Important Formulas for CAT 2023 to Score 99+ Percentile

Important Formulas for CAT 2023: CAT (Common Admission Test) is a very tough exam for MBA studies in India. Candidates, to do well in the Quantitative Aptitude and Data Interpretation sections, it's important to understand important CAT formulas and rules very well. You can find these formulas in books, online, and coaching materials. However, remembering these formulas and knowing how to use them during the actual CAT exam is very important.

Important Formulas for CAT 2023 to Score 99+ Percentile

Also Read: How to Prepare for CAT Exam

Profit, Loss, and Discount

• Profit % = Profit/(C P)×100
• Profit/Gain = (S.P.) – (C.P.)
• C P = 100/(100+gain %)×S P
• S P = (100+ gain %) /100 × C P
• Loss = C P – S P
• Loss % = Loss/(C P) × 100
• S P = (100 – loss %)/100 × C P
• C P = 100/(100 – loss %) × S P
• Discount = Marked Price – Selling Price
• Discount Percentage = (Discount/Marked price) x 100

Number System

• (a + b)² = (a² + b² + 2ab)
• (a – b)² = (a² + b² – 2ab)
• (a + b)(a – b) = (a² – b²)
• (a + b)² = (a² + b² + 2ab)
• (a³ – b³) = (a – b)(a² + ab + b²)
• (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
• (a³ + b³) = (a + b)(a² – ab + b²)
• When a + b + c = 0, then a³ + b³ + c³ = 3abc
• (a³ + b³ + c³ – 3abc) = (a + b + c)(a² + b² + c² – ab – bc – ac)

LCM and HCF

• HCF × LCM = Product of the two numbers.
• LCM of co-prime numbers = Product of the numbers.
• HCF of co-prime numbers = 1.
• HCF of two numbers a and b = HCF of (a-b, b).
• LCM of two numbers a and b = (a x b) / HCF(a, b).
• HCF of three numbers a, b, and c = HCF of (HCF(a, b), c).
• LCM of three numbers a, b, and c = LCM of (LCM(a, b), c).

Time and Work

• Work Done = Time Taken × Rate of Work
• Rate of Work = 1 / Time Taken
• Time Taken = 1 / Rate of Work
• If a piece of work is done by A in n days, then A’s 1 day’s work = 1/n
• If A’s 1 day’s work = 1/n, then A can finish the work in n days
• If A can do a piece of work in x days and B can do the same work in y days, then the work done by both A and B in one day = 1/x + 1/y
• If A can do a piece of work in x days and B can do the same work in y days, then the time taken by both A and B to complete the work together = (xy) / (x + y)

Averages

• Average = (Sum of observations) / (Number of observations)
• If the average of n numbers is A, then the sum of the n numbers is nA.
• If the average of n numbers is A and m more numbers are added to the list, then the new average becomes (nA + mB) / (n + m), where B is the average of the m numbers.
• If the average of n numbers is A and each number is increased by x, then the new average becomes (nA + nx) / n.
• If the average of n numbers is A and each number is decreased by x, then the new average becomes (nA – nx) / n.

Logarithm

• Definition of a logarithm: If x>0 and b is a constant (b≠1), then y=logbx if and only if x=by.
• Logarithmic identities:
  • logb(xy) = logbx + logby
  • logb(x/y) = logbx – logby
  • logb(x^p) = p logbx
  • logb1 = 0
  • logbb = 1
  • logb(x) = 1 / logx(b)
• Change of base formula: logb(x) = loga(x) / loga(b)
• Common logarithm: log10(x) = log(x)
• Natural logarithm: loge(x) = ln(x)

Surds and Indices

• Product rule: a^m × a^n = a^(m+n)
• Quotient rule: a^m / a^n = a^(m-n)
• Power rule: (a^m)^n = a^(m×n)
• Negative exponent rule: a^(-m) = 1 / a^m
• Rational exponent rule: a^(m/n) = nth root of a^m
• Fractional exponent rule: a^(p/q) = qth root of a^p
• Surds multiplication rule: √a × √b = √(ab)
• Surds division rule: √a / √b = √(a/b)
• Surds addition rule: √a + √b ≠ √(a+b)
• Surds subtraction rule: √a – √b ≠ √(a-b)

Set Theory & Function

• De Morgan’s Law: (A ∩ B)’ = A’ U B’ and (A U B)’ = A’ ∩ B’
• Cardinality of a set: The number of elements in a set is called its cardinality.
• Union of sets: A U B = {x: x ∈ A or x ∈ B}
• Intersection of sets: A ∩ B = {x: x ∈ A and x ∈ B}
• Complement of a set: A’ = {x: x ∉ A}
• Difference of sets: A – B = {x: x ∈ A and x ∉ B}
• Cartesian product of sets: A × B = {(a, b): a ∈ A and b ∈ B}

Trigonometry

• Sine = Opposite/Hypotenuse
• Cosine = Adjacent/Hypotenuse
• Tangent = Opposite/Adjacent
• Secant = Hypotenuse/Adjacent
• Cosecant = Hypotenuse/Opposite
• Cotangent = Adjacent/Opposite
• Sum and Difference Formulas:
  • sin(u + v) = sin(u)cos(v) + cos(u)sin(v)
  • cos(u + v) = cos(u)cos(v) – sin(u)sin(v)
  • sin(u – v) = sin(u)cos(v) – cos(u)sin(v)
  • cos(u – v) = cos(u)cos(v) + sin(u)sin(v)
• Reciprocal Identities:
  • cosec θ = 1/sin θ
  • sec θ = 1/ cos θ
  • cot θ = 1/tan θ
  • sin θ = 1/cosec θ
  • cos θ = 1/sec θ
  • tan θ = 1/cot θ

Coordinate Geometry

• Distance Formula: The distance between two points P(x1, y1) and Q(x2, y2) is given by √[(x2 – x1)^2 + (y2 – y1)^2].
• Slope Formula: The slope of a line passing through two points P(x1, y1) and Q(x2, y2) is given by (y2 – y1) / (x2 – x1).
• Equation of a Line: The equation of a line passing through a point (x1, y1) with slope m is given by y – y1 = m(x – x1) or y = mx + (y1 – mx1).
• Midpoint Formula: The midpoint of a line segment joining two points P(x1, y1) and Q(x2, y2) is given by [(x1 + x2) / 2, (y1 + y2) / 2].
• Section Formula: If a point R(x, y) divides the line segment joining two points P(x1, y1) and Q(x2, y2) in the ratio m:n, then x = [(nx2) + (mx1)] / (m + n) and y = [(ny2) + (my1)] / (m + n).
• Slope of a Parallel Line: The slope of a line parallel to a line with slope m is also m.
• Slope of a Perpendicular Line: The slope of a line perpendicular to a line with slope m is -1/m.

Mensuration

• Area of a triangle = 1/2 × base × height
• Area of a rectangle = length × breadth
• Area of a square = side × side
• Area of a parallelogram = base × height
• Area of a trapezium = 1/2 × (sum of parallel sides) × height
• Circumference of a circle = 2πr
• Area of a circle = πr^2
• Volume of a cube = side^3
• Surface area of a cube = 6 × side^2
• Volume of a cuboid = length × breadth × height
• Surface area of a cuboid = 2(lb + bh + hl)
• Volume of a cylinder = πr^2h
• Surface area of a cylinder = 2πrh + 2πr^2
• Volume of a sphere = 4/3 × πr^3
• Surface area of a sphere = 4πr^2

Number Systems

• Sum of first n natural numbers = n(n+1)/2
• Sum of squares of first n natural numbers = n(n+1)(2n+1)/6
• Sum of cubes of first n natural numbers = [n(n+1)/2]^2
• Sum of first n odd numbers = n^2
• Dividend = (Divisor x Quotient) + Remainder

Sum of Series

• 1 + 2 + 3 + … + n = n(n + 1)/2, and more.

Geometry

• Area of sector, Cube Volume, Surface Area, Diagonal, Volume of cuboid, and more.