₌ 100 / (100 ₊ Profit Percentage) x Selling Price.

⇒ Loss ₌ Cost Price (CP) - Selling Price (SP).

⇒ Loss Percentage (%) ₌ (Loss / Cost Price x 100).

⇒ Selling Price ₌ (100 - Loss Percentage / 100) x Cost Price

⇒ Cost Price ₌ 100 / (100 - Loss Percentage) x Selling Price.

### (3) LCM (Least common multiple) and HCF (Highest common factor)

LCM - The LCM of two or more given numbers is the least number to be exactly divisible by each of them.

Multiples of 25 are 25, 50, 75, 100, 125, 150,....

Multiples of 30 are 30, 60, 90, 120, 150, 180,...

The least common factor (LCM) is 150.

HCF - The highest common factor of two or more given numbers is the largest of their common factors.

Factors of 20 are 1, 2, 4, 5, 10, 20.

Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.

Common factors are 1, 2, and 4.

Highest common factor (HCF) is 4.

#### Formulas of LCM and HCF

⇒ LCM x HCF ₌ Products of the numbers.

⇒ LCM of Co - prime numbers ₌ Products of the numbers.

### (4) Speed, Time and Distance

Distance is measured in metres, Kilometres or miles.

Time is measured in hours, minutes or seconds.

Speed is measured in kilometre per hour (kmph), metre per hour (mph) or metre per second (mps).

#### Formulas of Speed, Time and Distance

⇒ Speed ₌ Distance / Time

⇒ Time ₌ Distance / Speed.

⇒ Distance ₌ Speed x Time.  ### (5) Percentages

It is used to express how longer or smaller one quantity is relative to another quantity. Percent means ‘per hundred’ it is given by ‘%’ symbol

#### Formulas of Percentage

⇒ Percentage Increased ₌ (Increased / Original Value x 100) %

⇒ Percentage decrease ₌ (decreased / Original Value x 100) %

⇒ If the price of a commodity increases by r%, then the reduction in consumption, so as not to increase the expenditure is {r / (100 ₊ r) x 100} %.

⇒ If the price of a commodity decreases by r%, then the increase in consumption, so as not to decrease the expenditure is {r / (100 - r) x 100} %.

### (6) Time and Work

Work to be done is generally considered as one unit, it may be digging a bench, constructing or painting a wall, filling up or emptying a tank, reservoir or a cistern.

#### Formulas of Time and Work

⇒ If A can do a piece of work in ‘n’ days, then work done by A in 1 day ₌ 1 / n.

⇒ If A’s 1 day work ₌ 1 / n, then A can finish the whole work in ‘n’ days

⇒ If A is twice as good a workman as B, then

Ratio of work done by A and B ₌ 2 : 1.

Ration of time taken by A and B ₌ 1 : 2.

⇒ If two persons A and B can individually do some work in ‘a’ and ‘b’ days respectively, then A and B together can complete the same work in (ab / a ₊ b) days.

⇒ If two persons A and B together can complete the same work in ‘a’ days and A (or B) can individually do some work in ‘b’ days then B (or A) can complete the work in (ab / b - a) days.

### (7) Averages

If all the given quantities have the same value, then the number itself is the average.

Formulas of Averages

The average of a given number of quantities of the same kind is expressed as

⇒ Average ₌ Sum of Quantities / Number of Quantities.

Average is also called the Arithmetic Mean. Also,

⇒ Sum of quantities ₌ Average x Number of Quantities.

⇒ Number of Quantities ₌ Sum of Quantities / Average.

### (8) Simple and Compound Interest

Simple Interest - If the interest is calculated on the original principal at any rate of interest for any period of time, then it is called simple interest.

Compound Interest - The interest for the future period is calculated not only on the principal, but also on the interest earned until the previous period, known as compound interest.

#### Formulas of Simple and Compound Interest

When interest is compounded annually:

⇒ Amount ₌ Principal (1 ₊ R / 100)ⁿ

When interest is compounded half - yearly::

⇒ Amount ₌ Principal (1 ₊ R/2 / 100)²ⁿ

When interest is compounded Quarterly:

⇒ Amount ₌ Principal (1 ₊ R/4 / 100)⁴ⁿ

When interest is compounded annually but time is in fraction, say 3 ⅖ years.:

⇒ Amount ₌ Principal (1 ₊ R / 100)³ x (1 ₊ ⅖ R / 100)

When Rates are different for different years, say R₁ %, R₂ %, R₃ % for 1st, 2nd, and 3rd year respectively:

Then,

⇒ Amount ₌ Principal (1 ₊ R₁ / 100) (1 ₊ R₂ / 100) (1 ₊ R₃ / 100).

Present worth of Rs. 𝑥 due n years hence is given by:

⇒ Present worth ₌ 𝑥 / (1 ₊ R / 100).

(9). Logarithm

#### Formulas of Logarithm:

⇒ logₐ (xy) ₌ logₐ x ₊ logₐ y

⇒ logₐ (x / y) ₌ logₐ x - logₐ y

⇒ logₓ x ₌ 1.

⇒ logₐ 1 ₌ 0.

⇒ logₐ (xⁿ) ₌ n (logₐ x)

⇒ logₐ x ₌ 1 / logₓ a

⇒ logₐ x ₌ logₑ x / logₑ a ₌ log x / log a.

### (10) Probability

Sample Space: When we perform an experiment, then the set S of all possible outcomes is called the sample space.

Event: Any subset of a sample space is called an event.

The Probability of Occurrence of an Event:

Let S be the sample and let E be an event.

Therefore, P(E) n(E) / n(S).

### (11) Surds and Indices

Root of any number is called surds e.g., 2, 3, ⁸5, ³7and etc.If P be a rational number and 𝒎 is positive integer, then ‴P is a surd of order 𝒎𝒏. When a number P is multiplied by itself 𝒏 times, then the product is called 𝒏th power of P and is written as Pⁿ. Here, P is called the basis and 𝒏 is known as the index of the power. (Here, the plural of index is called indices).

#### Formulas of Surds and Indices:

Law of Indices:

⇒ 𝑎 ‴ x 𝑎 ″ ₌ 𝑎 ‴ ⁺ ″

⇒ 𝑎 ‴ ÷ 𝑎 ″ ₌ 𝑎 ‴ ⁻ ″

⇒ (𝑎 ‴) ″ ₌ 𝑎 ‴″

⇒ (𝑎𝑏) ″ ₌ 𝑎 ″ 𝑏 ″

⇒ (𝑎 / 𝑏) ″ ₌ 𝑎 ″ / 𝑏 ″

⇒ 𝑎 ⁰ ₌ 1

Law of Surds:

⇒ ″ √ 𝑎 ₌ 𝑎 ¹ / ⁿ

⇒ (″ √ 𝑎) ″ ₌ 𝑎

⇒ ″ √ 𝑎𝑏 ₌ ″ √ 𝑎 x ″ √ 𝑏

⇒ ‴ √ ″ √ 𝑎 ₌ ‴″ √ 𝑎 ₌ ″ √ ‴ √ 𝑎

⇒ ″ √ 𝑎 / 𝑏 ₌ ″ √ 𝑎 / ″ √ 𝑏

⇒ (″ √ 𝑎) ‴ ₌ ″ √ 𝑎‴

### (12) Set Theory and Function

The Demorgan’s Law is the basic and most important formula for sets, which is defined as

(A ∩ B) ‘ = A’ U B’ and (A U B)’ = A’ ∩ B’

The relation R⊂A×AR⊂A×A is said to be called as:

⇒ Reflexive Relation: If a R a ∀∀ a ∈∈ A.

⇒ Symmetric Relation: If aRb, then bRa ∀∀ a, b ∈∈ A.

⇒ Transitive Relation: If aRb, bRc, then aRc ∀∀ a, b, c ∈∈ A.

If any relation R is reflexive, symmetric and transitive in a given set A, then that relation is known as an equivalence relation.

### (13) Permutation and Combination

Permutation and Combination are the ways to represent a group of objects by selecting them in a set and forming subsets. It defines the various ways to arrange a certain group of data. When we select the data or objects from a certain group, it is said to be permutations, whereas the order in which they are represented is called combination. Both concepts are very important in Mathematics.

Formula of Permutation is:

A permutation is the choice of r things from a set of n things without replacement and where the order matters.

nPr = (n!) / (n-r)!

Formula of Combination is:

A combination is the choice of r things from a set of n things without replacement and where order doesn't matter.

ɴCᵣ = (n/r) = ɴPᵣ / r! = n! / r! (n - r)!

### (14) Mixtures and Allegation

When two or more than two substances are mixed in any ratio to produce a product, then the product is known as a mixture. The process to produce a product is known as alligation.

The cost price of a unit quantity of the mixture is called the mean price.

Formula of Mixture:

⇒ Quantity of cheaper article / Quantity of costly article ₌ (Cost price of a unit of costly article - Average price) / (Average price - Cost price of a unit of cheaper article).

### (15) Trigonometry

Trigonometric Identities:

⇒ Sine ₌ Opposite / Hypotenuse

⇒ Secant ₌ Hypotenuse / Adjacent

⇒ Cosine ₌ Adjacent / Hypotenuse

⇒ Tangent ₌ Opposite / Adjacent

⇒ Co−Secant ₌ Hypotenuse / Opposite

⇒ Co−Tangent ₌ Adjacent / Opposite

The reciprocal identities are given as:

⇒ CosecΘ ₌ 1 / sinΘ

⇒ secΘ ₌ 1 / cosΘ

⇒ cotΘ ₌ 1 / tanΘ

⇒ sinΘ ₌ 1 / CosecΘ

⇒ cosΘ ₌ 1 / secΘ

⇒ tanΘ ₌ 1 / cotΘ

### (16) Coordinate Geometry

The Distance Between two Points A and B:

⇒ AB ² ₌ (Bx – Ax) ² ₊ (By – Ay) ²

The Midpoint of a Line Joining Two Points

The midpoint of the line joining the points (x1, y1) and (x2, y2) is:

⇒ [½ (x1 ₊ x2), ½ (y1 ₊ y2)]

The Equation of a Line Using One Point and the Gradient

The equation of a line which has gradient m and which passes through the point (x1, y1) is:

⇒ y – y1 ₌ m (x – x1).

### (17) Mensuration

Area - Area of a two dimensional figure is the amount of surface enclosed by its boundary. It is measured in square units.

Perimeter - Perimeter of a two dimensional figure is the length of its boundary. It is measured in units.

Volume - Volume of a 3D figure is the amount of space occupied by it. It is measured in cubic units.

Surface Area - Surface area of a 3D figure is the total area of all of its surfaces. It is measured in square units.

#### Formulas of Mensuration:

(1). Triangle:

Perimeter ₌ a ₊ b ₊ c (sum of all side).

⇒ Area ₌ ½ x Base x Height ₌ ½ b x h (if base and height are given).

(a). Scalene Triangle:

Perimeter ₌ a ₊ b ₊ c (sum of all side).

⇒ Area ₌ √s (s - a) (s - b) (s - c)

Where, s ₌ a ₊ b ₊ c / 2.

(b). Isosceles Triangle:

Perimeter ₌ a ₊ a ₊ b.

⇒ Area ₌ √s (s - a) (s - b) (s - c) or ½ x b x h

where , h ₌ √a ² - (b / 2) ²

a ₌ Equal side

b ₌ Unequal side

(c). Equilateral Triangle:

Perimeter ₌ a ₊ a ₊ a ₌ 3a.

⇒ Area ₌ √3/4 a ²

a ₌ Side

h ₌ √3 / 2 a.

(d). Right angled Triangle:

Perimeter ₌ a ₊ b ₊ c

⇒ Area ₌ ½ x base x height ₌ ½ x b x a.

Perimeter ₌ AB ₊ BC ₊ CD ₊ AD

⇒ Area ₌ ½ x d (h₁ ₊ h₂).

(3). Trapezium:

Perimeter ₌ a ₊ b ₊ c ₊ d

⇒ Area ₌ ½ x (sum of parallel side) x (Distance between parallel sides) ₌ ½ x (a ₊ b) x h.

(4). Parallelogram:

Perimeter ₌ a ₊ b ₊ a ₊ b ₌ 2 (a ₊ b).

⇒ Area ₌ Base x Height or 2 (Area of one Triangle) ₌ 2 x √s (s - a) (s - b) (s - c).

(5). Rectangle:

Perimeter ₌ 2 (a ₊ b).

⇒ Area ₌ Length x Breadth ₌ L x B.

(6). Rhombus:

Perimeter ₌ 4a

⇒ Area ₌ ½ x d₁ x d₂

d₁ and d₂ ₌ Diagonals.

(7). Square:

Perimeter ₌ 4a

⇒ Area ₌ a ²

(8). Circle:

Perimeter ₌ 2ᴫ𝑟

⇒ Area ₌ ᴫ𝑟 ²

(9). Semi - Circle:

Perimeter ₌ ᴫ𝑟 ₊ 2𝑟

⇒ Area ₌ ½ ᴫ𝑟 ²

(10). Cuboid:

⇒ Curved / Lateral Surface Area (C) ₌ 2 (LH ₊ BH)

⇒ Total Surface Area (S) ₌ 2 (LB ₊ BH ₊ HL)

⇒ Base (B) ₌ LB

⇒ Volume ₌ L x B x H

(11). Cube:

⇒ Curved / Lateral Surface Area (C) ₌ 4 a ²

⇒ Total Surface Area (S) ₌ 6 a ²

⇒ Base (B) ₌ a ²

⇒ Volume ₌ a ³

(12). Right Prism:

⇒ Curved / Lateral Surface Area (C) ₌ Height of Prism x Perimeter of Base.

⇒ Total Surface Area (S) ₌ C x 2 B

⇒ Base (B) ₌ Depends on the shapes of bases

⇒ Volume ₌ Base area x Height.

(13). Cylinder:

⇒ Curved / Lateral Surface Area (C) ₌ 2ᴫ𝑟h

⇒ Total Surface Area (S) ₌ 2ᴫ𝑟 (r ₊ h)

⇒ Base (B) ₌ ᴫ𝑟 ²

⇒ Volume ₌ ᴫ𝑟 ² h.

(14). Cone:

⇒ Curved / Lateral Surface Area (C) ₌ ᴫ𝑟l where, l ₌ √ (h ² ₊ 𝑟 ²)

⇒ Total Surface Area (S) ₌ ᴫ𝑟 (r ₊ l)

⇒ Base (B) ₌ ᴫ𝑟 ²

⇒ Volume ₌ ⅓ ᴫ𝑟 ² h.

(15). Frustum of Cone:

⇒ Curved / Lateral Surface Area (C) ₌ ᴫ (R ₊ 𝑟) l

⇒ Total Surface Area (S) ₌ ᴫl (R ₊ 𝑟) ₊ ᴫR ² ₊ ᴫ𝑟 ²

⇒ Base (B) ₌ ᴫ𝑟 ² or ᴫR ²

⇒ Volume ₌ ⅓ ᴫh (R ² ₊ 𝑟 ₊ R𝑟).

(16). Sphere:

⇒ Curved / Lateral Surface Area (C) ₌ 4 ᴫ𝑟 ²

⇒ Total Surface Area (S) ₌ 4 ᴫ𝑟 ²

⇒ Volume ₌ 4/3 ᴫ𝑟 ³

(17). Hemisphere:

⇒ Curved / Lateral Surface Area (C) ₌ 2 ᴫ𝑟 ²

⇒ Total Surface Area (S) ₌ 3 ᴫ𝑟 ²

⇒ Base (B) ₌ ᴫ𝑟 ²

⇒ Volume ₌ ⅔ ᴫ𝑟 ³

Some of the general preparation tips to crack the Maths section are:

Try to understand the concepts of the topics. Do not go cramming.

Do not try to get into some new topics at the end day of preparation.

Revise the old topics. Go through formulas, and try practicing them.

During preparation, check short-cuts too.

As per the new Exam Pattern released by the CLAT authority, 13 to 17 questions will be asked in the Elementary Mathematics section of the CLAT 2022.
Based on the previous year's CLAT Exam Analysis, the Math Section's difficulty level was easy to moderate.
No, in AILET Exam the questions are directly asked. There are no comprehension-based questions.
Yes. With a proper preparation strategy, it is easy to prepare for the AILET Mathematics section.
It solely depends on you whether you need to take AILET coaching or not. If your fundamentals are strong, then it is easy to crack the exam just by solving previous year sample papers and Mock tests. However, if you want to improve your fundamentals then opting for coaching is beneficial.
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• Important Maths Form...

# Important Maths Formulae for Quant Preparation for Law Entrance Exams

Author : Palak Khanna

Updated On : April 20, 2022

SHARE

Law entrance exams in India for LLB courses like CLAT and SLAT test a candidate's skill in Quantitative Aptitude.

The questions in the Quantitative Aptitude Section for the Law entrance examination are from various topics like

• Number Systems
• Profit, Loss and Discount
• LCM and HCF
• Time and Work
• Averages
• Surds and Indices
• Probability
• Set Theory & Function
• Permutation & Combination
• Coordinate Geometry
• Mensuration, etc.

Thus, it becomes crucial that candidates should be well prepared with the basic formulas required for solving questions with higher accuracy and in the least time possible.

Here is a list of essential formulas candidates need to successfully crack the Common Law Admission Test (CLAT).

This article will learn about the important maths formulas for the Quantitative Aptitude section to prepare for the Law entrance examination.

## List of Important Maths Formulas for Quant Preparation

Here below you will find all the important maths formulas and important topics which a candidate should know and practice to excel in the Law entrance examination.

(1) Number Systems

(2) Profit, Loss, and Discount

(3) LCM (Least common multiple) and HCF (Highest common factor)

(4) Speed, Time, and Distance.

(5) Percentages.

(6) Time and Work

(7) Averages.

(8) Simple and Compound Interest.

(9) Logarithm.

(10) Probability

(11) Surds and Indices

(12) Set Theory and Functions

(13) Permutation and Combination.

(14) Mixtures and Allegations

(15) Trigonometry

(16) Coordinate Geometry.

(17) Mensuration.  ### (1) Number Systems

In the Indian system, numbers are expressed by means of symbols, namely 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. We called them digits. Here, 0 is called an insignificant digit whereas 1, 2, 3, 4, 5, 6, 7, 8, and 9 are called significant digits.

We can express a number in two ways i.e. Notation and Numeration.

Notation - Representing a number in figures is known as Notation. For example, 250, 300, 1000, 12000.

Numeration - Representing a number in words is known as Numeration.

Enhance your CLAT Maths preparation by learning the important formulas that are given in the post below.

#### Different Types of Numbers

(1). Natural Numbers - (N) If N is the set of natural numbers, then we write N ₌ {1, 2, 3, 4, 5, 6,....}. The smallest natural number is 1.

(2). Whole Numbers - (W) If W is the set of whole numbers, then we write W ₌ {0, 1, 2, 3, 4, 5,.....}. The smallest whole number is 0.

(3). Integers - (I) If I is the set of integers, then we write I ₌ {...., -3, -2, -1, 0, 1, 2, 3,..,}. Where, {1, 2, 3, …} is the set of positive integers. {-1, -2, -3,..} is the set of negative integers and 0 is neither positive nor negative.

(4). Rational Numbers - (Q) Any number which can be expressed in the form of p/q, where both p and q are integers and q ≠ 0 is called a rational number.

(5). Irrational Numbers - Non - recurring and non - terminating decimals are called irrational numbers. These numbers cannot be expressed in the form of p/q and q ≠ 0.

(6). Real Number - It includes both rational and irrational numbers.

#### Formulas of Number Systems

⇒ Sum of the first ‘n’ natural numbers i.e. 1 ₊ 2 ₊ 3 ₊ 4 ₊ 5 ₊ .. ₊ n ₌ n (n ₊ 1) / 2.

⇒ Sum of the squares of the first ‘n’ natural numbers i.e. 1² ₊ 2² ₊ 3² ₊ 4² ₊ …. ₊ n² ₌ n (n ₊ 1) (2n ₊ 1) / 6.

⇒ Sum of the cubes of first ‘n’ natural numbers i.e. 1³ ₊ 2³ ₊ 3³ ₊ … ₊ n³ ₌ {n (n ₊ 1) / 2}².

⇒ Sum of first ‘n’ odd numbers ₌ n².

⇒ Sum of first ‘n’ even numbers ₌ n (n ₊ 1).

#### Mathematical Formulas:

⇒ (a – b)² ₌ (a² ₊ b² – 2ab)

⇒ (a ₊ b)² ₌ (a² ₊ b² ₊ 2ab)

⇒ (a ₊ b) (a – b) ₌ (a² – b²)

⇒ (a ₊ b)² ₌ (a² ₊ b² ₊ 2ab)

⇒ (a ₊ b ₊ c)² ₌ a² ₊ b² ₊ c² ₊ 2 (ab ₊ bc ₊ ca)

⇒ (a³ – b³) ₌ (a – b) (a² ₊ ab ₊ b²)

⇒ (a³ ₊ b³) ₌ (a ₊ b) (a² – ab ₊ b²)

⇒ (a³ ₊ b³ ₊ c³ – 3abc) ₌ (a ₊ b ₊ c) (a² ₊ b² ₊ c² – ab – bc – ac)

⇒ When a ₊ b ₊ c ₌ 0, then a³ ₊ b³ ₊ c³ ₌ 3abc

⇒ (a ₊ b) n ₌ an ₊ (nC1) an -1b ₊ (nC2) an - 2b² ₊ … ₊ (nCn-1) abn -1 ₊ bn

### (2) Profit, Loss, and Discount

It is a basic concept of arithmetic in which we study the gain or loss in a business transaction. Profit and loss are the terms related to transactions in trade and business. Whenever a purchase article is sold, then either profit is earned or loss is incurred.

#### Formulas of Profit, Loss, and Discount

⇒ Profit / Gain ₌ Selling Price (SP) - Cost Price (CP).

⇒ Profit Percentage (%) ₌ (Profit / Cost Price (CP) x 100)

⇒ Selling Price (SP) ₌ (100 ₊ Profit Percentage / 100) x Cost Price.

⇒ Cost Price ₌ 100 / (100 ₊ Profit Percentage) x Selling Price.

⇒ Loss ₌ Cost Price (CP) - Selling Price (SP).

⇒ Loss Percentage (%) ₌ (Loss / Cost Price x 100).

⇒ Selling Price ₌ (100 - Loss Percentage / 100) x Cost Price

⇒ Cost Price ₌ 100 / (100 - Loss Percentage) x Selling Price.

### (3) LCM (Least common multiple) and HCF (Highest common factor)

LCM - The LCM of two or more given numbers is the least number to be exactly divisible by each of them.

Multiples of 25 are 25, 50, 75, 100, 125, 150,....

Multiples of 30 are 30, 60, 90, 120, 150, 180,...

The least common factor (LCM) is 150.

HCF - The highest common factor of two or more given numbers is the largest of their common factors.

Factors of 20 are 1, 2, 4, 5, 10, 20.

Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.

Common factors are 1, 2, and 4.

Highest common factor (HCF) is 4.

#### Formulas of LCM and HCF

⇒ LCM x HCF ₌ Products of the numbers.

⇒ LCM of Co - prime numbers ₌ Products of the numbers.

### (4) Speed, Time and Distance

Distance is measured in metres, Kilometres or miles.

Time is measured in hours, minutes or seconds.

Speed is measured in kilometre per hour (kmph), metre per hour (mph) or metre per second (mps).

#### Formulas of Speed, Time and Distance

⇒ Speed ₌ Distance / Time

⇒ Time ₌ Distance / Speed.

⇒ Distance ₌ Speed x Time.  ### (5) Percentages

It is used to express how longer or smaller one quantity is relative to another quantity. Percent means ‘per hundred’ it is given by ‘%’ symbol

#### Formulas of Percentage

⇒ Percentage Increased ₌ (Increased / Original Value x 100) %

⇒ Percentage decrease ₌ (decreased / Original Value x 100) %

⇒ If the price of a commodity increases by r%, then the reduction in consumption, so as not to increase the expenditure is {r / (100 ₊ r) x 100} %.

⇒ If the price of a commodity decreases by r%, then the increase in consumption, so as not to decrease the expenditure is {r / (100 - r) x 100} %.

### (6) Time and Work

Work to be done is generally considered as one unit, it may be digging a bench, constructing or painting a wall, filling up or emptying a tank, reservoir or a cistern.

#### Formulas of Time and Work

⇒ If A can do a piece of work in ‘n’ days, then work done by A in 1 day ₌ 1 / n.

⇒ If A’s 1 day work ₌ 1 / n, then A can finish the whole work in ‘n’ days

⇒ If A is twice as good a workman as B, then

Ratio of work done by A and B ₌ 2 : 1.

Ration of time taken by A and B ₌ 1 : 2.

⇒ If two persons A and B can individually do some work in ‘a’ and ‘b’ days respectively, then A and B together can complete the same work in (ab / a ₊ b) days.

⇒ If two persons A and B together can complete the same work in ‘a’ days and A (or B) can individually do some work in ‘b’ days then B (or A) can complete the work in (ab / b - a) days.

### (7) Averages

If all the given quantities have the same value, then the number itself is the average.

Formulas of Averages

The average of a given number of quantities of the same kind is expressed as

⇒ Average ₌ Sum of Quantities / Number of Quantities.

Average is also called the Arithmetic Mean. Also,

⇒ Sum of quantities ₌ Average x Number of Quantities.

⇒ Number of Quantities ₌ Sum of Quantities / Average.

### (8) Simple and Compound Interest

Simple Interest - If the interest is calculated on the original principal at any rate of interest for any period of time, then it is called simple interest.

Compound Interest - The interest for the future period is calculated not only on the principal, but also on the interest earned until the previous period, known as compound interest.

#### Formulas of Simple and Compound Interest

When interest is compounded annually:

⇒ Amount ₌ Principal (1 ₊ R / 100)ⁿ

When interest is compounded half - yearly::

⇒ Amount ₌ Principal (1 ₊ R/2 / 100)²ⁿ

When interest is compounded Quarterly:

⇒ Amount ₌ Principal (1 ₊ R/4 / 100)⁴ⁿ

When interest is compounded annually but time is in fraction, say 3 ⅖ years.:

⇒ Amount ₌ Principal (1 ₊ R / 100)³ x (1 ₊ ⅖ R / 100)

When Rates are different for different years, say R₁ %, R₂ %, R₃ % for 1st, 2nd, and 3rd year respectively:

Then,

⇒ Amount ₌ Principal (1 ₊ R₁ / 100) (1 ₊ R₂ / 100) (1 ₊ R₃ / 100).

Present worth of Rs. 𝑥 due n years hence is given by:

⇒ Present worth ₌ 𝑥 / (1 ₊ R / 100).

(9). Logarithm

#### Formulas of Logarithm:

⇒ logₐ (xy) ₌ logₐ x ₊ logₐ y

⇒ logₐ (x / y) ₌ logₐ x - logₐ y

⇒ logₓ x ₌ 1.

⇒ logₐ 1 ₌ 0.

⇒ logₐ (xⁿ) ₌ n (logₐ x)

⇒ logₐ x ₌ 1 / logₓ a

⇒ logₐ x ₌ logₑ x / logₑ a ₌ log x / log a.

### (10) Probability

Sample Space: When we perform an experiment, then the set S of all possible outcomes is called the sample space.

Event: Any subset of a sample space is called an event.

The Probability of Occurrence of an Event:

Let S be the sample and let E be an event.

Therefore, P(E) n(E) / n(S).

### (11) Surds and Indices

Root of any number is called surds e.g., 2, 3, ⁸5, ³7and etc.If P be a rational number and 𝒎 is positive integer, then ‴P is a surd of order 𝒎𝒏. When a number P is multiplied by itself 𝒏 times, then the product is called 𝒏th power of P and is written as Pⁿ. Here, P is called the basis and 𝒏 is known as the index of the power. (Here, the plural of index is called indices).

#### Formulas of Surds and Indices:

Law of Indices:

⇒ 𝑎 ‴ x 𝑎 ″ ₌ 𝑎 ‴ ⁺ ″

⇒ 𝑎 ‴ ÷ 𝑎 ″ ₌ 𝑎 ‴ ⁻ ″

⇒ (𝑎 ‴) ″ ₌ 𝑎 ‴″

⇒ (𝑎𝑏) ″ ₌ 𝑎 ″ 𝑏 ″

⇒ (𝑎 / 𝑏) ″ ₌ 𝑎 ″ / 𝑏 ″

⇒ 𝑎 ⁰ ₌ 1

Law of Surds:

⇒ ″ √ 𝑎 ₌ 𝑎 ¹ / ⁿ

⇒ (″ √ 𝑎) ″ ₌ 𝑎

⇒ ″ √ 𝑎𝑏 ₌ ″ √ 𝑎 x ″ √ 𝑏

⇒ ‴ √ ″ √ 𝑎 ₌ ‴″ √ 𝑎 ₌ ″ √ ‴ √ 𝑎

⇒ ″ √ 𝑎 / 𝑏 ₌ ″ √ 𝑎 / ″ √ 𝑏

⇒ (″ √ 𝑎) ‴ ₌ ″ √ 𝑎‴

### (12) Set Theory and Function

The Demorgan’s Law is the basic and most important formula for sets, which is defined as

(A ∩ B) ‘ = A’ U B’ and (A U B)’ = A’ ∩ B’

The relation R⊂A×AR⊂A×A is said to be called as:

⇒ Reflexive Relation: If a R a ∀∀ a ∈∈ A.

⇒ Symmetric Relation: If aRb, then bRa ∀∀ a, b ∈∈ A.

⇒ Transitive Relation: If aRb, bRc, then aRc ∀∀ a, b, c ∈∈ A.

If any relation R is reflexive, symmetric and transitive in a given set A, then that relation is known as an equivalence relation.

### (13) Permutation and Combination

Permutation and Combination are the ways to represent a group of objects by selecting them in a set and forming subsets. It defines the various ways to arrange a certain group of data. When we select the data or objects from a certain group, it is said to be permutations, whereas the order in which they are represented is called combination. Both concepts are very important in Mathematics.

Formula of Permutation is:

A permutation is the choice of r things from a set of n things without replacement and where the order matters.

nPr = (n!) / (n-r)!

Formula of Combination is:

A combination is the choice of r things from a set of n things without replacement and where order doesn't matter.

ɴCᵣ = (n/r) = ɴPᵣ / r! = n! / r! (n - r)!

### (14) Mixtures and Allegation

When two or more than two substances are mixed in any ratio to produce a product, then the product is known as a mixture. The process to produce a product is known as alligation.

The cost price of a unit quantity of the mixture is called the mean price.

Formula of Mixture:

⇒ Quantity of cheaper article / Quantity of costly article ₌ (Cost price of a unit of costly article - Average price) / (Average price - Cost price of a unit of cheaper article).

### (15) Trigonometry

Trigonometric Identities:

⇒ Sine ₌ Opposite / Hypotenuse

⇒ Secant ₌ Hypotenuse / Adjacent

⇒ Cosine ₌ Adjacent / Hypotenuse

⇒ Tangent ₌ Opposite / Adjacent

⇒ Co−Secant ₌ Hypotenuse / Opposite

⇒ Co−Tangent ₌ Adjacent / Opposite

The reciprocal identities are given as:

⇒ CosecΘ ₌ 1 / sinΘ

⇒ secΘ ₌ 1 / cosΘ

⇒ cotΘ ₌ 1 / tanΘ

⇒ sinΘ ₌ 1 / CosecΘ

⇒ cosΘ ₌ 1 / secΘ

⇒ tanΘ ₌ 1 / cotΘ

### (16) Coordinate Geometry

The Distance Between two Points A and B:

⇒ AB ² ₌ (Bx – Ax) ² ₊ (By – Ay) ²

The Midpoint of a Line Joining Two Points

The midpoint of the line joining the points (x1, y1) and (x2, y2) is:

⇒ [½ (x1 ₊ x2), ½ (y1 ₊ y2)]

The Equation of a Line Using One Point and the Gradient

The equation of a line which has gradient m and which passes through the point (x1, y1) is:

⇒ y – y1 ₌ m (x – x1).

### (17) Mensuration

Area - Area of a two dimensional figure is the amount of surface enclosed by its boundary. It is measured in square units.

Perimeter - Perimeter of a two dimensional figure is the length of its boundary. It is measured in units.

Volume - Volume of a 3D figure is the amount of space occupied by it. It is measured in cubic units.

Surface Area - Surface area of a 3D figure is the total area of all of its surfaces. It is measured in square units.

#### Formulas of Mensuration:

(1). Triangle:

Perimeter ₌ a ₊ b ₊ c (sum of all side).

⇒ Area ₌ ½ x Base x Height ₌ ½ b x h (if base and height are given).

(a). Scalene Triangle:

Perimeter ₌ a ₊ b ₊ c (sum of all side).

⇒ Area ₌ √s (s - a) (s - b) (s - c)

Where, s ₌ a ₊ b ₊ c / 2.

(b). Isosceles Triangle:

Perimeter ₌ a ₊ a ₊ b.

⇒ Area ₌ √s (s - a) (s - b) (s - c) or ½ x b x h

where , h ₌ √a ² - (b / 2) ²

a ₌ Equal side

b ₌ Unequal side

(c). Equilateral Triangle:

Perimeter ₌ a ₊ a ₊ a ₌ 3a.

⇒ Area ₌ √3/4 a ²

a ₌ Side

h ₌ √3 / 2 a.

(d). Right angled Triangle:

Perimeter ₌ a ₊ b ₊ c

⇒ Area ₌ ½ x base x height ₌ ½ x b x a.

Perimeter ₌ AB ₊ BC ₊ CD ₊ AD

⇒ Area ₌ ½ x d (h₁ ₊ h₂).

(3). Trapezium:

Perimeter ₌ a ₊ b ₊ c ₊ d

⇒ Area ₌ ½ x (sum of parallel side) x (Distance between parallel sides) ₌ ½ x (a ₊ b) x h.

(4). Parallelogram:

Perimeter ₌ a ₊ b ₊ a ₊ b ₌ 2 (a ₊ b).

⇒ Area ₌ Base x Height or 2 (Area of one Triangle) ₌ 2 x √s (s - a) (s - b) (s - c).

(5). Rectangle:

Perimeter ₌ 2 (a ₊ b).

⇒ Area ₌ Length x Breadth ₌ L x B.

(6). Rhombus:

Perimeter ₌ 4a

⇒ Area ₌ ½ x d₁ x d₂

d₁ and d₂ ₌ Diagonals.

(7). Square:

Perimeter ₌ 4a

⇒ Area ₌ a ²

(8). Circle:

Perimeter ₌ 2ᴫ𝑟

⇒ Area ₌ ᴫ𝑟 ²

(9). Semi - Circle:

Perimeter ₌ ᴫ𝑟 ₊ 2𝑟

⇒ Area ₌ ½ ᴫ𝑟 ²

(10). Cuboid:

⇒ Curved / Lateral Surface Area (C) ₌ 2 (LH ₊ BH)

⇒ Total Surface Area (S) ₌ 2 (LB ₊ BH ₊ HL)

⇒ Base (B) ₌ LB

⇒ Volume ₌ L x B x H

(11). Cube:

⇒ Curved / Lateral Surface Area (C) ₌ 4 a ²

⇒ Total Surface Area (S) ₌ 6 a ²

⇒ Base (B) ₌ a ²

⇒ Volume ₌ a ³

(12). Right Prism:

⇒ Curved / Lateral Surface Area (C) ₌ Height of Prism x Perimeter of Base.

⇒ Total Surface Area (S) ₌ C x 2 B

⇒ Base (B) ₌ Depends on the shapes of bases

⇒ Volume ₌ Base area x Height.

(13). Cylinder:

⇒ Curved / Lateral Surface Area (C) ₌ 2ᴫ𝑟h

⇒ Total Surface Area (S) ₌ 2ᴫ𝑟 (r ₊ h)

⇒ Base (B) ₌ ᴫ𝑟 ²

⇒ Volume ₌ ᴫ𝑟 ² h.

(14). Cone:

⇒ Curved / Lateral Surface Area (C) ₌ ᴫ𝑟l where, l ₌ √ (h ² ₊ 𝑟 ²)

⇒ Total Surface Area (S) ₌ ᴫ𝑟 (r ₊ l)

⇒ Base (B) ₌ ᴫ𝑟 ²

⇒ Volume ₌ ⅓ ᴫ𝑟 ² h.

(15). Frustum of Cone:

⇒ Curved / Lateral Surface Area (C) ₌ ᴫ (R ₊ 𝑟) l

⇒ Total Surface Area (S) ₌ ᴫl (R ₊ 𝑟) ₊ ᴫR ² ₊ ᴫ𝑟 ²

⇒ Base (B) ₌ ᴫ𝑟 ² or ᴫR ²

⇒ Volume ₌ ⅓ ᴫh (R ² ₊ 𝑟 ₊ R𝑟).

(16). Sphere:

⇒ Curved / Lateral Surface Area (C) ₌ 4 ᴫ𝑟 ²

⇒ Total Surface Area (S) ₌ 4 ᴫ𝑟 ²

⇒ Volume ₌ 4/3 ᴫ𝑟 ³

(17). Hemisphere:

⇒ Curved / Lateral Surface Area (C) ₌ 2 ᴫ𝑟 ²

⇒ Total Surface Area (S) ₌ 3 ᴫ𝑟 ²

⇒ Base (B) ₌ ᴫ𝑟 ²

⇒ Volume ₌ ⅔ ᴫ𝑟 ³

Some of the general preparation tips to crack the Maths section are:

Try to understand the concepts of the topics. Do not go cramming.

Do not try to get into some new topics at the end day of preparation.

Revise the old topics. Go through formulas, and try practicing them.

During preparation, check short-cuts too.

As per the new Exam Pattern released by the CLAT authority, 13 to 17 questions will be asked in the Elementary Mathematics section of the CLAT 2022.
Based on the previous year's CLAT Exam Analysis, the Math Section's difficulty level was easy to moderate.
No, in AILET Exam the questions are directly asked. There are no comprehension-based questions.
Yes. With a proper preparation strategy, it is easy to prepare for the AILET Mathematics section.
It solely depends on you whether you need to take AILET coaching or not. If your fundamentals are strong, then it is easy to crack the exam just by solving previous year sample papers and Mock tests. However, if you want to improve your fundamentals then opting for coaching is beneficial.