Comprehensive list of essential mathematics formulas required for most competitive exams.

Posted On June 20, 2025

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1. Arithmetic

Percentages:

• Percentage = (Part / Whole) × 100
• Increase % = [(New Value – Original Value) / Original Value] × 100
• Decrease % = [(Original Value – New Value) / Original Value] × 100
• Converting Fraction to Percentage: (Numerator/Denominator) × 100
• Converting Percentage to Fraction: Percentage ÷ 100

Profit and Loss:

• Profit = Selling Price (SP) – Cost Price (CP)
• Loss = Cost Price (CP) – Selling Price (SP)
• Profit % = (Profit / Cost Price) × 100
• Loss % = (Loss / Cost Price) × 100
• Selling Price = Cost Price × (1 + Profit%/100)
• Cost Price = Selling Price / (1 + Profit%/100)

Simple Interest (SI):

• SI = (P × R × T) / 100
• Where: P = Principal, R = Rate of interest (% per annum), T = Time in years

Compound Interest (CI):

• CI = P × [(1 + R/100)^T – 1]
• Amount = P × (1 + R/100)^T
• If compounded half-yearly: CI = P × [(1 + R/200)^(2T) – 1]
• If compounded quarterly: CI = P × [(1 + R/400)^(4T) – 1]

Ratio and Proportion:

• Ratio: a:b = a / b
• Proportion: a:b = c:d ⇒ a/b = c/d
• Continued Proportion: If a/b = b/c, then b is the mean proportion.
• Mean Proportion = √(ab)
• Duplicate Ratio = Square of the given ratio
• Triplicate Ratio = Cube of the given ratio

Average:

• Average = (Sum of quantities) / (Number of quantities)
• Weighted Average = (w₁x₁ + w₂x₂ + … + wₙxₙ) / (w₁ + w₂ + … + wₙ)

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2. Algebra

Algebraic Identities:

• (a + b)^2 = a^2 + 2ab + b^2
• (a – b)^2 = a^2 – 2ab + b^2
• a^2 – b^2 = (a – b)(a + b)
• (x + a)(x + b) = x^2 + (a + b)x + ab
• (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)
• a^3 + b^3 = (a + b)(a^2 – ab + b^2)
• a^3 – b^3 = (a – b)(a^2 + ab + b^2)

Quadratic Equations:

• General form: ax^2 + bx + c = 0
• Roots = [-b ± √(b^2 – 4ac)] / 2a
• Nature of roots:
  • D = b^2 – 4ac
  • If D > 0 ⇒ Real & Distinct
  • If D = 0 ⇒ Real & Equal
  • If D < 0 ⇒ Complex Roots
• Sum of roots = -b/a
• Product of roots = c/a

Progressions:

• Arithmetic Progression (AP):
  • Tₙ = a + (n – 1)d
  • Sₙ = n/2 × [2a + (n – 1)d]
• Geometric Progression (GP):
  • Tₙ = a × r^(n – 1)
  • Sₙ = a(1 – r^n) / (1 – r), for r ≠ 1
  • Infinite GP: S∞ = a / (1 – r), when |r| < 1
• Harmonic Progression (HP):
  • HP is the reciprocal of AP

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3. Geometry

Triangles:

• Area = (1/2) × base × height
• Heron’s Formula: √[s(s – a)(s – b)(s – c)] where s = (a + b + c)/2
• Types: Equilateral (all sides equal), Isosceles (two sides equal), Scalene (no sides equal)
• Angle sum property: Sum of interior angles = 180°
• Exterior angle = sum of opposite interior angles

Circles:

• Circumference = 2πr
• Area = πr^2
• Sector Area = (θ/360) × πr^2
• Arc Length = (θ/360) × 2πr
• Diameter = 2r

Quadrilaterals:

• Rectangle: Area = l × b; Perimeter = 2(l + b)
• Square: Area = a^2; Perimeter = 4a
• Parallelogram: Area = b × h
• Rhombus: Area = (1/2) × d1 × d2
• Trapezium: Area = (1/2) × (a + b) × h

Polygons:

• Sum of interior angles = (n – 2) × 180°
• Measure of each angle (regular polygon) = [(n – 2) × 180°]/n

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4. Mensuration

2D Figures:

• Triangle, Square, Rectangle, Parallelogram, Circle: Use basic area and perimeter formulas

3D Figures:

• Cube:
  • Surface Area = 6a^2
  • Volume = a^3
• Cuboid:
  • Surface Area = 2(lb + bh + hl)
  • Volume = l × b × h
• Cylinder:
  • CSA = 2πrh
  • TSA = 2πr(h + r)
  • Volume = πr^2h
• Cone:
  • Slant height l = √(r^2 + h^2)
  • CSA = πrl
  • TSA = πr(l + r)
  • Volume = (1/3)πr^2h
• Sphere:
  • Surface Area = 4πr^2
  • Volume = (4/3)πr^3
• Hemisphere:
  • CSA = 2πr^2
  • TSA = 3πr^2
  • Volume = (2/3)πr^3

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5. Coordinate Geometry

• Distance = √[(x2 – x1)^2 + (y2 – y1)^2]
• Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
• Slope = (y2 – y1)/(x2 – x1)
• Equation of line with slope m: y – y1 = m(x – x1)
• General form: Ax + By + C = 0
• Circle: (x – h)^2 + (y – k)^2 = r^2

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6. Trigonometry

Basic Ratios:

• sinθ = Opposite/Hypotenuse
• cosθ = Adjacent/Hypotenuse
• tanθ = Opposite/Adjacent
• cotθ = 1/tanθ
• secθ = 1/cosθ
• cscθ = 1/sinθ

Identities:

• sin^2θ + cos^2θ = 1
• tan^2θ + 1 = sec^2θ
• 1 + cot^2θ = csc^2θ

Complementary angles:

• sin(90° – θ) = cosθ
• cos(90° – θ) = sinθ
• tan(90° – θ) = cotθ

Trigonometric values of standard angles (0°, 30°, 45°, 60°, 90°)

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7. Calculus (Basics)

Differentiation:

• Power Rule: d/dx (x^n) = n × x^(n – 1)
• d/dx (e^x) = e^x
• d/dx (ln x) = 1/x
• d/dx (sin x) = cos x
• d/dx (cos x) = -sin x
• d/dx (tan x) = sec^2 x

Integration:

• ∫ x^n dx = x^(n+1)/(n+1) + C (n ≠ -1)
• ∫ 1/x dx = ln|x| + C
• ∫ e^x dx = e^x + C
• ∫ sin x dx = -cos x + C
• ∫ cos x dx = sin x + C

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8. Probability and Statistics

Probability:

• P(E) = Number of favorable outcomes / Total outcomes
• P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
• P(A’) = 1 – P(A)
• If A and B are independent: P(A ∩ B) = P(A) × P(B)

Permutations & Combinations:

• n! = n × (n – 1) × (n – 2) × … × 1
• Permutations (nPr) = n! / (n – r)!
• Combinations (nCr) = n! / [r! × (n – r)!]

Statistics:

• Mean = Sum of observations / Number of observations
• Median: Arrange data in order, take middle value (or average of two middle values)
• Mode = Most frequently occurring value
• Range = Maximum value – Minimum value
• Standard Deviation (σ) = √(Σ(x – μ)^2 / N)

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This formula sheet is highly useful for quick reference in preparation for competitive exams like SSC, CDS, Bank PO, CAT, and other aptitude-based assessments.