Complete Aptitude Formulas Cheat Sheet 2026 — All Formulas in One Page

Last Updated: March 2026

This comprehensive cheat sheet contains every formula you need for aptitude tests, placement exams, and competitive examinations. Organized by topic with quick tricks and shortcuts. Printable format for last-minute revision.

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Table of Contents

1. Percentages
2. Profit & Loss
3. Simple & Compound Interest
4. Time, Work & Pipes
5. Time, Speed & Distance
6. Ratio & Proportion
7. Averages
8. Permutation & Combination
9. Probability
10. Number System
11. Geometry
12. Mensuration

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Percentages

Basic Formulas

Concept	Formula	Shortcut
Percentage to Fraction	x% = x/100	50% = 1/2, 25% = 1/4
Fraction to Percentage	a/b = (a/b) × 100%	1/5 = 20%, 1/8 = 12.5%
Increase by x%	New value = Original × (100+x)/100	20% increase = ×1.2
Decrease by x%	New value = Original × (100-x)/100	20% decrease = ×0.8

Quick Tricks

• x% of y = y% of x → 16% of 25 = 25% of 16 = 4
• Successive % changes: a + b + (a×b)/100
  • +20% then -20% = -4% net change
• Percentage change: (Change/Original) × 100
• Population after n years: P(1 ± R/100)ⁿ

Common Fractions to Remember

1/2 = 50%      1/3 = 33.33%   1/4 = 25%      1/5 = 20%
1/6 = 16.67%   1/7 = 14.28%   1/8 = 12.5%    1/9 = 11.11%
1/10 = 10%     1/11 = 9.09%   1/12 = 8.33%   1/16 = 6.25%

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Profit & Loss

Core Formulas

Scenario	Formula
Profit	SP - CP
Loss	CP - SP
Profit %	(Profit/CP) × 100
Loss %	(Loss/CP) × 100
Selling Price (Profit)	CP × (100+P)/100
Selling Price (Loss)	CP × (100-L)/100
Cost Price (Profit)	SP × 100/(100+P)
Cost Price (Loss)	SP × 100/(100-L)

Advanced Tricks

• Dishonest Shopkeeper: Gain% = (Error/(True Value - Error)) × 100
• Same SP with x% Profit & Loss: Loss% = x²/100
• Successive Discounts: a + b - (a×b)/100
• Marked Price: MP = CP × (100+Desired%)/(100-Discount%)

Golden Rules

1. Always calculate % on Cost Price unless specified
2. When SP is same: CP₁/CP₂ = (100+L%)/(100+P%)
3. Break-even: No profit, no loss → SP = CP

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Simple & Compound Interest

Simple Interest (SI)

SI = (P × R × T) / 100
Amount = P + SI = P(1 + RT/100)

Trick: If amount becomes n times in T years at SI, Rate = 100(n-1)/T

Compound Interest (CI)

Amount = P(1 + R/100)ᵀ
CI = P[(1 + R/100)ᵀ - 1]

Compounding Variations

Type	Formula
Half-yearly	Amount = P(1 + R/200)^(2T)
Quarterly	Amount = P(1 + R/400)^(4T)
Monthly	Amount = P(1 + R/1200)^(12T)

Difference between CI & SI

Period	Difference
2 years	P(R/100)²
3 years	P(R/100)² × (3 + R/100)

Quick Reference

• If CI = SI for 2 years: CI - SI = P(r/100)²
• Rate when amount doubles: R ≈ 72/n (Rule of 72)
• Installments: P = x/(1+r/100) + x/(1+r/100)² + ...

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Time, Work & Pipes

Fundamental Rules

1. Work Rate: If A can do work in n days → Work/day = 1/n
2. Combined Work: 1/A + 1/B = 1/Time together
3. LCM Method: Assume total work = LCM of days

Quick Formulas

Scenario	Formula
A does in x days, B in y days	Together: xy/(x+y) days
A+B in x days, B+C in y days, C+A in z days	A+B+C = 2xyz/(xy+yz+zx)
A is k times faster than B	Time ratio = 1:k, Work ratio = k:1
A leaves after m days	Work remaining = 1 - m/x

Pipes & Cisterns

• Filling pipe: Positive work
• Emptying pipe: Negative work
• Net rate: (Sum of filling) - (Sum of emptying)
• Time to fill: Total capacity / Net rate

Man-Work-Hour Formula

M₁ × D₁ × H₁ / W₁ = M₂ × D₂ × H₂ / W₂

Where M=Men, D=Days, H=Hours, W=Work

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Time, Speed & Distance

Basic Formula

Speed = Distance / Time
Distance = Speed × Time
Time = Distance / Speed

Unit Conversions

From	To	Multiply by
km/h	m/s	5/18
m/s	km/h	18/5

Average Speed

• Same distance: 2xy/(x+y) (when speeds are x and y)
• Different distances: Total Distance / Total Time
• Three equal distances: 3xyz/(xy+yz+zx)

Relative Speed

Case	Formula
Same direction	S₁ - S₂
Opposite direction	S₁ + S₂

Trains

Scenario	Formula
Crossing a pole/person	Time = Length/Speed
Crossing a platform	Time = (Train + Platform)/Speed
Crossing another train	Time = Sum of lengths / Relative speed

Boats & Streams

Downstream speed = B + S (Boat + Stream)
Upstream speed = B - S (Boat - Stream)
Speed of boat = (Down + Up)/2
Speed of stream = (Down - Up)/2

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Ratio & Proportion

Basic Concepts

• Ratio: a:b or a/b (comparison of same units)
• Proportion: a:b = c:d → a×d = b×c
• Fourth proportional: a:b = c:x → x = bc/a
• Third proportional: a:b = b:x → x = b²/a
• Mean proportional: a:x = x:b → x = √ab

Dividing Amounts

• In ratio a:b: Parts = a/(a+b) and b/(a+b)
• In ratio a:b:c: Parts = a/(a+b+c), etc.

Compound Ratio

(a:b), (c:d), (e:f) → Compound = (ace:bdf)
Duplicate = a²:b²
Triplicate = a³:b³
Sub-duplicate = √a:√b

Variations

• Direct: x ∝ y → x = ky (x₁/y₁ = x₂/y₂)
• Inverse: x ∝ 1/y → xy = k (x₁y₁ = x₂y₂)

Partnership

Profit ratio = (Investment × Time) ratio

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Averages

Basic Formula

Average = Sum of quantities / Number of quantities

Weighted Average

Weighted Avg = (w₁x₁ + w₂x₂ + ... + wₙxₙ) / (w₁ + w₂ + ... + wₙ)

Quick Tricks

Scenario	Shortcut
Average of n consecutive numbers	(First + Last)/2
Average of first n natural numbers	(n+1)/2
Average of first n even numbers	n+1
Average of first n odd numbers	n
Average of squares of first n natural numbers	(n+1)(2n+1)/6
Average of cubes of first n natural numbers	n(n+1)²/4

Replacement Formula

If average of n items is x, and one item y is replaced by z:
New average = x + (z-y)/n

Adding/Removing Items

• Adding new item: New Avg = (n×Old + New)/(n+1)
• Removing item: New Avg = (n×Old - Removed)/(n-1)

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Permutation & Combination

Factorial

n! = n × (n-1) × (n-2) × ... × 1
0! = 1

Combination (Selection)

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

Remember: nCr = nC(n-r)

Permutation (Arrangement)

nPr = n! / (n-r)!

Key Differences

Combination	Permutation
Selection only	Selection + Arrangement
ABC = ACB = BAC	ABC ≠ ACB ≠ BAC
nCr ways	nCr × r! = nPr ways

Special Cases

Scenario	Formula
Circular permutation	(n-1)!
Circular (necklace/ring)	(n-1)!/2
With repetition	nʳ
All not together	Total - Together
At least one	Total - None

Distribution

• n identical to r distinct: (n+r-1)C(r-1)
• n distinct to r distinct: rⁿ
• n distinct to r identical: Sum of Stirling numbers

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Probability

Basic Formula

Probability = Favorable outcomes / Total possible outcomes

Properties

• 0 ≤ P(E) ≤ 1
• P(E) + P(Not E) = 1
• P(Sure event) = 1
• P(Impossible event) = 0

Types of Events

Event Type	Formula
Mutually Exclusive	P(A or B) = P(A) + P(B)
Independent	P(A and B) = P(A) × P(B)
Dependent	P(A and B) = P(A) × P(B
Complementary	P(A') = 1 - P(A)

Card Deck (52 cards)

Property	Count
Total cards	52
Suits	4 (Spade♠, Heart♥, Diamond♦, Club♣)
Cards per suit	13
Face cards	12 (J, Q, K in each suit)
Aces	4
Red cards	26 (Hearts + Diamonds)
Black cards	26 (Spades + Clubs)

Coin & Dice

Item	Outcomes
1 coin	2 (H, T)
2 coins	4 (HH, HT, TH, TT)
1 die	6 (1-6)
2 dice	36 combinations

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Number System

Divisibility Rules

Divisor	Rule
2	Last digit even (0,2,4,6,8)
3	Sum of digits divisible by 3
4	Last 2 digits divisible by 4
5	Last digit 0 or 5
6	Divisible by both 2 and 3
8	Last 3 digits divisible by 8
9	Sum of digits divisible by 9
11	(Sum odd positions) - (Sum even positions) = 0 or divisible by 11
12	Divisible by both 3 and 4

Formulas

Concept	Formula
Sum of first n natural numbers	n(n+1)/2
Sum of first n even numbers	n(n+1)
Sum of first n odd numbers	n²
Sum of squares	n(n+1)(2n+1)/6
Sum of cubes	[n(n+1)/2]²

Remainder Theorems

• Basic: Dividend = Divisor × Quotient + Remainder
• Euler's: a^φ(n) ≡ 1 (mod n) when gcd(a,n)=1
• Fermat's: a^(p-1) ≡ 1 (mod p) when p is prime

HCF & LCM

HCF × LCM = Product of two numbers
HCF of fractions = HCF(numerators)/LCM(denominators)
LCM of fractions = LCM(numerators)/HCF(denominators)

Unit Digit Patterns

Base	Pattern	Cycle
0,1,5,6	Same digit	1
4,9	4,6 / 9,1	2
2,3,7,8	2,4,8,6 / 3,9,7,1	4

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Geometry

Lines & Angles

Angle Type	Property
Straight line	180°
Complete rotation	360°
Complementary	Sum = 90°
Supplementary	Sum = 180°
Vertically opposite	Equal
Alternate angles	Equal (parallel lines)
Corresponding angles	Equal (parallel lines)
Co-interior angles	Sum = 180°

Triangles

Types by sides: Equilateral, Isosceles, Scalene
Types by angles: Acute, Right, Obtuse

Property	Formula
Sum of angles	180°
Exterior angle	Sum of two opposite interior angles
Area (base×height)	(1/2) × b × h
Area (Heron's)	√[s(s-a)(s-b)(s-c)], s=(a+b+c)/2
Equilateral area	(√3/4) × a²

Triangle Centers

Center	Intersection of
Centroid	Medians (divides 2:1)
Circumcenter	Perpendicular bisectors
Incenter	Angle bisectors
Orthocenter	Altitudes

Pythagoras Theorem

Right triangle: a² + b² = c² (c = hypotenuse)
Common triplets: (3,4,5), (5,12,13), (8,15,17), (7,24,25), (9,40,41)

Polygons

Polygon	Sides	Sum of interior angles
Triangle	3	180°
Quadrilateral	4	360°
Pentagon	5	540°
Hexagon	6	720°
n-sided	n	(n-2) × 180°

Interior angle of regular n-gon: (n-2) × 180°/n
Exterior angle of regular n-gon: 360°/n

Circles

Property	Formula
Circumference	2πr
Area	πr²
Arc length	(θ/360°) × 2πr
Sector area	(θ/360°) × πr²
Chord length	2r sin(θ/2)

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Mensuration

2D Shapes

Rectangle

• Area = l × b
• Perimeter = 2(l + b)
• Diagonal = √(l² + b²)

Square (side = a)

• Area = a²
• Perimeter = 4a
• Diagonal = a√2

Parallelogram

• Area = base × height
• Perimeter = 2(sum of adjacent sides)

Rhombus (diagonals d₁, d₂)

• Area = (d₁ × d₂)/2
• Perimeter = 4a (a = side)

Trapezium

• Area = (1/2) × (a + b) × h

Circle (radius r)

• Area = πr²
• Circumference = 2πr
• Arc = 2πr × (θ/360°)

3D Shapes

Cube (side = a)

Property	Formula
Volume	a³
LSA	4a²
TSA	6a²
Space diagonal	a√3

Cuboid (l × b × h)

Property	Formula
Volume	l × b × h
LSA	2h(l + b)
TSA	2(lb + bh + hl)
Diagonal	√(l² + b² + h²)

Cylinder (radius r, height h)

Property	Formula
Volume	πr²h
CSA	2πrh
TSA	2πr(r + h)

Cone (radius r, height h, slant l)

Property	Formula
Volume	(1/3)πr²h
CSA	πrl
TSA	πr(l + r)
Slant height	l = √(r² + h²)

Sphere (radius r)

Property	Formula
Volume	(4/3)πr³
Surface Area	4πr²

Hemisphere (radius r)

Property	Formula
Volume	(2/3)πr³
CSA	2πr²
TSA	3πr²

Frustum of Cone

Volume = (πh/3)(R² + Rr + r²)
CSA = π(R + r)l
TSA = π[R² + r² + (R+r)l]

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Quick Reference Tables

Squares (1-30)

n	n²	n³
1	1	1
2	4	8
3	9	27
...	...	...
25	625	15625

Powers of 2

2⁵=32    2⁶=64    2⁷=128   2⁸=256
2⁹=512   2¹⁰=1024 2¹¹=2048 2¹²=4096

Common Roots

√2 ≈ 1.414   √3 ≈ 1.732   √5 ≈ 2.236
√6 ≈ 2.449   √7 ≈ 2.646   √8 ≈ 2.828
√10 ≈ 3.162

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FAQ

Q: How do I solve percentage problems quickly?

A: Convert percentages to fractions. Use x% of y = y% of x. For successive changes, use the formula a + b + ab/100.

Q: What's the fastest way to calculate compound interest?

A: For 2 years, use: CI - SI = P(R/100)². For quick estimates, remember the Rule of 72 for doubling time.

Q: How do I remember all these formulas?

A: Practice with this cheat sheet daily. Focus on one topic per day. Derive formulas instead of memorizing blindly.

Q: Which formulas are most important for placements?

A: Percentages, TSD, Time & Work, and P&C are asked most frequently. Master these first.

Q: Can I use this for CAT/GATE as well?

A: Yes! These formulas cover 90% of quantitative aptitude sections in all competitive exams.

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Pro Tip: Keep this page handy during practice. The more you use these formulas, the faster you'll recall them during actual tests!

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Good luck with your placements! 🎯