Compound Interest Formulas & Shortcuts

💰 Basic Compound Interest Formula

Main Formula

Amount = P × (1 + r/n)^(nt)

Where:
P = Principal amount
r = Annual interest rate (in decimal)
n = Number of times interest compounded per year
t = Time in years
CI = Amount - Principal

Simple Case (Compounded Annually)

Amount = P × (1 + r/100)^t
CI = P × [(1 + r/100)^t - 1]

Example: P = 1000, r = 10%, t = 2 years Amount = 1000 × (1 + 10/100)² = 1000 × 1.21 = 1210 CI = 1210 - 1000 = 210

📅 Different Compounding Periods

Half-Yearly Compounding

Amount = P × (1 + r/200)^(2t)
Rate per half-year = r/2
Time in half-years = 2t

Quarterly Compounding

Amount = P × (1 + r/400)^(4t)
Rate per quarter = r/4
Time in quarters = 4t

Monthly Compounding

Amount = P × (1 + r/1200)^(12t)
Rate per month = r/12
Time in months = 12t

🚀 Effective Interest Rate

When Compounding Multiple Times

Effective Rate = (1 + r/n)^n - 1

Example: 12% compounded quarterly Effective Rate = (1 + 12/400)^4 - 1 = 1.1255 - 1 = 12.55%

🔗 Compound Interest vs Simple Interest

Relationship

CI > SI when P, r, t > 0
Difference increases with time

Quick Comparison Formula

For 2 years:
CI - SI = P × (r/100)²

For 3 years:
CI - SI = P × (r/100)² × (3 + r/100)

📈 Special Cases and Shortcuts

Case 1: Rate Changes Each Year

Amount = P × (1 + r₁/100) × (1 + r₂/100) × ... × (1 + rₙ/100)

Case 2: Different Compounding Frequencies

Annual to Half-Yearly: Rate halves, Time doubles
Annual to Quarterly: Rate quarters, Time quadruples

Case 3: Population Growth/Depreciation

Population: A = P × (1 + r/100)^t
Depreciation: A = P × (1 - r/100)^t

🧮 Rule of 72 (Doubling Time)

Quick Estimation

Time to double = 72/Interest Rate

Example: At 8% interest, money doubles in 72/8 = 9 years

More Accurate Formula

Time to double = ln(2)/ln(1 + r/100)

🎯 Important Properties

Compound Interest Factors

Factor for n years at r% = (1 + r/100)^n

Common Factors to Remember:

• 10% for 2 years: 1.21
• 10% for 3 years: 1.331
• 20% for 2 years: 1.44
• 5% for 2 years: 1.1025

Present Value and Future Value

Future Value = PV × (1 + r/100)^t
Present Value = FV / (1 + r/100)^t

📊 Comparison Tables

SI vs CI at 10% for different time periods

Effective Annual Rates

🎪 Installment Problems

When Amount is Paid in Installments

Let P = Principal, n = number of installments
Each installment = P × (1 + r/100)^n / [Sum of factors]

Equal Annual Installments

Each installment = P × r(1 + r/100)^n / [(1 + r/100)^n - 1]

🔢 Series and Patterns

Compound Interest Series

For r% interest:
Year 1: P × r/100
Year 2: P × r/100 × (1 + r/100)
Year 3: P × r/100 × (1 + r/100)²
...
Total CI = Sum of all yearly interests

Geometric Progression in CI

Compound interests form a GP
First term = P × r/100
Common ratio = 1 + r/100

📝 Quick Calculation Methods

Method 1: Successive Percentage

For 2 years at r%: Overall increase = r + r + (r²/100)

Example: 10% for 2 years = 10 + 10 + (100/100) = 21%

Method 2: Using Known Values

Memorize common compound factors:

Method 3: Approximation

For small rates and short periods:

(1 + r/100)^n ≈ 1 + nr/100 (when r is small and n is small)

🎯 Problem Types and Solutions

Type 1: Find Amount

Question: Find amount on 5000 at 8% CI for 3 years Solution: A = 5000 × (1.08)³ = 5000 × 1.2597 = 6298.50

Type 2: Find Principal

Question: What sum amounts to 1331 at 10% CI in 3 years? Solution: P = 1331 ÷ (1.1)³ = 1331 ÷ 1.331 = 1000

Type 3: Find Rate

Question: At what rate will 800 amount to 968 in 2 years? Solution: (968/800)^(1/2) - 1 = √1.21 - 1 = 1.1 - 1 = 0.1 = 10%

Type 4: Find Time

Question: In how many years will 1000 amount to 1331 at 10%? Solution: (1331/1000) = (1.1)^t → 1.331 = (1.1)^t → t = 3 years

🔗 Related Topics

• Simple Interest - Simple Interest calculations
• Percentage - Percentage concepts
• Time and Work - Time and Work problems
• Partnership - Partnership calculations

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