Aptitude Formulas Quick Reference
Everything you need day‑to‑day – quantitative aptitude for competitive exams.
Percentages
% = (Value / Total) × 100
Key Conversions
- 1 = 100%
- 0.5 = 50%
- 0.25 = 25%
- 0.75 = 75%
- 0.1 = 10%
- 0.01 = 1%
Common Fractions to %
- 1/2 = 50%
- 1/3 = 33.33%
- 2/3 = 66.67%
- 1/4 = 25%
- 3/4 = 75%
- 1/5 = 20%
- 1/8 = 12.5%
- 1/10 = 10%
Percentage Change
% Change = (New Value - Old Value) / Old Value × 100
Successive Percentage Change
Net Change = a + b + (a × b) / 100
Percentage Increase/Decrease in Product
If A changes by a% and B changes by b%, product changes by a + b + (a × b)/100
Profit & Loss
Profit = SP - CP
Loss = CP - SP
Profit % = (Profit / CP) × 100
Loss % = (Loss / CP) × 100
SP = CP × (100 + Profit%) / 100
SP = CP × (100 - Loss%) / 100
CP = SP × 100 / (100 + Profit%)
CP = SP × 100 / (100 - Loss%)
Discount
Discount = MP - SP
Discount % = (Discount / MP) × 100
SP = MP × (100 - Discount%) / 100
Successive Discount
Net Discount = a + b - (a × b)/100
Profit/Loss with Successive Changes
Net = a + b + (a × b)/100
Ratio & Proportion
a : b = a/b
a : b = c : d → a × d = b × c
Componendo & Dividendo
If a/b = c/d, then:
- (a + b)/b = (c + d)/d
- (a - b)/b = (c - d)/d
- (a + b)/(a - b) = (c + d)/(c - d)
Partnership
Profit Ratio = Capital × Time
Time & Work
Work = Rate × Time
Rate = 1 / Time (if work is 1 unit)
Time = 1 / Rate
Combined Work
1/T = 1/T₁ + 1/T₂ + ...
Where T₁, T₂ are individual times
Work Done Together
Time = (T₁ × T₂) / (T₁ + T₂)
Efficiency
Efficiency ∝ 1 / Time
Work = Efficiency × Time
Men/Women/Children Ratio
M₁ × D₁ × H₁ / W₁ = M₂ × D₂ × H₂ / W₂
Pipes & Cisterns
Net Fill Rate = Rate(in) - Rate(out)
Time = 1 / Net Rate
Time, Speed & Distance
Speed = Distance / Time
Distance = Speed × Time
Time = Distance / Speed
Conversions
- km/h → m/s: × 5/18
- m/s → km/h: × 18/5
Relative Speed
Same Direction: Relative Speed = S₁ - S₂
Opposite Direction: Relative Speed = S₁ + S₂
Average Speed
Avg Speed = 2xy / (x + y) [equal distances]
Avg Speed = Total Distance / Total Time
Boats & Streams
Downstream Speed = Speed in Still Water + Stream Speed
Upstream Speed = Speed in Still Water - Stream Speed
Speed in Still Water = (Downstream + Upstream)/2
Stream Speed = (Downstream - Upstream)/2
Trains
- Time to cross pole = Length / Speed
- Time to cross platform = (Length of Train + Length of Platform) / Speed
- Time to cross another train = Sum of Lengths / Relative Speed
Permutations & Combinations
Factorials
n! = n × (n-1) × ... × 1
- 0! = 1
- 1! = 1
Permutations (Order Matters)
nPr = n! / (n - r)!
nPn = n!
Combinations (Order Doesn't Matter)
nCr = n! / (r! × (n - r)!)
nC0 = nCn = 1
nCr = nC(n-r)
Important Results
- Number of ways to arrange n distinct objects = n!
- Number of ways to arrange n objects with identical objects = n! / (a! × b! × ...)
- Number of ways to select r objects from n = nCr
- Number of ways to distribute n objects among r groups = rⁿ
- Circular permutations = (n-1)!
Probability
P(E) = Number of Favorable Outcomes / Total Outcomes
P(E) + P(not E) = 1
Probability of Events
- Mutually Exclusive: P(A ∪ B) = P(A) + P(B)
- Not Mutually Exclusive: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
- Independent Events: P(A ∩ B) = P(A) × P(B)
- Conditional Probability: P(A|B) = P(A ∩ B) / P(B)
Bayes' Theorem
P(A|B) = (P(B|A) × P(A)) / P(B)
Simple & Compound Interest
Simple Interest (SI)
SI = (P × R × T) / 100
Amount = P + SI
Compound Interest (CI)
A = P × (1 + R/100)ⁿ
CI = A - P
CI with Different Compounding Frequencies
A = P × (1 + R/(100 × m))^(n × m)
Where m = number of times compounded per year
Difference between CI and SI
For 2 years: CI - SI = P × (R/100)²
For 3 years: CI - SI = P × (R/100)² × (3 + R/100)
Geometry & Mensuration
Triangles
- Area = ½ × Base × Height
- Heron's Formula: Area = √(s(s-a)(s-b)(s-c)), s = (a+b+c)/2
- Pythagoras: a² + b² = c² (right triangle)
- Equilateral Triangle Area = (√3/4) × side²
Circles
- Area = πr²
- Circumference = 2πr
- Diameter = 2r
Squares & Rectangles
- Square Area = side², Perimeter = 4 × side
- Rectangle Area = l × b, Perimeter = 2(l + b)
3D Shapes
- Cube: Volume = a³, SA = 6a²
- Cuboid: Volume = l × b × h, SA = 2(lb + bh + lh)
- Sphere: Volume = (4/3)πr³, SA = 4πr²
- Cylinder: Volume = πr²h, CSA = 2πrh, TSA = 2πr(r + h)
- Cone: Volume = (1/3)πr²h, CSA = πrl, TSA = πr(l + r)
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 2 and 3 |
| 7 | Double last digit, subtract from rest, repeat |
| 8 | Last 3 digits divisible by 8 |
| 9 | Sum of digits divisible by 9 |
| 10 | Last digit 0 |
| 11 | Difference of sum of digits at odd/even positions is 0 or multiple of 11 |
LCM & HCF
LCM × HCF = Product of two numbers
- HCF (GCD) – greatest common divisor
- LCM – least common multiple
- Co‑prime numbers: HCF = 1
Average
Average = Sum of Observations / Number of Observations
Age Problems
- Age difference remains constant
- Ratio of ages changes over time
Quick Reference Table
| Topic | Formula |
|---|---|
| Percentage | (Value / Total) × 100 |
| Profit % | (SP - CP) / CP × 100 |
| Loss % | (CP - SP) / CP × 100 |
| Discount % | (MP - SP) / MP × 100 |
| Simple Interest | (P × R × T) / 100 |
| Compound Interest | P × (1 + R/100)ⁿ - P |
| Speed | Distance / Time |
| Relative Speed (same) | S₁ - S₂ |
| Relative Speed (opposite) | S₁ + S₂ |
| Average Speed | Total Distance / Total Time |
| Work | Rate × Time |
| Combined Work | 1/T = 1/T₁ + 1/T₂ |
| nPr | n! / (n-r)! |
| nCr | n! / (r! × (n-r)!) |
| Probability | Favorable / Total |
| LCM × HCF | Product of numbers |
| Average | Sum / Count |
📌 Quick Reference
Percentage: (Value/Total) × 100
Profit/Loss: SP - CP / CP × 100
SI: (P × R × T) / 100
CI: P(1 + R/100)ⁿ - P
Speed: D/T
Work: 1/T = Σ(1/Tᵢ)
nPr: n!/(n-r)!
nCr: n!/(r!(n-r)!)
Probability: Fav/Total
LCM × HCF: Product of numbers
Profit/Loss: SP - CP / CP × 100
SI: (P × R × T) / 100
CI: P(1 + R/100)ⁿ - P
Speed: D/T
Work: 1/T = Σ(1/Tᵢ)
nPr: n!/(n-r)!
nCr: n!/(r!(n-r)!)
Probability: Fav/Total
LCM × HCF: Product of numbers