CAT Quant Formula List 2025: Important Formulas, Shortcuts & Last-Minute Revision Guide

CAT Arithmetic Important Formulas

The Arithmetic section is the most important section in the Quantitative Aptitude Section, which is also useful to solve the Data Interpretation problems. The following are some 50+ Important Formulas for CAT Preparation of this section, which are given in this CAT Formula Sheet:

Percentage:

Following are some basic and Important Formulas for CAT 2025 related to percentages:

Profit & Loss:

Following are some basic and Important Formulas for CAT 2025 related to Profit and Loss:

Simple Interest (S.I.) & Compound Interest (C.I.):

Following are some basic and Important Formulas for CAT 2025 related to Simple Interest and Compound Interest:

Time, Speed & Distance:

Following are some basic and Important Formulas for CAT 2025 related to Time, Speed and Distance:

Circular Track:

If two people start from the same point on a circular track of length $D$ km with speeds $a$ & $b$ kmph.

Clocks

Time and Work

Following are some basic and Important Formulas for CAT 2025 related to Time and Work:

CAT Algebra Formulas

The Algebra section is a critical part of the Quantitative Aptitude section in the CAT exam. Below are over 50 important formulas for CAT preparation in this section, which are provided in this comprehensive CAT Formula Sheet:

Quadratic Equations

General form: $ ax^2 + bx + c = 0 $

Roots of the equation (Shreedhara Acharya's Formula): $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $

Sum of roots: $ \text{Sum} = -\frac{b}{a} $

Product of roots: $ \text{Product} = \frac{c}{a} $

Discriminant: $ D = b^2 - 4ac $

• If $D > 0$, roots are real and distinct

• If $D = 0$, roots are real and equal

• If $D < 0$, roots are imaginary and distinct

Progression & Series

Indices & Surds

Product Rule: $ a^m \cdot a^n = a^{m+n} $

Quotient Rule: $ \frac{a^m}{a^n} = a^{m-n} $

Power of a Power: $ (a^m)^n = a^{mn} $

Power of a Product: $ (ab)^n = a^n \cdot b^n $

Power of a Quotient: $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $

Negative Exponent: $ a^{-n} = \frac{1}{a^n} $

Logarithmic Rules and Properties

Definition of Logarithm:
$ \log_b a = x ;;\Longleftrightarrow;; b^x = a $

Log of 1: $ \log_b 1 = 0 \quad (b>0, ; b \neq 1) $

Log of the base itself: $ \log_b b = 1 $

Log of a product: $ \log_b (mn) = \log_b m + \log_b n $

Log of a quotient: $ \log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n $

Log of a power: $ \log_b (m^n) = n \cdot \log_b m $

Change of base formula: $ \log_b a = \frac{\log_k a}{\log_k b} $

Base switch rule: $ \log_a b = \frac{1}{\log_b a} $

HCF and LCM

• HCF of given fraction = $\frac{\text{HCF of numerators}}{\text{LCM of denominators}}$

• LCM of given fraction = $\frac{\text{LCM of numerators}}{\text{HCF of denominators}}$

• HCF x LCM= Product of the two numbers

• If the H.C.F of the numbers a, b, c is K, then a, b, c can be written as multiples of K (Kx, Ky, Kz, where x, y, z are some numbers). K divides the numbers a, b, c, so the given numbers can be written as the multiples of K.

• If the H.C.F of the numbers a, b is K, then the numbers (a + b), (a -b) is also divisible by K.

• The numbers a and b can be written as the multiples of K, a = Kx, b = Ky.

(a + b) = (Kx + Ky) = K (x + y)

(a - b) = (Kx - Ky) = K(x – y)

Therefore, (a + b) and (a - b) is also divisible by the H.C.F of a, b.

CAT Geometry and Mensuration Formulas

Mastering important geometry formulas is crucial for cracking the CAT exam, as geometry and mensuration is a key topic in the Quantitative Aptitude section. The following section covers all essential CAT geometry and mensuration formulas:

Triangles:

Properties of Triangles:

• The sum of any two sides is always greater than the third one and the difference of any two sides is less than the third one.

• Let $a, b, c$ be the sides of a triangle, then $|b-c| < a < b+c$

• In a scalene triangle the greatest side is always greater than one-third of the perimeter and less than half of the perimeter.

• Let $a, b, c$ be the sides of the triangle and $a$ be the greatest side. Let the perimeter be $P$. Then

$\dfrac{P}{3} < a < \dfrac{P}{2}$

• For $a, b, c$ sides of a triangle and $a$ the greatest side:

If $a^2 < b^2 + c^2$, then the triangle is acute angled.

If $a^2 = b^2 + c^2$, then the triangle is right angled (Pythagoras theorem).

If $a^2 > b^2 + c^2$, then the triangle is obtuse angled.

Properties related to median

(Here $D$ is the midpoint of side $AC$, so $AD = DC$)

Midpoint of a triangle

• Length of the Median –
  BD = $\frac12 \sqrt{2(AB^2+BC^2)−AC^2}$

• 3× (Sum of squares of sides) =4× (Sum of squares of medians), that is,
  $3(a^2+b^2+c^2)=4({M_a}^2+{M_b}^2+{M_c}^2)$
  where a,b, and c are the sides of the triangle and Ma, Mb, and Mc are the medians.

• In a right-angled triangle, Median of Hypotenuse = Half of Hypotenuse
  That is,
  CD = AB/2

If all the medians are drawn in the triangle, then the 6 small triangles are generated in the triangle, which are equal in the Area.

Area of Triangle:

Heron’s Formula: Let $a, b, c$ be the sides of the triangle,
Area $= \sqrt{s(s-a)(s-b)(s-c)}$ where $s = \dfrac{a+b+c}{2}$ is the semiperimeter.

If two sides and the included angle are given:

Area $= \dfrac{1}{2} \times \text{(Product of given sides)} \times \sin(\text{included angle})$

Area $= \dfrac{1}{2} \times a \times b \times \sin C$

(Example: sides $a, b$ and included angle $C$ are given)

If a side and its respective altitude (perpendicular drawn from the opposite vertex) is given, then:

Area of the triangle $= \dfrac{1}{2} \times \text{Base} \times \text{Height (Altitude)}$

Area of an equilateral triangle = $ \frac{\sqrt{3}}{4} a^2 $

Height (Altitude) of an equilateral triangle = $ \frac{\sqrt{3}}{2} a $

Area of a triangle = $ r \times s $ (where $r$ is the inradius and $s$ is the semiperimeter)

Area of a triangle = $ \frac{abc}{4R} $ (where $a$, $b$, $c$ are sides and $R$ is the circumradius)

Quadrilaterals:

Circle:

Circumference of a circle $= 2\pi r$

Area of a circle $= \pi r^2$

For a semi-circle:

Circumference of a semi-circle $= \pi r$

Perimeter of a semi-circle $= \pi r + 2r$

Area of a semi-circle $= \dfrac{\pi r^2}{2}$

Sector & Segment of a circle

$OAXC$ is called the sector of the circle and $AXC$ is called the segment.

Length of arc $AXC = \dfrac{\theta}{360} \times 2\pi r$ (where $r$ is the radius)

Area of sector $OAXC = \dfrac{\theta}{360} \times \pi r^2$

$2 \times \text{Area of sector} = \text{Length of arc} \times \text{Radius}$

Area of segment $AXC = \text{Area of sector } OAXC - \text{Area of } \triangle OAC$

$A = \dfrac{\theta}{360} \pi r^2 - \dfrac{1}{2} r^2 \sin \theta$

Where:

$\theta$ = angle subtended at the center (in degrees)

$r$ = radius of the circle

Common Tangent

$PQ$ and $RS$ are the direct common tangents of the circles, which are equal in length.

Length of direct common tangent $(L)$:

$L^2 = d^2 - (r_1 - r_2)^2$

Where:

$d =$ distance between centers of the circles

$r_1, r_2 =$ radii of the circles

$PQ$ and $RS$ are the transverse common tangents of the circles, which are equal in length.

Length of transverse common tangent $(L)$:

$L^2 = d^2 - (r_1 + r_2)^2$

Where:

$L =$ length of the transverse common tangent

$d =$ distance between the centres of the two circles

$r_1, r_2 =$ radii of the two circles

Mensuration:

CAT Important Formulas of Trigonometry

Trigonometry plays a vital role in the CAT Quantitative Aptitude section, making it essential to learn and memorise key formulas. This comprehensive list of important CAT trigonometry formulas helps aspirants solve complex problems with speed, accuracy, and confidence during the exam.

1. Basic Trigonometric Ratios

These are defined in relation to a right-angled triangle:

$\sin \theta = \frac{\text{Opposite side}}{\text{Hypotenuse}}$

$\cos \theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}}$

$\tan \theta = \frac{\text{Opposite side}}{\text{Adjacent side}}$

$\csc \theta = \frac{\text{Hypotenuse}}{\text{Opposite side}}$

$\sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent side}}$

$\cot \theta = \frac{\text{Adjacent side}}{\text{Opposite side}}$

2. Pythagorean Identities

$\sin^2 \theta + \cos^2 \theta = 1$

$1 + \tan^2 \theta = \sec^2 \theta$

$1 + \cot^2 \theta = \csc^2 \theta$

3. Trigonometric Functions of Negative Angles

$\sin(-\theta) = -\sin \theta$

$\cos(-\theta) = \cos \theta$

$\tan(-\theta) = -\tan \theta$

$\csc(-\theta) = -\csc \theta$

$\sec(-\theta) = \sec \theta$

$\cot(-\theta) = -\cot \theta$

4. Angle Sum and Difference Formulas

$\sin(A + B) = \sin A \cos B + \cos A \sin B$

$\sin(A - B) = \sin A \cos B - \cos A \sin B$

$\cos(A + B) = \cos A \cos B - \sin A \sin B$

$\cos(A - B) = \cos A \cos B + \sin A \sin B$

$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$

$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$

CAT Permutation& Combination and Probability Key Formulas

Permutation and Combination:

1. Number of permutations (or arrangements) of n different things taken all at a time = n!

2. Number of permutations of n things out of which $P_1$ are alike and are of one type, $P_2$ are alike and are of a second type and $P_3$ are alike and are of a third type and the rest are all different = $\frac{n!}{P_1!P_2!P_3!}$

3. Number of permutations of n different things taken r at a time when repetition is allowed = n × n × n ×… (r times) = n^r.

4. Total number of selections of zero or more things out of k identical things = k + 1.

5. Total number of selections of zero or more things out of n different things =

$n_(C_o ) + n_(C_1 ) + n_(C_2 ) + … + n_(C_n ) = 2^n$

6. The number of selections of 1 or more things out of n different things = $n_(C_1 ) + n_(C_2 ) + … + n_(C_n ) = 2^n-1$

7. Number of ways of arranging n people on a circular track (circular arrangement)

= (n – 1)!

8. Number of squares in a square of n × n side = $1^2 + 2^2 + 3^2 + 4^2 + … n^2$

9. Number of rectangles in a square of n × n side = $1^3 + 2^3 +3^3 + 4^3 + … n^3$

10. A rectangle having m rows and n columns:

The number of squares is given by:

m.n + (m – 1)(n – 1) + (m – 2)(n – 2) + ….. until any of (m –x) or (n – x) comes to 1.

The number of rectangles is given by: (1 + 2 + …… + m)(1 + 2 + …… + n)

Probability:

1. Probability of an Event

Let E be an event and S be the sample space. Then the probability of the event E can be defined as

P(E)=(n(E))/(n(S))

where P(E) = Probability of the event E, n(E) = number of ways in which the event can occur and n(S) = Total number of outcomes possible

2. 0 ≤ P (E) ≤ 1

P(ϕ) = 0 (∵ Probability of occurrence of an impossible event = 0)

3. Addition Theorem

Let A and B be two events associated with a random experiment. Then

P(A U B) = P(A) + P(B) – P(A ∩ B)

If A and B are mutually exclusive events, then P(A U B) = P(A) + P(B) because for mutually exclusive events, P(A ∩ B) = 0

4. If A and B are two independent events, then

P(A ∩ B) = P(A).P(B)

5. Odds on an event

Let E be an event associated with a random experiment. Let x outcomes are favourable to E and y outcomes are not favourable to E, then

Odds in favour of E are x:y,i.e.,x/y, and

Odds against E are y:x,i.e.y/x

P(E) = x/(x+y)

P(not E )=x/(x+y)

6. Conditional Probability

Let A and B be two events associated with a random experiment. Then, the probability of the occurrence of A given that B has already occurred is called conditional probability and denoted by P(A/B)

Example: A bag contains 5 black and 4 blue balls. Two balls are drawn from the bag one by one without replacement. What is the probability of drawing a blue ball in the second draw if a black ball is already drawn in the first draw?

Let A be the event of drawing a black ball in the first draw and B be the event of drawing a blue ball in the second draw. Then, P(B/A) = Probability of drawing a blue ball in the second draw given that a black ball is already drawn in the first draw.

Total Balls = 5 + 4 = 9

Since a black ball is drawn already,

Total number of balls left after the first draw = 8

Total number of blue balls after the first draw = 4

P(B/A) = 4/8=1/2

Download Now: CAT 2025 Quantitative Aptitude Cheat Sheet PDF

CAT 2025 Formula Sheet: Why You Need It?

In the last stage of preparation CAT 2025 formula sheet will help you to recall key concepts, save revision time, and improve accuracy in problem-solving. Having all important formulas from Arithmetic, Algebra, Geometry, and Modern Math in one place boosts your speed and confidence while attempting Quantitative Aptitude questions in the exam. It is helpful in the following ways:

• Saves Time During CAT Revision

• Strengthens Conceptual Clarity

• Minimises Errors and Improves Accuracy

• Enhances Mock Test Performance

• Builds Confidence Before the Exam

CAT 2025 Recommended eBooks and Study Materials

Explore the best CAT 2025 eBooks and study materials recommended by experts for complete preparation. These resources will help aspirants revise efficiently and boost exam preparation.