Updated On : February 22, 2024

**Reader's Digest:** Preparing for law entrance exams often involves honing various skills, including logical reasoning, analytical abilities, and basic mathematics or quantitative aptitude. Mathematics sections in these exams typically cover high school level topics and are designed to test your problem-solving skills and numerical ability.

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.

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.

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.

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.

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.

**(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.

⇒ 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).

**Read more**: **Important number system questions for CLAT**

⇒ (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

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.

⇒ 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.

**Read more**: **Data Interpretation Questions for CLAT Exam**

**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.

⇒ LCM x HCF ₌ Products of the numbers.

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

**Read more**: **Most important Maths questions with answers for CLAT exam**

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).

⇒ Speed ₌ Distance / Time

⇒ Time ₌ Distance / Speed.

⇒ Distance ₌ Speed x Time

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

⇒ 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} %.

**Read more**: **Short tricks to enhance your CLAT logical reasoning preparation**

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.

⇒ 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.

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.

**Read more**: **Short tricks to enhance your time management skills for CLAT**

**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.

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

⇒ 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.

**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).**

Root of any number is called surds e.g., **√**2, **√**3, ⁸**√**5, ³**√**7and 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).

**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‴

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.*

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.

⇒ **n****P****r**** = (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)!**

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).

**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

**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).

**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.

**(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.

**(2). Quadrilateral:**

**⇒** **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

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February 22, 2024

**Reader's Digest:** Preparing for law entrance exams often involves honing various skills, including logical reasoning, analytical abilities, and basic mathematics or quantitative aptitude. Mathematics sections in these exams typically cover high school level topics and are designed to test your problem-solving skills and numerical ability.

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.

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.

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.

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.

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.

**(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.

⇒ 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).

**Read more**: **Important number system questions for CLAT**

⇒ (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

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.

⇒ 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.

**Read more**: **Data Interpretation Questions for CLAT Exam**

**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.

⇒ LCM x HCF ₌ Products of the numbers.

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

**Read more**: **Most important Maths questions with answers for CLAT exam**

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).

⇒ Speed ₌ Distance / Time

⇒ Time ₌ Distance / Speed.

⇒ Distance ₌ Speed x Time

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

⇒ 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} %.

**Read more**: **Short tricks to enhance your CLAT logical reasoning preparation**

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.

⇒ 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.

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.

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**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.

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

⇒ 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.

**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).**

Root of any number is called surds e.g., **√**2, **√**3, ⁸**√**5, ³**√**7and 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).

**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‴

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.*

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.

⇒ **n****P****r**** = (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)!**

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).

**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

**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).

**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.

**(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.

**(2). Quadrilateral:**

**⇒** **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

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